ON THE SMOOTHNESS OF THERMAL POTENTIALS

L. I. KAMYNIN

I. Thermal potentials on surfaces of type \(\Lambda^{0,1,\frac{1+\alpha}{2}}_{1,\alpha,\alpha/2}\)

The paper considers the thermal potentials of the simple layer \(U(x,t)\), the double layer \(W(x,t)\), and the normal derivative of the thermal potential of the simple layer \(V(x,t)\), with densities distributed on a noncylindrical surface \(\Gamma\) of type \(\Lambda^{0,1,\frac{1+\alpha}{2}}_{1,\alpha,\alpha/2}\) (see the definitions in § 1). The dependence of the smoothness of these potentials on the smoothness of their densities is investigated. The paper consists of 5 sections. In § 1 definitions are given and auxiliary information on Lyapunov surfaces needed in what follows is presented. § 2 contains the statements of the main results (Theorems 1–8). In § 3 Theorems 2–4 are proved concerning the smoothness of the direct values of the thermal potentials \(\overline V\) and \(\overline W\) on \(\Gamma\). § 4 contains the proofs of Theorems 5–7, establishing jump formulas (cf. [1], [2]) for \(V\) and \(W\) when passing through the surface \(\Gamma\) and refining the behavior of these potentials near \(\Gamma\). § 5 is set aside for the proof of a Lyapunov-type theorem (cf. [3]) on the first derivatives of the thermal potential of the simple layer. The proofs of all the theorems are carried out by classical methods, close to the methods of the ordinary theory of potentials as presented in [3]. A formulation of a number of results of the present paper is contained in the author’s note [10].

§ 1. DEFINITIONS AND AUXILIARY INFORMATION

Let a function \(f(x,t) \equiv f(x_1,x_2,\ldots,x_k;t)\) be defined in a domain \(B\) of the Euclidean space \(\{x_1,\ldots,x_k;t\}\).

Definition 1. If \(f(x,t)\) is such that

\[ \left| f(x_1,\ldots,x_k;t+\Delta t)-f(x_1,\ldots,x_k;t)\right| \leq H_{0,t}(f)|\Delta t|^{\frac{1+\alpha_3}{2}}, \tag{1.1} \]

\[ \left| \frac{\partial f(x_1+\Delta x_1,\ldots,x_k+\Delta x_k;t+\Delta t)}{\partial x_l} - \frac{\partial f(x_1,\ldots,x_k;t)}{\partial x_l} \right| \leq \]

\[ \leq H_{1,x}(f)\sum_{j=1}^{k}|\Delta x_j|^{\alpha_1} + H_{1,t}(f)|\Delta t|^{\frac{\alpha_2}{2}}, \quad l=1,2,\ldots,k, \tag{1.2} \]

then we shall say that \(f(x,t)\) belongs in the domain \(B\) to the class

\[ H^{0,1,\frac{1+\alpha_3}{2}}_{1,\alpha_1,\frac{\alpha_2}{2}}(B) \quad (0<\alpha_i\leq 1,\ i=1,2,3). \]

Definition 2. If \(f(x,t)\) is such that

\[ |f(x_1+\Delta x_1,\ldots,x_k+\Delta x_k;t+\Delta t)-f(x_1,\ldots,x_k;t)| \]

\[ \leq H_{0,x}(f)\sum_{j=1}^{k}|\Delta x_j|^{\alpha_1} +H_{0,t}(f)|\Delta t|^{\alpha_2/2}, \tag{1.3} \]

then we shall say that the function \(f(x,t)\) belongs in the domain \(B\) to the class

\[ H^{0,\alpha_1,\frac{\alpha_2}{2}}(B)\qquad (0<\alpha_i\leq 1,\ i=1,2). \]

Let, in the \((n+1)\)-dimensional Euclidean space of variables \((x,t)=(x_1,\ldots,x_n;t)\), there be given a closed surface \(\Gamma\), lying between the hyperplanes \(t=0\) and \(t=T>0\). By \(D_T^{\,B}\) we shall denote the domain lying between the hyperplanes \(t=0\) and \(t=T\) and having \(\Gamma\) as its lateral boundary. By \(D_T^{\,H}\) we denote the complement of \(D_T^{\,B}\cup\Gamma\) in the strip

\[ D_T=\{(x,t),\ |x_i|<+\infty,\ i=1,2,\ldots,n;\ 0<t<T\}. \tag{1.4} \]

Suppose there exists a constant \(R\), independent of the choice of \((x^0,t^0)\in\Gamma\), such that in every \((n+1)\)-dimensional ball \(S_R(x^0,t^0)\) with center at the point \((x^0,t^0)\in\Gamma\) and radius \(R\), the part of the surface \(\Gamma\cap S_R(x^0,t^0)\) admits a representation of the form

\[ x_i=\psi(x_1,\ldots,x_{i-1},x_{i+1},\ldots,x_n;t), \]

\[ (x_1,\ldots,x_{i-1},x_{i+1},\ldots,x_n;t)\in B_R(x^0,t^0). \tag{1.5} \]

Without loss of generality we shall henceforth assume \(i=n\).

Definition 3. If \(\psi(x,t)\) from (1.5) belongs in its domain of definition \(B_R(x^0,t^0)\) to the class \(H_{1,\alpha,\alpha/2}^{0,1,\frac{1+\alpha}{2}}(B_R)\), and the Hölder constants \(L_{0,t}\), \(L_{1,x}\), and \(L_{1,t}\) from (1.1), (1.2) do not depend on the choice of the point \((x^0,t^0)\in\Gamma\), then we shall say that the surface \(\Gamma\) is of type

\[ \mathcal{L}_{1,\alpha,\alpha/2}^{0,1,\frac{1+\alpha}{2}}. \]

Definition 4. A function \(\varphi(x,t)\) belongs to the class \(H^{0,\alpha_1,\alpha_2/2}(\Gamma)\) if, in the domain \(B_R(x^0,t^0)\), the conditions (1.3) are satisfied for \(\varphi(x_1,\ldots,x_{n-1},t)\) with constants \(H_{0,x}\) and \(H_{0,t}\) independent of the choice of \((x^0,t^0)\in\Gamma\).

For a surface \(\Gamma\) of type \(\mathcal{L}_{1,\alpha,\alpha/2}^{0,1,\frac{1+\alpha}{2}}\), the section \(\Gamma_\tau=\Gamma\cap\{t=\tau\}\) is a surface of Lyapunov type (see [3], Ch. 1, § 1); therefore at each point \((x,\tau)\in\Gamma_\tau\) there exists a definite inward normal \(N(x,\tau)\), lying in the plane \(t=\tau\), and if \((N(x,\tau),N(y,\tau))\) is the angle between the normals at the points \((x,\tau)\), \((y,\tau)\in\Gamma_\tau\), and \(r(x,y)\) is the distance between the points \((x,\tau)\) and \((y,\tau)\), then by virtue of (1.2)

\[ |(N(x,\tau),N(y,\tau))|\leq L r^{\alpha}(x,y). \tag{1.6} \]

Furthermore, there exists a number \(3d>0\), the same for all points of \(\Gamma_\tau\) and all \(\tau\) \((0\leq\tau\leq T)\), possessing the following property: parallels to the normal \(N(x,\tau)\) intersect in no more than one point the part of the surface \(\Gamma_\tau\) located inside the \(n\)-dimensional Lyapunov ball \(S_{3d}(x,\tau)\) of radius \(3d\) with center at the point \((x,\tau)\) and lying on the hyperplane \(t=\tau\). Finally, if we consider a point \((x,t)\in\Gamma_t\) and the ball \(S_R(x,t)\), cutting out from \(\Gamma\) a piece

\(\Gamma_R(x,t)\), admitting the representation (1.5), then from conditions (1.1), (1.2) it follows that for \((x^{(1)}, t_1), (x^{(2)}, t_2) \in \Gamma_R(x,t)\) the inequality holds

\[ \left| (N(x^{(1)}, t_1),\, N(x^{(2)}, t_2)) \right| \leq L\left(r^\alpha(x^{(1)}, x^{(2)}) + |t_1-t_2|^{\alpha/2}\right). \tag{1.7} \]

The constant \(L\) in (1.6), (1.7) depends on \(L_{1,x}\), \(L_{1,t}\) from (1.2). Without loss of generality we shall henceforth assume \(n=3\). From (1.1), (1.2) (for \(\psi\)) there follows the existence of numbers \(d>0\), \(d_1>0\), depending only on \(R\), such that if \(x^*=(x_1,x_2,\psi(x_1,x_2,\tau))\), then \(s_{3d}(x^*,\tau)\subset S_R(x,t)\) for \(|t-\tau|\leq d_1\). Put

\[ S_{dd_1}(x,t)=\bigcup_{|t-\tau|\leq d_1} s_{3d}(x^*,\tau), \qquad \Gamma_{dd_1}(x,t)=\Gamma\cap S_{dd_1}(x,t). \]

Let \(d\) and \(d_1\) be so small that (cf. (1.1), (1.2), (1.6))

\[ L_{0,t}d_1^{\frac{1+\alpha}{2}}\leq \frac d2, \qquad L\left((3d)^\alpha+(d_1)^{\alpha/2}\right)<\frac12 . \tag{1.8} \]

Then for \((y,\tau)\in \Gamma_{dd_1}(x,t)\), from (1.6) it follows that

\[ 0\leq \gamma=(N(y,\tau),N(x,t)) \leq L\left(r^\alpha(y,x)+|t-\tau|^{\alpha/2}\right)<\frac12, \]

\[ 1\geq \cos\gamma \geq 1-\frac{\gamma^2}{2}>\frac12 . \tag{1.9} \]

Let \(\Gamma_\tau(x,r)\) be the part of the surface \(\Gamma_\tau\cap \Gamma_{dd_1}(x,t)\) contained in the right circular cylinder \(Ц_r^\tau(x,t)\) of radius \(r>0\) with axis of symmetry parallel to the normal \(N(x,t)\) and passing through the point \((x^*,\tau)\in \Gamma_\tau\). We shall take \(r>0\) so small that, for \(|t-\tau|\leq d_1\),

\[ \Gamma_\tau(x,r)\subset \Gamma_{dd_1}(x,t)\subset S_{dd_1}(x,t)\subset S_R(x,t). \]

Lemma 1 (cf. [3], Ch. 1, § 1). If the surface \(\Gamma\) is of type \(Л^{0,1,\frac{1+\alpha}{2}}_{1,\alpha/2}\) \((0<\alpha\leq 1)\), then, when the inequalities (1.8) are satisfied, \(r\leq d\) and \(|t-\tau|\leq d_1\), the projection of \(\Gamma_\tau(x,r)\) onto the base of the cylinder \(Ц_r^\tau(x,t)\) completely fills this base.

Proof. Let \(M(y,\tau)\) be a point of \(\Gamma_\tau\) lying on the surface of the Lyapunov sphere \(s_{3d}(x^*,\tau)\) and least removed from the normal \(N(x^*,\tau)\). Then from the Lyapunov conditions (cf. [3], Ch. 1, § 1) it follows that \(M(y,\tau)\) is projected onto the plane orthogonal to \(N(x^*,\tau)\) at a point \(M_1\), removed from \(M_0(x^*,\tau)\) by a distance greater than \(\frac79(3d)\).

Let \(\gamma_1=(N(x,t),N(x^*,\tau))\), and let \(\gamma_2\) be the angle between \(\overline{M_0M}\) and the plane orthogonal to \(N(x^*,\tau)\). Obviously, \(\cos\gamma_2>\frac79\), \(\sin\gamma_2\leq \frac{\sqrt{32}}9\).

Denoting by \(b\) the distance of the projection of the point \(M(y,\tau)\) onto the plane orthogonal to \(N(x,t)\) from the point \(M_0(x^*,\tau)\), we see that, by virtue of (1.7), (1.8),

\[ b=3d\cos(\gamma_1+\gamma_2)\geq 3d\left(\frac79\cos\gamma_1-\frac{\sqrt{32}}9\sin\gamma_1\right)\geq \frac{25}{72}(3d)>d. \]

In the Lyapunov sphere \(S_R(x,t)\) with fixed center \((x,t)\in\Gamma_t\), introduce a local Cartesian coordinate system \(\{\xi,\tau\}=\{\xi_1,\xi_2,\xi_3,\tau\}\) with origin at the point \((x_1,x_2,\psi(x_1,x_2,t),0)\). Let the axis \(O\xi_3\) be directed along the normal \(N(x,t)\) to the section \(\Gamma_t\) at the point \((x,t)\), and let the axes \(O\xi_i\) \((i=1,2)\) be parallel to the tangent plane to the section \(\Gamma_t\) at the point \((x,t)\). In the system \(\{\xi,\tau\}\) we have

\[ x_1=x_2=0,\quad x_3=\psi(x_1,x_2,t)=0,\quad \frac{\partial\psi(x_1,x_2,t)}{\partial x_i}=0,\quad i=1,2. \tag{1.10} \]

We shall also consider cylindrical coordinates \(\{\rho,\vartheta,\xi_3,\tau\}\) in the system \(\{\xi,\tau\}\):

\[ \xi_1=x_1+\rho\cos\vartheta,\quad \xi_2=x_2+\rho\sin\vartheta. \tag{1.11} \]

Since \(\Gamma\) is of type \(Л_{1,\alpha,\alpha/2}^{0,1,\frac{1+\alpha}{2}}\), it follows from (1.1), (1.2) that

\[ \left|\psi(\xi_1,\xi_2,\tau+\Delta\tau)-\psi(\xi_1,\xi_2,\tau)\right| \leq L_{0,t}|\Delta\tau|^{\frac{1+\alpha}{2}}, \tag{1.12} \]

\[ \left| \frac{\partial\psi(\xi_1+\Delta\xi_1,\xi_2+\Delta\xi_2,\tau+\Delta\tau)}{\partial \xi_j} - \frac{\partial\psi(\xi_1,\xi_2,\tau)}{\partial \xi_j} \right| \leq L_{1,x}\sum_{i=1}^{2}|\Delta\xi_i|^\alpha + L_{1,t}|\Delta\tau|^{\alpha/2}, \quad j=1,2. \tag{1.13} \]

Introduce the notation (cf. [8], [9]):

\[ [\psi]_0= \sup_{\substack{(x,t)\in B_R(x^0,t^0),\\ (x^0,t^0)\in\Gamma}} |\psi(x,t)|,\quad [\psi]_1=\sup\left|\frac{\partial\psi(x,t)}{\partial x_i}\right|, \]

\[ [\psi]_{1+\alpha}=\max(L_{1,x};\,L_{1,t};\,L_{0,t}),\quad |\psi|_{1+\alpha}=[\psi]_0+[\psi]_1+[\psi]_{1+\alpha}. \]

By virtue of the choice of the local system \(\{\xi,\tau\}\), from (1.10), (1.13) it follows that

\[ \left| \frac{\partial\psi(\xi_1,\xi_2,\tau)}{\partial \xi_i} \right| \leq [\psi]_{1+\alpha}\bigl(\rho^\alpha+(t-\tau)^{\alpha/2}\bigr), \quad i=1,2. \tag{1.14} \]

Moreover, we have

\[ \left|\psi(\xi_1,\xi_2,\tau)-\psi(x_1,x_2,t)\right| \leq [\psi]_{1+\alpha}\left(\rho^{1+\alpha}+(t-\tau)^{\frac{1+\alpha}{2}}\right), \tag{1.15} \]

which follows, with the aid of (1.10), (1.14), and (1.11), from the representation

\[ \psi(\xi_1,\xi_2,t)-\psi(x_1,x_2,t)= \]

\[ = \sum_{i=1}^{2}(\xi_i-x_i) \int_{0}^{1} \frac{ \partial\psi\bigl(x_1+z(\xi_1-x_1),\,x_2+z(\xi_2-x_2),\,t\bigr) }{\partial x_i}\,dz \tag{1.16} \]

and (1.12) (where \(\xi_i=x_i,\ \tau+\Delta\tau=t\)).

Let us note that, for a function \(\varphi\in H^{0,\alpha,\alpha/2}(\Gamma)\), in the local system \(\{\xi,\tau\}\), by virtue of (1.3) we have

\[ \left|\varphi(\xi_1+\Delta \xi_1,\ \xi_2+\Delta \xi_2,\ \tau+\Delta \tau) -\varphi(\xi_1,\xi_2,\tau)\right| \leq H_{0,x}\sum_{i=1}^{2}|\Delta \xi_i|^\alpha+ \]
\[ +H_{0,t}|\Delta \tau|^{\alpha/2}. \tag{1.17} \]

We shall also write (cf. [8], [9])

\[ [\varphi]_0=\sup_{\Gamma}|\varphi|,\qquad [\varphi]_\alpha=\max(H_{0,x};\,H_{0,t}),\qquad |\varphi|_\alpha=[\varphi]_0+[\varphi]_\alpha . \]

Lemma 2. Suppose (1.8) is satisfied. Then, for any \(|t-\tau|\leq d_1\), there exists an angle \(\omega_0>\pi/3\) such that every straight line lying in the Lyapunov ball \(S_{3d}(x^*;\tau)\) and making with \(N(x,t)\) an angle smaller than \(\omega_0\) intersects \(\Gamma_t\cap S_{3d}(x^*;\tau)\) in at most one point; moreover one may put

\[ \omega_0=\arccos\left[ L\bigl((3d)^\alpha+d_1^{\alpha/2}\bigr) \left[ 1+\frac14 L^4\bigl((3d)^\alpha+d_1^{\alpha/2}\bigr) \right]^{-1/2} \right]. \]

The proof of Lemma 2 is carried out according to the scheme indicated in [3] (Ch. 1, § 1, p. 14), using inequalities (1.9) and

\[ \left[ \sum_{i=1}^{2} \left(\frac{\partial\psi(\xi_1,\xi_2,\tau)}{\partial \xi_i}\right)^2 \right]^{1/2} =\operatorname{tg}(N(x,t),\,N(\xi,\tau))\leq \]

\[ \leq L\bigl((3d)^\alpha+d_1^{\alpha/2}\bigr) \left[ 1-\frac12 L^2\bigl((3d)^\alpha+d_1^{\alpha/2}\bigr)^2 \right]^{-1}. \]

Lemma 3 (cf. [4], and also [5]). If \(f(\tau)\) satisfies, with respect to \(\tau\), a Hölder condition with exponent \(\beta_1\) \((0<\beta_1<1)\), and \(|f(\tau)|\leq [f]_{\beta_1}\tau^{\beta_1}\), then the function

\[ F(t)=\int_{t_1}^{t}(t-\tau)^{-1+\beta_0}f(\tau)\,d\tau,\qquad 0<\beta_0<1,\quad 0\leq t_1<t \]

satisfies, with respect to \(t\), a Hölder condition with exponent \(\beta_0+\beta_1\), provided that \(0<\beta_0+\beta_1<1\).

Let us introduce some notation. If \((x,t)\) and \((y,\tau)\) are two points, with \((y,\tau)\in \Gamma_\tau\), then by \(r_{xy}(\tau)\) we shall denote the vector directed from the point \((x,\tau)\) to the point \((y,\tau)\), of length

\[ r(x,y)=\sqrt{\sum_{i=1}^{n}(x_i-y_i)^2}. \tag{1.18} \]

In the local system \(\{\xi,\tau\}\) associated with the point \((x,t)\in \Gamma_t\) \((\xi,\tau)\in\Gamma_\tau,\ \bar x=(\bar x_1,\bar x_2,\bar x_3)\),

\[ r(\bar x,\xi)=\sqrt{|\bar x-\xi|^2+\bigl(\bar x_3-\psi(\xi_1,\xi_2,\tau)\bigr)^2}, \]

where

\[ |\bar x-\xi|=\sqrt{\sum_{i=1}^{2}(\bar x_i-\xi_i)^2}. \tag{1.19} \]

Next, put

\[ g^{*}_{p,q}(a,b)=a^p b^{-q}\exp\left\{-\frac{a^2}{4b}\right\}\quad \text{for } a>0,\ b>0. \tag{1.20} \]

In the local system \(\{\xi,\tau\}\), for \((y,t)\in\Gamma_t,\ (\xi,\tau)\in\Gamma_\tau\),

\[ g(\psi;y,t)=g_{0,0}\bigl(|\psi(\xi_1,\xi_2,\tau)-\psi(y_1,y_2,t)|,\ t-\tau\bigr), \]

\[ g(\psi;y,\rho,t)=g_{0,0}\bigl(|\psi(y_1+\rho\cos\vartheta,\ y_2+\rho\sin\vartheta,\tau)- \psi(y_1,y_2,t)|,\ t-\tau\bigr), \tag{1.21} \]

\[ g^{(0)}(\psi;y,t)=g_{0,0}\bigl(|\psi(y_1,y_2,\tau)-\psi(y_1,y_2,t)|,\ t-\tau\bigr). \]

Lemma 4. For \(0\le z<+\infty\) and \(m>0\), the estimates

\[ 0\le g_{m,\frac m2}(z,t-\tau)\le (2me^{-1})^{\frac m2}, \tag{1.22} \]

\[ 0\le g^{*}_{m,\frac m2}(z,t-\tau)\le (4me^{-1})^{\frac m2}g_{0,0}(z,2(t-\tau)). \tag{1.23} \]

hold.

The proof of Lemma 4 is elementary.

Corollary. For \(0\le \tau<t\) and \(0<\Delta t\),

\[ |g_{0,m}(\rho,t+\Delta t-\tau)-g_{0,m}(\rho,t-\tau)|\le \]

\[ \le C_m\,\Delta t\,(t-\tau)^{-m-1}g_{0,0}\bigl(\rho,2(t+\Delta t-\tau)\bigr). \tag{1.24} \]

Lemma 5. Let the surface \(\Gamma\) be of type \(J^{0,1,\frac{1+\alpha}{2}}_{1,\alpha,\alpha/2}\) \((0<\alpha\le 1)\), and let (1.10), (1.11) be fulfilled in the local system \(\{\xi,\tau\}\) associated with \((x,t)\in\Gamma_t\). Then, for \(0\le t-\tau\le d_1,\ 0<\Delta t<d_1\), the estimates

\[ \left\{ \begin{array}{l} |g(\psi;x,\rho,t+\Delta t)-g(\psi;x,\rho,t)|\\ |g(\psi;x,t+\Delta t)-g(\psi;x,t)| \end{array} \right\}\le \]

\[ \le C[\psi]_{1+\alpha}\left[\Delta t\left(\rho^{1+\alpha}+(t-\tau)^{\frac{1+\alpha}{2}}\right)^2 (t-\tau)^{-1}(t+\Delta t-\tau)^{-1}+\right. \]

\[ \left.+(\Delta t)^{\frac{1+\alpha}{2}}(t+\Delta t-\tau)^{-1} \left(\rho^{1+\alpha}+|t+\Delta t-\tau|^{\frac{1+\alpha}{2}}\right)\right], \tag{1.25} \]

\[ |g^{(0)}(\psi;x,t+\Delta t)-g^{(0)}(\psi;x,t)|\le \]

\[ \le C[\psi]_{1+\alpha}\left[\Delta t\,(t-\tau)^{\alpha-1} +(\Delta t)^{\frac{1+\alpha}{2}}(t-\tau)^{\frac{\alpha-1}{2}}\right], \tag{1.26} \]

\[ \left\{ \begin{array}{l} |g(\psi;x,t+l_i\Delta t)-g^{(0)}(\psi;x,t+l_i\Delta t)|\\ |g(\psi;x,\rho,t+l_i\Delta t)-g^{(0)}(\psi;x,t+l_i\Delta t)| \end{array} \right\}\le \]

\[ \le C[\psi]_{1+\alpha}(t+l_i\Delta t-\tau)^{-\frac12} \left(\rho^{1+\alpha}+\right. \]

\[ \left.+(t+l_i\Delta t-\tau)^{\frac{1+\alpha}{2}}\right),\quad l_1=0,\ l_2=1. \tag{1.27} \]

If \((y,t)\in\Gamma_t\) and \(|y-x|\le \dfrac d4\), then

\[ |g(\psi; y,\rho,t)-g^{(0)}(\psi; y,t)| \leq C[\psi]_1 (t-\tau)^{-1/2}\rho, \tag{1.28} \]

\[ |g^{(0)}(\psi; y,t)-g^{(0)}(\psi; x,t)| \leq C[\psi]_{1+\alpha}(t-\tau)^{\frac{\alpha-1}{2}}|y-x|, \tag{1.29} \]

\[ |g(\psi; y,\rho,t)-g(\psi; x,\rho,t)| \leq \]

\[ \leq C[\psi]_{1+\alpha}(t-\tau)^{-1/2}\bigl(\rho^\alpha+(t-\tau)^{\alpha/2}\bigr)|y-x|, \tag{1.30} \]

\[ \begin{aligned} &\bigl|(g(\psi; y,\rho,t)-g(\psi; x,\rho,t))-(g^{(0)}(\psi; y,t)-g^{(0)}(\psi; x,t))\bigr| \leq \\[2mm] &\leq C|\psi|_{1+\alpha}\bigl[(t-\tau)^{-1}\rho\bigl(\rho^\alpha+(t-\tau)^{\alpha/2}\bigr) +(t-\tau)^{-1/2}\rho^\alpha\bigr]|y-x|. \end{aligned} \tag{1.31} \]

The constants \(C\) in (1.25)—(1.31) do not depend on \(\psi\).

Proof. The estimates (1.25), (1.26) are obtained from (1.12) (where \(\xi_i=x_i,\ \tau=t,\ \Delta\tau=\Delta t\)), (1.15), by means of the mean-value theorem and the identity (cf. [1])

\[ \begin{aligned} &\frac{(\psi(\xi_1,\xi_2,\tau)-\psi(x_1,x_2,t+\Delta t))^2}{t+\Delta t-\tau} -\frac{(\psi(\xi_1,\xi_2,\tau)-\psi(x_1,x_2,t))^2}{t-\tau} \\[1mm] &=- \frac{(\psi(\xi_1,\xi_2,\tau)-\psi(x_1,x_2,t))^2}{(t+\Delta t-\tau)(t-\tau)}\Delta t +\frac{\psi(x_1,x_2,t+\Delta t)-\psi(x_1,x_2,t)}{t+\Delta t-\tau}\times \\[1mm] &\qquad \times \bigl(\psi(x_1,x_2,t+\Delta t)-2\psi(\xi_1,\xi_2,\tau)+\psi(x_1,x_2,t)\bigr), \end{aligned} \tag{1.32} \]

and also the identity obtained by replacing in (1.32) \(\xi_i\) by \(x_i\). The estimates (1.27) follow from the mean-value theorem with the aid of (1.22) and (1.15). The estimate (1.28) follows from the mean-value theorem and (1.22). To derive the estimate (1.29), note the relation

\[ \begin{aligned} &\bigl|(\psi(y_1,y_2,\tau)-\psi(x_1,x_2,\tau))-(\psi(y_1,y_2,t)-\psi(x_1,x_2,t))\bigr| = \\[1mm] &=\left|\sum_{i=1}^{2}(y_i-x_i) \int_{0}^{1}\left( \frac{\partial\psi(x_1+z(y_1-x_1),\,x_2+z(y_2-x_2),\,\tau)}{\partial \xi_i} -\right.\right. \\[1mm] &\qquad\qquad\left.\left. -\frac{\partial\psi(x_1+z(y_1-x_1),\,x_2+z(y_2-x_2),\,t)}{\partial \xi_i} \right)\,dz\right|. \end{aligned} \tag{1.33} \]

Then the estimate (1.29) follows easily from (1.13) by means of the mean-value theorem and (1.22). In a similar way, (1.30) is also derived. Finally, to derive (1.31), we note that the left-hand side of (1.31) does not exceed, by virtue of the mean-value theorem, the quantity \(C(A_1|\lambda_1|+A_2|\lambda_1-\lambda_2|)\), where \((x_i=0)\):

\[ A_1=\int_{0}^{1}\left|g_{1,1}(\lambda_1(z),t-\tau)-g_{1,1}(\lambda_2(z),t-\tau)\right|\,dz,\quad A_2= \]

\[ =\int_{0}^{1} g_{1,1}(|\lambda_2(z)|,t-\tau)\,dz; \]

\[ \lambda_1(z)=\psi(\rho\cos\vartheta,\rho\sin\vartheta,\tau)-\psi(0,0,t)+z\lambda_1, \]

\[ \lambda_2(z)=\psi(0,0,\tau)-\psi(0,0,t)+z\lambda_2, \]

\[ \lambda_1=[\psi(y_1+\rho\cos\vartheta,\ y_2+\rho\sin\vartheta,\ \tau)-\psi(\rho\cos\vartheta,\ \rho\sin\vartheta,\ \tau)]- \]
\[ -[\psi(y_1,\ y_2,\ t)-\psi(0,\ 0,\ t)]; \]
\[ \lambda_2=[\psi(y_1,\ y_2,\ \tau)-\psi(0,\ 0,\ \tau)]-[\psi(y_1,\ y_2,\ t)-\psi(0,\ 0,\ t)]. \]

With the aid of relations of the form (1.33), using (1.13) and (1.22), it is easy to obtain the inequalities
\[ |\lambda_1|\le C[\psi]_{1+\alpha}(\rho^\alpha+(t-\tau)^{\alpha/2})|y-x|, \]
\[ |\lambda_1-\lambda_2|\le C[\psi]_{1+\alpha}\rho^\alpha |y-x|,\quad A_2\le C(t-\tau)^{-\frac12}, \]
\[ A_1\le C(t-\tau)^{-1}\int_0^1|\lambda_1(z)-\lambda_2(z)|\,dz \le C[\psi]_1(t-\tau)^{-1}\rho. \]

From this (1.31) follows.

Let \((x,t)\in\Gamma_t\), \((\bar x,\bar t)\in\bar\Gamma_t\), and let the function \(\varphi(y,\tau)\) be given on \(\Gamma\). With the aid of \(g_{0,\frac n2}(r(\bar x,y),\bar t-\tau)\)—the fundamental solution of the heat equation
\[ \sum_{i=1}^n \frac{\partial^2 u}{\partial x_i^2}-\frac{\partial u}{\partial t}=0 \tag{1.34} \]
we introduce the functions
\[ U(\bar x,\bar t)=\int_0^{\bar t}d\tau\iint_{\Gamma_\tau} g_{0,\frac n2}(r(\bar x,y),\bar t-\tau)\varphi(y,\tau)\,d\sigma_y(\tau), \tag{1.35} \]
\[ V(\bar x,\bar t)=\frac{\partial U(\bar x,\bar t)}{\partial N(x,t)}, \tag{1.36} \]
\[ W(\bar x,\bar t)=\int_0^{\bar t}d\tau\iint_{\Gamma_\tau} \frac{\partial g_{0,\frac n2}(r(\bar x,y),\bar t-\tau)}{\partial N(y,\tau)} \varphi(y,\tau)\,d\sigma_y(\tau), \tag{1.37} \]
where \(d\sigma_y(\tau)\) is the surface element of the section \(\Gamma_\tau\). The function \(U(\bar x,\bar t)\) will be called the heat potential of a simple layer, \(V(\bar x,\bar t)\) the normal derivative of the heat potential of a simple layer, and \(W(\bar x,\bar t)\) the heat potential of a double layer. The function \(\varphi(y,\tau)\), defined on \(\Gamma\), will be called the density of the corresponding heat potential. If \((x,t)\in\Gamma_t\), then we introduce the functions \(\bar U(x,t)\), \(\bar V(x,t)\), \(\bar W(x,t)\), obtained from \(U(\bar x,\bar t)\), \(V(\bar x,\bar t)\), and \(W(\bar x,\bar t)\), respectively, by replacing in (1.35)—(1.37) \((\bar x,\bar t)\) by \((x,t)\). The functions \(\bar U(x,t)\), \(\bar V(x,t)\), and \(\bar W(x,t)\) will be called (cf. [3]) the direct values (on \(\Gamma\)) of the corresponding heat potentials.

§ 2. FORMULATION OF THE MAIN RESULTS ON THE SMOOTHNESS OF THE HEAT POTENTIALS \(U\), \(V\), \(W\) ON A SURFACE OF TYPE

\[ \Lambda^{0,1,\frac{1+\alpha}{2}}_{1,\alpha,\alpha/2}. \]

We introduce the notation:
\[ \alpha_0\text{—any number satisfying the condition }0<\alpha_0<1, \tag{2.1} \]
\[ \alpha^0=\alpha\text{ for }0<\alpha<1\text{ and }\alpha^0=\alpha_0\text{ (from (2.1)) for }\alpha=1, \tag{2.2} \]
\[ \alpha'\text{—any number satisfying the condition }0<\alpha'<\alpha. \tag{2.3} \]

Theorem 1. If the surface \(\Gamma\) is of type \(J^{0,1,\frac{1+\alpha}{2}}_{1,\alpha,\alpha/2}\) \((0<\alpha\leqslant 1)\) and the density \(\varphi\) is a bounded function integrable on \(\Gamma\), then the function \(U(\bar x,\bar t)\) (see (1.35))—the heat potential of a simple layer—belongs to the class \(H^{0,\alpha_0,\frac12}(D_T)\), where the Hölder constants from (1.3) \(H_{0,x}(U)\) and \(H_{0,t}(U)\) have the form \((C)[\varphi]_0\). Moreover,

\[ |U(\bar x,\bar t)|\leqslant (C)[\varphi]_0 V^{\bar t},\qquad (\bar x,\bar t)\in D_T . \tag{2.4} \]

Remark 1. Throughout the paper, by \((C)\) we shall denote, without specifying their concrete form, constants which may depend on \(\Gamma\) (i.e., on \(\alpha\), \(T\), and \(|\psi|_{1+\alpha}\)), but do not depend on the density \(\varphi\).

Remark 2. For a bounded density \(\varphi\), the membership of \(U\) in the class \(H^{0,\alpha_0,\frac{\alpha_0}{2}}(D_T)\) was proved in [6] for cylindrical (with generators parallel to the axis \(Ot\)) surfaces \(\Gamma\) of Lyapunov type. In [7] this result was generalized to noncylindrical surfaces \(\Gamma\) of Lyapunov type in \(D_T\). We note that from \(\Gamma\in J^{0,1,\frac{1+\alpha}{2}}_{1,\alpha,\alpha/2}\) it does not follow that \(\Gamma\) is a Lyapunov surface in \(D_T\) (see § 1).

Theorem 2. If the hypotheses of Theorem 1 are satisfied for \(\Gamma\) and \(\varphi\), then the function \(\overline W(x,t)\) (see (1.37), where \((\bar x,\bar t)\equiv (x,t)\in\Gamma_t\))—the direct value of the heat double-layer potential—belongs to the class

\[ H^{0,\alpha_0,\frac{\alpha_0}{2}}(\Gamma)\quad \text{(see (2.2))}, \]

where \(H_{0,x}(\overline W)\) and \(H_{0,t}(\overline W)\) have the form \((C)[\varphi]_0\); moreover,

\[ |\overline W(x,t)|\leqslant (C)[\varphi]_0\,t^{\alpha/2},\qquad (x,t)\in\Gamma . \tag{2.5} \]

Theorem 3. If the hypotheses of Theorem 1 are satisfied for \(\Gamma\) and \(\varphi\), then the function \(\overline V(x,t)\) (see (1.36), where \((\bar x,\bar t)\equiv (x,t)\in\Gamma_t\))—the direct value of the normal derivative of the heat simple-layer potential—belongs to the class

\[ H^{0,\alpha',\frac{\alpha'}{2}}(\Gamma)\quad \text{(see (2.3))}, \]

where \(H_{0,x}(\overline V)\) and \(H_{0,t}(\overline V)\) have the form \((C)[\varphi]_0\), and, moreover,

\[ |\overline V(x,t)|\leqslant (C)[\varphi]_0\,t^{\alpha/2},\qquad (x,t)\in\Gamma_t . \tag{2.6} \]

Theorem 4. If the surface \(\Gamma\) is of type \(J^{0,1,\frac{1+\alpha}{2}}_{1,\alpha,\alpha/2}\) \((0<\alpha\leqslant 1)\) and the density \(\varphi\) is of class \(H^{0,\alpha,\alpha/2}(\Gamma)\), then

\[ \overline V(x,t)\in H^{0,\alpha_0,\alpha/2}(\Gamma)\quad \text{(see (2.2))}, \]

and \(H_{0,x}(\overline V)\) and \(H_{0,t}(\overline V)\) have the form \((C)|\varphi|_\alpha\).

Theorem 5. Let \(\Gamma\in J^{0,1,\frac{1+\alpha}{2}}_{1,\alpha,\alpha/2}\) \((0<\alpha\leqslant 1)\). If the density \(\varphi\) is continuous on \(\Gamma\) and the point \((\bar x,\bar t)\) tends to the point \((x,t)\in\Gamma_t\), then (cf. [1], [2])

\[ \lim_{(\bar x,\bar t)\to (x,t)\in\Gamma_t} W(\bar x,\bar t) = \overline W(x,t) \mp \frac12(2\sqrt{\pi})^n\varphi(x,t), \tag{2.7} \]

where the sign \(-\) (or \(+\)) is taken when \((\bar x,\bar t)\in D_T^{\mathrm{в}}\) (or \((\bar x,\bar t)\in D_T^{\mathrm{н}}\)). If, moreover, \(\varphi\in H^{0,\alpha,\alpha/2}(\Gamma)\), and

\[ \varphi(y,0)\equiv 0\quad \text{for }(y,0)\in\Gamma_0, \tag{2.8} \]

then (see (2.2))

\[ \left| W(\bar{x}, \bar{t})-\bar{W}(x,t)\pm \frac{(2\sqrt{\pi})^n}{2}\,\varphi(x,t)\right| \leq (C)[\varphi]_{\alpha}\bigl(r^{\alpha^0}(\bar{x},x)+|\bar{t}-t|^{\alpha/2}\bigr). \tag{2.9} \]

Theorem 6. Suppose that \(\Gamma\) and \(\varphi\) satisfy the conditions of Theorem 4 and that (2.8) holds. Then \(W(\bar{x},\bar{t})\in H^{0,\alpha^0,\alpha/2}(\overline{D_T^k})\) \((k=v,n)\), where \(H_{0,x}(W)\) and \(H_{0,t}(W)\) have the form \((C)[\varphi]_{\alpha}\), and
\[ |W(\bar{x},\bar{t})|\leq (C)[\varphi]_{\alpha}\,t^{\alpha/2},\qquad (\bar{x},\bar{t})\in \overline{D_T^k}. \tag{2.10} \]

Theorem 7. Suppose that for \(\Gamma\) and \(\varphi\) the conditions of Theorem 5 are fulfilled. Suppose that \((\bar{x},\bar{t})\to(x,t)\) in tending to \((x,t)\in\Gamma_t\) lies on the normal \(N(x,t)\). Then for continuous \(\varphi\) (cf. [1], [2])
\[ \lim_{\substack{(\bar{x},\bar{t})\to(x,t)\in\Gamma_t}} V(\bar{x},\bar{t}) = \bar{V}(x,t)\mp \frac{(2\sqrt{\pi})^n}{2}\,\varphi(x,t), \tag{2.11} \]
the sign \(-\) (or \(+\)) being taken when \((\bar{x},\bar{t})\in D_T^k\), where \(k=v\) (or \(k=n\)). For \(\varphi\in H^{0,\alpha,\alpha/2}(\Gamma)\) and under (2.8),
\[ \left| V(\bar{x},t)-\bar{V}(x,t)\pm \frac{(2\sqrt{\pi})^n}{2}\,\varphi(x,t) \right| \leq (C)[\varphi]_{\alpha}\,r^{\alpha^0}(x,\bar{x}) \tag{2.12} \]
and
\[ |V(\bar{x},t)|\leq (C)|\varphi|_{\alpha}\,t^{\alpha/2},\qquad (\bar{x},t)\in \overline{D_T}. \tag{2.13} \]

Remark 3. (cf. [6], [7]). If \(\varphi\) is only continuous, then in equality (2.11) one may replace \((\bar{x},t)\) by \((\bar{x},\bar{t})\) under the condition that
\[ \lim_{\substack{(\bar{x},\bar{t})\to(x,t)\in\Gamma_t}} \bigl(r^{\alpha}(\bar{x}_t,x)+|\bar{t}-t|^{\alpha/2}\bigr) \left|\ln |r(\bar{x},\bar{x}_t)|\right|=0, \tag{2.14} \]
where \((\bar{x}_t,\bar{t})\in\Gamma_t\) is the point of \(\Gamma\) nearest to \((\bar{x},\bar{t})\).

Remark 4. From estimate (2.15) of Theorem 8 it follows that, for \(\varphi\in H^{0,\alpha,\alpha/2}(\Gamma)\) and (2.8), in (2.12) one may replace \((\bar{x},t)\) by \((\bar{x},\bar{t})\), without imposing conditions on the manner in which \((\bar{x},\bar{t})\) tends to \((x,t)\in\Gamma\); in this case on the right-hand side of (2.12) one must write \(r^{\alpha^0}(\bar{x},x)+|\bar{t}-t|^{\alpha/2}\) (see (2.2)).

Theorem 8. Suppose that \(\Gamma\in \Pi_{1,\alpha}^{0,1,(1+\alpha)/2}\) \((0<\alpha\leq 1)\), \(\varphi\in H^{0,\alpha,\alpha/2}(\Gamma)\), and that (2.8) holds. Then \(U(\bar{x},\bar{t})\in H_{1,\alpha',\alpha'/2}^{0,1,(1+\alpha^0)/2}(\overline{D_T^k})\) \((k=v,n)\), where the Hölder constants \(H_{0,t}(U)\), \(H_{1,x}(U)\), and \(H_{1,t}(U)\) (cf. (1.1), (1.2)) have the form \((C)|\varphi|_{\alpha}\), and moreover
\[ \left| \frac{\partial U(\bar{x},\bar{t})}{\partial x_j} \right| \leq (C)[\varphi]_{\alpha}\,t^{\alpha/2}, \qquad (\bar{x},\bar{t})\in \overline{D_T^k},\quad j=1,2,\ldots,n. \tag{2.15} \]

The proofs of Theorems 1–8 are deferred to §§ 3–5 and are carried out for \(n=3\), which, of course, does not impair the generality of the results.

§ 3. Proof of Theorems 1–4

The proof of Theorem 1 can be carried out by the methods of [1], [2], [6], [7], and therefore it will be omitted. Theorems 2–4 are proved

quite analogously. First we shall dwell on the proof of Theorems 3 and 4. Put, denoting \(t_1=t-d_1\) (if \(0<t\leq d_1\), then \(t_1=0\)),

\[ \overline V(x,t)=\sum_{i=1}^{3}\overline V_i^{(0)}(x,t), \]

where

\[ \overline V_1^{(0)}(x,t)= \int_{t_1}^{t} d\tau \iint_{\Gamma_\tau(x,d)} \left\{ \frac12 \cos\bigl(N(x,t),r_{xy}(\tau)\bigr)\times \right. \]

\[ \left. {}\times g_{1,\,5/2}(r(x,y),t-\tau)\,\varphi(y,\tau) \right\}d\sigma_y(\tau), \]

\[ \overline V_2^{(0)}(x,t)= \int_{t_1}^{t} d\tau \iint_{\Gamma_\tau\setminus\Gamma_\tau(x,d)} \{\ \}d\sigma_y(\tau), \tag{3.1} \]

\[ \overline V_3^{(0)}(x,t)= \int_{0}^{t_1} d\tau \iint_{\Gamma_\tau} \{\ \}d\sigma_y(\tau). \]

Taking into account the inequalities \(((x,t)\in\Gamma,\ (y,\tau)\in\Gamma,\ (\bar x,t)\in\Gamma,\ (x,t+\Delta t)\in\Gamma)\)

\[ \left|\cos\bigl(N(x,t+\Delta t),r_{xy}(\tau)\bigr) -\cos\bigl(N(x,t),r_{xy}(\tau)\bigr)\right| \leq (C)|\Delta t|^{\alpha/2}, \]

\[ \left|\cos\bigl(N(x,t),r_{xy}(\tau)\bigr) -\cos\bigl(N(\bar x,t),r_{\bar x y}(\tau)\bigr)\right| \leq (C)r^\alpha(x,\bar x), \]

which follow from the Lyapunov conditions (1.6), (1.7), we see that it is enough to prove Theorems 3, 4 for the functions in (3.1). Introduce on \(S_{dd_1}(x,t)\) a local coordinate system \(\{\xi,\tau\}\) connected with the point \((x,t)\in\Gamma_t\) (see § 1), and we shall also use the polar system (1.11). By Lemma 1, the inner integration in (3.1) will be carried out over the disk \(0\leq\rho\leq d\). We note that from the relations

\[ \cos\bigl(N(x,t),\xi_3\bigr)=b_0^{(0)}(x,t), \]

\[ \cos\bigl(N(x,t),\xi_j\bigr)= -\,b_0^{(0)}(x,t)\frac{\partial\psi(x_1,x_2,t)}{\partial x_j}, \]

\[ \cos\bigl(r_{x\xi}(\tau),\xi_3\bigr) =r^{-1}(x,\xi)\bigl(\psi(\xi_1,\xi_2,\tau)-\psi(x_1,x_2,t)\bigr), \]

\[ \cos\bigl(r_{x\xi}(\tau),\xi_j\bigr) =r^{-1}(x,\xi)(\xi_j-x_j) \]

\[ (j=1,2),\qquad b_0^{(0)}(x,t)= \left[ 1+\sum_{i=1}^{2}\left(\frac{\partial\psi(x_1,x_2,t)}{\partial x_i}\right)^2 \right]^{-1/2} \]

and the theorem on the mean it follows that

\[ r(x,\xi)\cos\bigl(N(x,t),r_{x\xi}(\tau)\bigr)= \]

\[ =r(x,\xi)\sum_{j=1}^{3} \cos\bigl(N(x,t),\xi_j\bigr)\cos\bigl(r_{x\xi}(\tau),\xi_j\bigr) = b_0^{(0)}(x,t)\sum_{j=1}^{3} b_j^{(0)}(x,t), \]

where

\[ b_1^{(0)}(x,t)=\psi(x_1,x_2,\tau)-\psi(x_1,x_2,t),\quad b_3^{(0)}(x,t)= \]

\[ = \sum_{j=1}^{2}\left(\frac{\partial \psi(x_1,x_2,\tau)}{\partial x_j} -\frac{\partial \psi(x_1,x_2,t)}{\partial x_j}\right)(\xi_j-x_j), \]

\[ \begin{aligned} b_2^{(0)}(x,t) &\equiv b_2^{(0)}(x) = \sum_{j=1}^{2}(\xi_j-x_j)\times \\ &\quad \times \int_{0}^{1}\left( \frac{\partial \psi\bigl(x_1+z(\xi_1-x_1),\,x_2+z(\xi_2-x_2),\,\tau\bigr)}{\partial \xi_j} -\frac{\partial \psi(x_1,x_2,\tau)}{\partial \xi_j} \right)\,dz . \end{aligned} \tag{3.2} \]

Introduce the functions

\[ \varphi^{(0)}(\xi_1,\xi_2,\tau) =\varphi(\xi_1,\xi_2,\tau) \left[1+\sum_{j=1}^{2}\left(\frac{\partial \psi(\xi_1,\xi_2,\tau)}{\partial \xi_j}\right)^2\right]^{1/2}; \]

\[ \varphi_0(\xi_1,\xi_2,\tau)=\varphi^{(0)}(\xi_1,\xi_2,\tau)\omega(\rho), \tag{3.3} \]

\[ \psi_1(\xi_1,\xi_2,\tau)=\psi(\xi_1,\xi_2,\tau)\omega(\rho);\qquad \varphi_1(\xi_1,\xi_2,\tau)=\varphi(\xi_1,\xi_2,\tau)\omega(\rho), \]

where \(\omega(\rho)\) \((0\leq \omega(\rho)\leq 1)\) is an infinitely differentiable function equal to \(0\) for \(\rho\geq d\) and to \(1\) for \(0\leq \rho\leq \frac{3}{4}d\). Obviously, \(\varphi_0,\varphi_1\) and \(\psi_1\) are defined on

\[ G_{d_1}=\{(\xi,\tau),\ |\xi_i|<+\infty,\ i=1,2;\ t_1\leq \tau\leq t_1+2d_1\} \]

and vanish for \(\rho\geq d\). Obviously, \(\psi_1\in H^{0,1,\frac{1+\alpha}{2}}_{1,\alpha,\alpha/2}(G_{d_1})\) with Hölder constants of the form \((C)\|\psi\|_{1+\alpha}\). If \(\varphi\in H^{0,\alpha,\alpha/2}(\Gamma)\), then \(\varphi_0,\varphi_1\in H^{0,\alpha,\alpha/2}(G_{d_1})\) with Hölder constants of the form \((C)|\varphi|_{\alpha}\). Denote by \(b_i(x,t)\) the functions obtained from \(b_i^{(0)}(x,t)\) in (3.2) by replacing \(\psi\) with \(\psi_1\).

Put (see (1.19)—(1.21) and (1.11))

\[ a_i(x,\xi,t)=\varphi_0(\xi_1,\xi_2,\tau)b_i(x,t)g_{0,5/2}(|x-\xi|,t-\tau)g(\psi_1;x,t), \]

\[ i=1,2,3,\quad b_j(x,\rho,t)\equiv b_j(x,t),\quad j=0,1, \]

\[ b_2(x,\rho,t)\equiv b_2(x,\rho)= \]

\[ =\rho\sum_{i=1}^{2}s_i\int_{0}^{1}\left( \frac{\partial \psi_1(x_1+z\rho s_1,\,x_2+z\rho s_2,\tau)}{\partial \xi_i} -\frac{\partial \psi_1(x_1,x_2,\tau)}{\partial \xi_i} \right)\,dz, \]

\[ b_3(x,\rho,t)=\rho\sum_{i=1}^{2}s_i\left( \frac{\partial \psi_1(x_1,x_2,\tau)}{\partial x_i} -\frac{\partial \psi_1(x_1,x_2,t)}{\partial x_i} \right), \tag{3.4} \]

\[ a_i(x,\rho,t)=\varphi_0(x_1+\rho s_1,x_2+\rho s_2,\tau)b_i(x,\rho,t)\times \]

\[ \times g_{1,5/2}(\rho,t-\tau)g(\psi_1;x,\rho,t),\quad i=1,2,3, \]

where

\[ s_1=\cos\vartheta,\quad s_2=\sin\vartheta. \tag{3.5} \]

Obviously, instead of the function (3.1) it suffices to investigate the function

\[ \overline V_1(x,t)=\frac{1}{2} b_0(x,t)\sum_{i=1}^{3}\overline V_{1i}(x,t) =\frac{1}{2} b_0(x,t)\sum_{i=1}^{3}\overline V^{*}_{1i}(x,t), \]

where

\[ \overline V_{1i}(x,t)=\int_{t_1}^{t}d\tau\int_{-\infty}^{+\infty}d\xi_1 \int_{-\infty}^{+\infty} a_i(x,\xi,t)\,d\xi_2; \]

\[ \overline V^{*}_{1i}(x,t)=\int_{t_1}^{t}d\tau\int_{0}^{2\pi}d\vartheta \int_{0}^{+\infty} a_i(x,\rho,t)\,d\rho,\qquad i=1,2,3. \]

From the estimates (1.12), (1.13) the inequalities follow

\[ |b_1(x,t)|\leq (C)(t-\tau)^{\frac{1+\alpha}{2}},\qquad |b_2(x,t)|\leq (C)\rho^{1+\alpha}, \]

\[ |b_3(x,\rho,t)|\leq (C)\rho(t-\tau)^{\alpha/2}, \tag{3.6} \]

whence, with the aid of inequality (1.23), it is easy to obtain the estimate

\[ |\overline V^{*}_{1i}(x,t)|\leq (C)[\varphi]_0 t^{\alpha/2},\qquad i=1,2,3, \tag{3.7} \]

from which, in particular, (2.6) follows; moreover, for \(t\leq |\Delta t|\),

\[ |\overline V^{*}_{1i}(x,t+\Delta t)-\overline V^{*}_{1i}(x,t)| \leq (C)[\varphi]_0|\Delta t|^{\alpha/2}. \]

In what follows we shall assume that \(|\Delta t|\leq d_1\) and

\[ 0<\Delta t\leq t. \tag{3.8} \]

Put

\[ \overline V^{*}_{1i}(x,t+\Delta t)-\overline V^{*}_{1i}(x,t) =\sum_{j=1}^{3}\overline V_{1ij},\qquad i=1,2,3, \]

where

\[ \overline V_{1i1}= \int_{t-\Delta t}^{t+\Delta t}d\tau\int_{0}^{2\pi}d\vartheta \int_{0}^{+\infty} a_i(x,\rho,t+\Delta t)\,d\rho, \]

\[ \overline V_{1i2}= -\int_{t-\Delta t}^{t}d\tau\int_{0}^{2\pi}d\vartheta \int_{0}^{+\infty} a_i(x,\rho,t)\,d\rho, \]

\[ \overline V_{1i3}= \int_{t_1}^{t-\Delta t}d\tau\int_{0}^{2\pi}d\vartheta \int_{0}^{+\infty} \left[a_i(x,\rho,t+\Delta t)-a_i(x,\rho,t)\right]\,d\rho. \tag{3.9} \]

From the estimates (3.6) it follows at once that

\[ |\overline V_{1ij}|\leq (C)[\varphi]_0|\Delta t|^{\alpha/2},\qquad i=1,2,3;\quad j=1,2. \]

Let us now put (with \(x_i=0,\ i=1,2\), by virtue of (1.10), and taking into account (3.5))

\[ a_1(x,\rho,t+\Delta t)-a_1(x,\rho,t) =\varphi_0(\rho s_1,\rho s_2,\tau)\sum_{j=1}^{3}a_{1j}, \]

where

\[ a_{11}=\left[b_1(0,t+\Delta t)-b_1(0,t)\right] g_{1,5/2}(\rho,t+\Delta t-\tau)\,g(\psi_1;0,\rho,t+\Delta t), \]

\[ a_{12}=b_1(0,t)\,[g_{1,5/2}(\rho,t+\Delta t-\tau)-g_{1,5/2}(\rho,t-\tau)]\times \]
\[ \times g(\psi_1;0,\rho,t+\Delta t), \tag{3.10} \]
\[ a_{13}=b_1(0,t)g_{1,5/2}(\rho,t-\tau)[g(\psi_1;0,\rho,t+\Delta t)-g(\psi_1;0,\rho,t)]. \]

From the estimates (1.12), (3.6), (1.24), (1.23), and (3.6) we have

\[ |a_{11}|\leq (C)g_{1,5/2}(\rho,t+\Delta t-\tau)(\Delta t)^{\frac{1+\alpha}{2}}, \]

\[ |a_{12}|\leq (C)(t-\tau)^{-3+\frac{\alpha}{2}}g_{1,0}(\rho,2(t+\Delta t-\tau))\Delta t, \]

\[ |a_{13}|\leq (C)g_{1,2-\frac{\alpha}{2}}(\rho,2(t-\tau))(\Delta t)^\alpha. \]

Therefore, making substitutions of the form \(\frac18\rho^2(t-\tau)^{-1}=s\), we obtain, in view of (3.8),

\[ |\overline V_{113}|\leq (C)[\varphi]_0 \int_0^{t-\Delta t} \left[(\Delta t)^{\frac{1+\alpha}{2}}(t+\Delta t-\tau)^{-\frac32} +\right. \]

\[ \left. +\Delta t(t-\tau)^{-2+\frac{\alpha}{2}} +(\Delta t)^2(t-\tau)^{-3+\frac{\alpha}{2}} +\right. \]

\[ \left. +(\Delta t)^\alpha(t-\tau)^{-1+\frac{\alpha}{2}} \right]d\tau \leq (C)[\varphi]_0\times \]

\[ \times\left(3+\left(\frac{\Delta t}{t}\right)^{1/2} +\left(\frac{\Delta t}{t}\right)^{1-\frac{\alpha}{2}} +\left(\frac{\Delta t}{t}\right)^{2-\frac{\alpha}{2}}\right)(\Delta t)^{\alpha/2}\leq \]

\[ \leq (C)[\varphi]_0|\Delta t|^{\alpha/2}. \tag{3.11} \]

In an analogous way, with the aid of (3.6), the estimate

\[ |\overline V_{123}|\leq (C)[\varphi]_0|\Delta t|^{\alpha/2}. \]

is proved.

When estimating \(|\overline V_{133}|\), we shall distinguish two cases: a) \(\varphi_0\) is only bounded (Theorem 3) and b) \(\varphi_0\in H^{0,\alpha,\alpha/2}(G_{d_1})\) (Theorem 4). In case a) we set

\[ a_3(x,\rho,t+\Delta t)-a_3(x,\rho,t) =\varphi_0(\rho s_1,\rho s_2,\tau)\sum_{j=1}^3 a_{3j}, \]

where \(a_{3j}\) \((j=1,2,3)\) are obtained from \(a_{1j}\) by replacing in (3.10) the function \(b_1(0,t)\) by \(b_3(0,\rho,t)\). From the estimates (3.8), (3.6), (1.13), (1.24), (1.25), and (1.23) we obtain

\[ |a_{31}|\leq (C)g_{2,5/2}(\rho,t+\Delta t-\tau)(\Delta t)^{\alpha/2}, \]

\[ |a_{32}|\leq (C)(t-\tau)^{-\frac{7-\alpha}{2}}g_{2,0}(\rho,2(t+\Delta t-\tau))\Delta t, \]

\[ |a_{33}|\leq (C)g_{2,\frac{5-\alpha}{2}}(\rho,2(t-\tau))(\Delta t)^\alpha. \]

Therefore, comparing the derivation of (3.11), taking (3.9) into account (see (2.3)), we have

\[ |\bar V_{133}| \leq (C)[\varphi]_0 \left(\int_0^{t-\Delta t} (t+\Delta t-\tau)^{-1}\,d\tau+1\right)(\Delta t)^{\alpha/2} \leq \]

\[ \leq (C)[\varphi]_0\left(|\Delta t|^{\frac{\alpha-\alpha'}{2}}\times \left(1+\left|\ln\frac{\Delta t}{t}\right|\right)\right)(\Delta t)^{\alpha'/2} \leq (C)[\varphi]_0\,t^{\frac{\alpha-\alpha'}{2}}|\Delta t|^{\alpha'/2}. \tag{3.12} \]

Taking (3.7) and (see (1.13)) into account,

\[ |b_0(x,t+\Delta t)-b_0(x,t)| \leq (C)|\Delta t|^{\alpha/2}, \tag{3.13} \]

we see that in case a) (see (2.3))

\[ |\bar V_1(x,t+\Delta t)-\bar V_1(x,t)| \leq (C)[\varphi]_0|\Delta t|^{\alpha'/2}. \tag{3.14} \]

In case b) we put

\[ a_3(x,\rho,t)=\sum_{i=1}^{2} a_{3i}(x,\rho,t), \]

where (see (1.21))

\[ a_{31}(x,\rho,t)=\varphi_0(0,0,\tau)g^{(0)}(\psi_1;0,t)b_3(0,\rho,t)g_{1,\,5/2}(\rho,t-\tau), \]

\[ a_{32}(x,\rho,t)= [\varphi_0(\rho S_1,\rho S_2,\tau)g(\psi_1;0,\rho,t)- \]

\[ -\varphi_0(0,0,\tau)g^{(0)}(\psi_1;0,t)]b_3(0,\rho,t)g_{1,\,5/2}(\rho,t-\tau). \]

Obviously,

\[ \int_{t_1}^{t-\Delta t} d\tau \int_0^{2\pi} d\vartheta \int_0^{+\infty} a_{31}(x,\rho,t+l_i\Delta t)\,d\rho=0, \qquad l_1=0,\ l_2=1. \]

Therefore it is sufficient to estimate

\[ a_{32}(x,\rho,t+\Delta t)-a_{32}(x,\rho,t)=\sum_{i=1}^{3} a_{32i}, \]

where

\[ a_{321}=[g_{1,\,5/2}(\rho,t+\Delta t-\tau)-g_{1,\,5/2}(\rho,t-\tau)]\times \]

\[ \times b_3(0,\rho,t+\Delta t)[\varphi_0(\rho S_1,\rho S_2,\tau)g(\psi_1;0,\rho,t+\Delta t)- \]

\[ -\varphi_0(0,0,\tau)g^{(0)}(\psi_1;0,t+\Delta t)], \]

\[ a_{322}=g_{1,\,5/2}(\rho,t-\tau)[b_3(0,\rho,t+\Delta t)-b_3(0,\rho,t)]\times \]

\[ \times[(\varphi_0(\rho S_1,\rho S_2,\tau)-\varphi_0(0,0,\tau))g(\psi_1;0,\rho,t+\Delta t)+ \]

\[ +\varphi_0(0,0,\tau)(g(\psi_1;0,t+\Delta t)-g^{(0)}(\psi_1;0,t+\Delta t))], \]

\[ a_{323}=g_{1,\,5/2}(\rho,t-\tau)b_3(0,\rho,t)[\varphi_0(\rho S_1,\rho S_2,\tau)\times \]

\[ \times(g(\psi_1;0,\rho,t+\Delta t)-g(\psi_1;0,\rho,t))- \]

\[ -\varphi_0(0,0,\tau)(g^{(0)}(\psi_1;0,t+\Delta t)-g^{(0)}(\psi_1;0,t))]. \]

From (3.8), (1.24), and (3.6) we have
\[ |a_{321}| \leq (C)[\varphi]_0 (t-\tau)^{-7/2}\, g_{2,-\frac{\alpha}{2}}(\rho,\,2(t+\Delta t-\tau))\Delta t . \]

From (3.8), (1.13), (1.17), (1.27) we have, by virtue of (1.23),
\[ |a_{322}| \leq (C)|\varphi|_{\alpha}\, g_{2,\frac{5-\alpha}{2}}(\rho,\,2(t-\tau))(\Delta t)^{\frac{\alpha}{2}} . \]

Finally, from (3.8), (3.6), (1.25), (1.26), by virtue of (1.23), we have
\[ |a_{323}| \leq (C)[\varphi]_0\, g_{2,\frac{5-\alpha}{2}}(\rho,\,2(t-\tau))(\Delta t)^{\alpha}. \]

Therefore, in view of (3.8),
\[ |\bar V_{133}| \leq (C)|\varphi|_{\alpha}|\Delta t|^{\frac{\alpha}{2}} . \]

Thus, in case b), in view of (3.13), we have
\[ |\bar V_1(x,t+\Delta t)-\bar V_1(x,t)| \leq (C)|\varphi|_{\alpha}|\Delta t|^{\frac{\alpha}{2}} . \tag{3.15} \]

Let now \(x_1=x_2=y_1=0;\ y_2=\delta>0\). We shall again distinguish cases a) and b). In case a) we put
\[ \bar V_{1i}(y,t)-\bar V_{1i}(x,t)=\sum_{j=1}^{3'} \bar V_{1i}^{(j)}, \]
where
\[ \bar V_{1i}^{(1)} =\int_{t_1}^{t} d\tau \iint_{0\leq \rho \leq 2\delta} a_i(y,\xi,t)\,d\xi_1\,d\xi_2, \]
\[ \bar V_{1i}^{(2)} =-\int_{t_1}^{t} d\tau \iint_{0\leq \rho \leq 2\delta} a_i(x,\xi,t)\,d\xi_1\,d\xi_2, \]
\[ \bar V_{1i}^{(3)} =\int_{t_1}^{t} d\tau \iint_{\rho>2\delta} \bigl[a_i(y,\xi,t)-a_i(x,\xi,t)\bigr]\,d\xi_1\,d\xi_2 . \tag{3.16} \]

Passing to polar coordinates (1.11) and making the substitutions \(t-\tau=\eta\), \(\frac14\rho^2\eta^{-1}=s\), \(\delta^2\eta^{-1}=z\), we obtain, by virtue of (3.6), the estimate
\[ |\bar V_{1i}^{(2)}| \leq (C)[\varphi]_0 \int_{0}^{t}\eta^{-1+\frac{\alpha}{2}} \left(1-\exp\{-\delta^2\eta^{-1}\}\right)d\eta \leq \tag{3.17} \]
\[ \leq (C)[\varphi]_0\delta^\alpha \int_{0}^{+\infty} z^{-1-\frac{\alpha}{2}}(1-e^{-z})\,dz \leq (C)[\varphi]_0\delta^\alpha,\qquad i=1,2,3. \]

Next we have
\[ |\bar V_{1i}^{(1)}| \leq \int_{t_1}^{t} d\tau \iint_{\rho_1\leq 3\delta} |a_i(y,\xi,t)|\,d\xi_1\,d\xi_2, \]
where
\[ \rho_1=\sqrt{\xi_1^2+(\xi_2-\delta)^2}, \tag{3.18} \]

whence, passing to polar coordinates

\[ \xi_1=\rho_1\cos\vartheta,\quad \xi_2=\delta+\rho_1\sin\vartheta, \tag{3.19} \]

we obtain

\[ \left|\overline V_{li}^{(1)}\right|\leq (C)[\varphi]_{0}\delta^\alpha . \tag{3.20} \]

Let us note that for \(\rho\geq 2\delta\) we have

\[ \frac{3}{2}\rho\geq (1-\mu)\rho+\mu\rho_1>\frac{\rho}{2} \quad\text{for any }0\leq \mu\leq 1. \tag{3.21} \]

Put

\[ a_i(y,\xi,t)-a_i(x,\xi,t)=\varphi_0(\xi_1,\xi_2,\tau)\sum_{j=1}^{2}a_i^{(j)},\quad i=1,2,3, \]

where

\[ \begin{aligned} a_i^{(1)}&=[b_i(y,t)-b_i(x,t)]\,g_{0,\,5/2}(|y-\xi|,t-\tau)\,g(\psi_1;y,t),\\ a_i^{(2)}&=b_i(x,t)\bigl[g_{0,\,5/2}(|y-\xi|,t-\tau)\,g(\psi_1;y,t)\\ &\qquad\qquad\qquad{}-g_{0,\,5/2}(|x-\xi|,t-\tau)\,g(\psi_1;x,t)\bigr]. \end{aligned} \tag{3.22} \]

From the representation

\[ b_1(y,t)-b_1(x,t)= \delta\int_{0}^{1} \left( \frac{\partial\psi_1(0,z\delta,\tau)}{\partial \xi_2} - \frac{\partial\psi_1(0,z\delta,t)}{\partial \xi_2} \right)\,dz \]

with the aid of (1.13) we obtain

\[ |b_1(y,t)-b_1(x,t)|\leq (C)(t-\tau)^{\alpha/2}\delta . \tag{3.23} \]

Moreover, from (3.21) and (1.13) it follows that

\[ |b_2(y)-b_2(x)|\leq (C)(\rho^\alpha\delta+\rho\delta^\alpha), \]

\[ |b_3(y,t)-b_3(x,t)|\leq (C)\bigl[(t-\tau)^{\alpha/2}\delta+\rho\delta^\alpha\bigr]. \]

Therefore, by virtue of (3.21),

\[ |a_1^{(1)}|\leq (C)g_{0,\,(5-\alpha)/2}(\rho/2,t-\tau)\delta, \]

\[ |a_i^{(1)}|\leq (C)\bigl[g_{0,\,(5-\alpha)/2}(\rho/2,t-\tau)\delta +g_{1,\,5/2}(\rho/2,t-\tau)\delta^\alpha\bigr],\quad i=2,3. \tag{3.24} \]

With the aid of the mean value theorem, (3.21), and (1.22) we obtain

\[ \begin{aligned} &\bigl|g_{0,\,5/2}(|y-\xi|,t-\tau)\,g(\psi_1;y,t) -g_{0,\,5/2}(|x-\xi|,\\ &\qquad t-\tau)\,g(\psi_1;x,t)\bigr| \leq (C)g_{0,\,3}\left(\frac{\rho}{2},t-\tau\right)\delta . \end{aligned} \tag{3.25} \]

From (3.22), with the aid of (3.24), (3.25), and (3.6), passing to polar coordinates (1.11) and making the substitutions \(t-\tau=\eta\), \(\frac{1}{16}\rho^2\eta^{-1}=s\), \(\frac{1}{4}\delta^2\eta^{-1}=z\), we easily obtain, using (1.23), the estimate (see (2.3)):

\[ \left|\overline{V}_{1i}^{(3)}\right|\leq (C)[\varphi]_0 \int_0^t \left(\delta \eta^{-\frac{3-\alpha}{2}}+\delta^\alpha \eta^{-1}\right) \exp\left\{-\frac{\delta^2}{4\eta}\right\}\,d\eta \leq \]
\[ \leq (C)[\varphi]_0 \int_{\frac{\delta^2}{4t}}^{+\infty} \left(z^{-\frac{1+\alpha}{2}}+z^{-1}\right)e^{-z}\,dz \leq (C)[\varphi]_0 t^{\frac{\alpha-\alpha'}{2}}\delta^{\alpha'} . \tag{3.26} \]

In deriving (3.26) we used the obvious inequality
\[ 0\leq \int_u^{+\infty} z^{-1}e^{-z}\,dz \leq (C)u^{-\beta},\qquad 0<u<+\infty, \tag{3.27} \]
valid for any \(\beta>0\). From (3.17), (3.20), (3.26), by virtue of the inequality (see (1.13))
\[ |b_0(y,t)-b_0(x,t)|\leq (C)\delta^\alpha \tag{3.28} \]
we obtain, in case a), the estimate (see (2.3))
\[ |\overline{V}_1(y,t)-\overline{V}_1(x,t)| \leq (C)[\varphi]_0\delta^{\alpha'} . \tag{3.29} \]

In case b), put
\[ \overline{V}_{1i}^{*}(y,t)-\overline{V}_{1i}^{*}(x,t) =\sum_{j=1}^3 \overline{V}_{1i}^{(j)}(\delta),\qquad i=1,2,3, \]
where
\[ \overline{V}_{1i}^{(1)}(\delta) =\int_{t_1}^{t}d\tau\int_0^{2\pi}d\vartheta\int_0^{2\delta} a_i(y,\rho,t)\,d\rho, \]
\[ \overline{V}_{1i}^{(2)}(\delta) =-\int_{t_1}^{t}d\tau\int_0^{2\pi}d\vartheta\int_0^{2\delta} a_i(x,\rho,t)\,d\rho, \]
\[ \overline{V}_{1i}^{(3)}(\delta) =\int_{t_1}^{t}d\tau\int_0^{2\pi}d\vartheta\int_{2\delta}^{+\infty} [a_i(y,\rho,t)-a_i(x,\rho,t)]\,d\rho,\qquad i=1,2,3. \tag{3.30} \]

From the estimates (3.6) we immediately obtain
\[ \left|\overline{V}_{1i}^{(j)}(\delta)\right| \leq (C)[\varphi]_0 \int_0^t \eta^{-1+\frac{\alpha}{2}} \left(1-\exp\{-\delta^2\eta^{-1}\}\right)\,d\eta \leq \]
\[ \leq (C)[\varphi]_0\delta^\alpha,\qquad j=1,2;\ i=1,2,3. \tag{3.31} \]

Now put (see (3.5))
\[ a_1(y,\rho,t)-a_1(x,\rho,t)=\sum_{j=1}^3 a_{1j}(\rho), \]
where
\[ a_{11}(\rho)=[b_1(y,t)-b_1(x,t)]\varphi_0(\rho s_1,\delta+\rho s_2,\tau)\times \]
\[ \times g_{1,\frac52}(\rho,t-\tau)g(\psi_1;y,\rho,t), \]
\[ a_{12}(\rho)=b_1(x,t)[\varphi_0(\rho s_1,\delta+\rho s_2,\tau)- \]

\[ -\varphi_0(\rho s_1,\rho s_2,\tau)g_{1,\frac52}(\rho,t-\tau)g(\psi_1;y,\rho,t), \]

\[ a_{13}(\rho)=b_1(x,t)\varphi_0(\rho s_1,\rho s_2,\tau)\times \]

\[ \times g_{1,\frac52}(\rho,t-\tau)[g(\psi_1;y,\rho,t)-g(\psi_1;x,\rho,t)]. \]

From (3.23), (3.6), (1.17), (1.30), in view of (1.23), we obtain

\[ |a_i(\rho)|\leq (C)[\varphi]_0\, g_{1,\frac{5-\alpha}{2}}(\rho,2(t-\tau))\,\delta,\qquad i=1,3, \]

\[ |a_{12}(\rho)|\leq (C)|\varphi|_\alpha\,g_{1,2-\frac{\alpha}{2}}(\rho,t-\tau)\delta^\alpha, \]

whence, with the substitutions \(t-\tau=\eta\), \(\frac18\rho^2\eta^{-1}=s\), \(\frac12\delta^2\eta^{-1}=z\) and (1.23) (using (3.27) for \(\alpha=1\)), we obtain (see (2.2))

\[ |\bar V_{13}^{(3)}(\delta)|\leq (C)|\varphi|_\alpha \int_0^t \left( \delta\eta^{-\frac{3-\alpha}{2}} +\delta^\alpha\eta^{-1+\frac{\alpha}{2}} \right) \exp\left\{-\frac{\delta^2\eta^{-1}}{2}\right\}\,d\eta \leq \]

\[ \leq (C)|\varphi|_\alpha\delta^{\alpha_0}. \tag{3.32} \]

Put

\[ a_2(y,\rho,t)-a_2(x,\rho,t)=\sum_{i=1}^{5}a_{2i}(\rho), \]

where (see (3.5))

\[ a_{21}(\rho)=[\varphi_0(0,\delta,\tau)-\varphi_0(0,0,\tau)]\times \]

\[ \times g(\psi_1;y,\rho,t)b_2(y,\rho)g_{1,\frac52}(\rho,t-\tau), \]

\[ a_{22}(\rho)=\varphi_0(0,0,\tau)g^{(0)}(\psi_1;0,t)[b_2(y,\rho)- \]

\[ -b_2(x,\rho)]g_{1,\frac52}(\rho,t-\tau), \]

\[ a_{23}(\rho)=[(\varphi_0(\rho s_1,\delta+\rho s_2,\tau)-\varphi_0(\rho s_1,\rho s_2,\tau))- \]

\[ -(\varphi_0(0,\delta,\tau)-\varphi_0(0,0,\tau))]g(\psi_1;y,\rho,t)b_2(y,\rho)g_{1,\frac52}(\rho,t-\tau), \]

\[ a_{24}(\rho)=\varphi_0(\rho s_1,\rho s_2,\tau)[g(\psi_1;y,\rho,t)- \]

\[ -g(\psi_1;x,\rho,t)]b_2(y,\rho)g_{1,\frac52}(\rho,t-\tau), \]

\[ a_{25}(\rho)=[(\varphi_0(\rho s_1,\rho s_2,\tau)-\varphi_0(0,0,\tau))g(\psi_1;x,\rho,t)+ \]

\[ +\varphi_0(0,0,\tau)(g(\psi_1;x,\rho,t)- \]

\[ -g^{(0)}(\psi_1;0,t))][b_2(y,\rho)-b_2(x,\rho)]g_{1,\frac52}(\rho,t-\tau). \]

From the estimates (1.17), (3.6), (1.30), (1.27) and (see (3.6), (1.13))

\[ |b_2(y,\rho)-b_2(x,\rho)|\leq (C)\rho\delta^{\alpha} \tag{3.33} \]

we obtain, by virtue of (1.23),

\[ |a_{2i}(\rho)| \leq (C)|\varphi|_{\alpha}\, g_{2+\alpha,\;5/2}(\rho,2(t-\tau))\,\delta^\alpha, \qquad i=1,3,5, \]

\[ |a_{24}(\rho)| \leq (C)[\varphi]_0\, g_{2+\alpha,\;3-\alpha/2}(\rho,2(t-\tau))\,\delta . \tag{3.34} \]

We note that

\[ \int_0^{2\pi} (b_2(y,\rho)-b_2(x,\rho))\,d\vartheta = \]

\[ = \delta \int_0^{2\pi} d\vartheta \int_0^1 \left( \frac{\partial \psi_1(\rho\cos\vartheta,z\delta+\rho\sin\vartheta,\tau)} {\partial x_2} - \frac{\partial \psi_1(0,z\delta,\tau)} {\partial x_2} \right)\,dz . \]

Therefore, estimating with the aid of (1.13) and making the substitutions \(t-\tau=\eta\), \(\frac14\rho^2\eta^{-1}=s\), \(\frac14\delta^2\eta^{-1}=u\), we obtain

\[ \left| \int_{t_1}^t d\tau \int_0^{2\pi} d\vartheta \int_{2\delta}^{+\infty} a_{22}(\rho)\,d\rho \right| \leq (C)[\varphi]_0\delta \int_0^t \eta^{-\frac{3-\alpha}{2}} \times \]

\[ \times \exp\{-\delta^2\eta^{-1}\}\,d\eta \leq (C)[\varphi]_0\delta^{\alpha_0}. \tag{3.35} \]

From (3.33)—(3.35), by virtue of (1.23), we have

\[ |\bar V_{12}^{(3)}(\delta)| \leq (C)|\varphi|_\alpha \left( \delta^{\alpha_0} + \int_0^t \left[ \delta^\alpha \eta^{-1+\frac{\alpha}{2}} + \delta\eta^{-\frac{3-\alpha}{2}} \right] \exp\left\{-\frac{\delta^2\eta^{-1}}{2}\right\}\,d\eta \right) \leq \]

\[ \leq (C)|\varphi|_\alpha\delta^{\alpha_0}. \tag{3.36} \]

Finally, put

\[ a_3(y,\rho,t)-a_3(x,\rho,t)=\sum_{i=1}^3 a_{3i}(\rho), \]

where

\[ a_{31}(\rho)= [(\varphi_0(\rho s_1,\delta+\rho s_2,\tau)-\varphi_0(0,\delta,\tau))\times \]

\[ \times b_3(y,\rho,t)g(\psi_1;y,\rho,t) - (\varphi_0(\rho s_1,\rho s_2,\tau)- \varphi_0(0,0,\tau))b_3(x,\rho,t)g(\psi_1;x,\rho,t)] g_{1,\,5/2}(\rho,t-\tau), \]

\[ a_{32}(\rho)= [\varphi_0(0,\delta,\tau)b_3(y,\rho,t) (g(\psi_1;y,\rho,t)-g^{(0)}(\psi_1;y,t))- \]

\[ -\varphi_0(0,0,\tau)b_3(x,\rho,t) (g(\psi_1;x,\rho,t)- g^{(0)}(\psi_1;x,t))] g_{1,\,5/2}(\rho,t-\tau), \]

\[ a_{33}(\rho)= [\varphi_0(0,\delta,\tau)b_3(y,\rho,t)g^{(0)}(\psi_1;y,t)- \]

\[ -\varphi_0(0,0,\tau)b_3(x,\rho,t)g^{(0)}(\psi_1;x,t)] g_{1,\,5/2}(\rho,t-\tau). \]

We note the equality

\[ \int_{t_1}^t d\tau \int_0^{2\pi} d\vartheta \int_{2\delta}^{+\infty} a_{33}(\rho)\,d\rho=0. \]

From the estimates (1.17), (3.6), (1.30), and

\[ |b_3(y,\rho,t)-b_3(x,\rho,t)|\leq (C)\rho\delta^\alpha \tag{3.37} \]

using the substitutions \(t-\tau=\eta,\ \dfrac{1}{4}\rho^2\eta^{-1}=s,\ \delta^2\eta^{-1}=z\), we obtain (cf. (3.36))

\[ \left| \int_{t_1}^{t} d\tau \int_{0}^{2\pi} d\vartheta \int_{2\delta}^{+\infty} a_{31}(\rho)\, d\rho \right| \leq (C)|\varphi|_{\alpha}\delta^{\alpha_0}. \tag{3.38} \]

From the estimates (1.17), (3.6), (1.28), (3.37), (1.31), in an analogous way we obtain

\[ \left| \int_{t_1}^{t} d\tau \int_{0}^{2\pi} d\vartheta \int_{2\delta}^{+\infty} a_{32}(\rho)\, d\rho \right| \leq (C)|\varphi|_{\alpha}\delta^{\alpha_0}. \tag{3.39} \]

From (3.31), (3.32), (3.36), (3.38), (3.39), in view of (3.28), we obtain for case b) the estimate

\[ |\bar V_1(y,t)-\bar V_1(x,t)|\leq (C)|\varphi|_{\alpha}\delta^{\alpha_0}. \tag{3.40} \]

The estimates (3.14), (3.29) in case a) and (3.15), (3.40) in case b) complete the proof of Theorems 3 and 4. Theorem 2 is proved according to the scheme of Theorem 3, but with some simplifications. Omitting the details, we indicate the main differences. Restricting ourselves to the study of the function \(\bar W_1^{(0)}(x,t)\), obtained from (3.1) by replacing \(N(x,t)\) by \(N(\xi,\tau)\), we introduce two representations in the local system \(\{\xi,\tau\}\), associated with \((x,t)\in \Gamma_t\) (see also (1.11)),

\[ a^{(0)}(x,\xi,t,\tau)=r(x,\xi)\cos(N(\xi,\tau),r_{x\xi}(\tau))= \]

\[ =\bar b_0^{(0)}(\xi,\tau)\sum_{j=1}^{2}\bar b_j^{(0)}(x,\rho,t,\tau) =\bar b_0^{(0)}(\xi,\tau)\sum_{j=1}^{2}\hat b_j^{(0)}(x,\xi,t,\tau), \tag{3.41} \]

where (see (3.5))

\[ \bar b_0^{(0)}(\xi,\tau)= \left[ 1+\sum_{j=1}^{2} \left( \frac{\partial \psi(\xi_1,\xi_2,\tau)}{\partial \xi_j} \right)^2 \right]^{-\frac12}, \]

\[ \bar b_1^{(0)}(x,\rho,t,\tau)\equiv \bar b_1^{(0)}(x,t,\tau) =\psi(x_1,x_2,\tau)-\psi(x_1,x_2,t), \tag{3.42} \]

\[ \bar b_2^{(0)}(x,\rho,t,\tau)\equiv \bar b_2^{(0)}(x,\rho,t)= \]

\[ =\rho\sum_{j=1}^{2}s_j \int_{0}^{1} \left( \frac{\partial \psi(x_1+z\rho s_1,x_2+z\rho s_2,\tau)}{\partial \xi_j} - \frac{\partial \psi(x_1+\rho s_1,x_2+\rho s_2,\tau)}{\partial \xi_j} \right)\,dz; \]

\[ \hat b_1^{(0)}(x,\xi,t,\tau)=\psi(\xi_1,\xi_2,\tau)-\psi(x_1,x_2,t); \]

\[ \hat b_2^{(0)}(x,\xi,t,\tau)\equiv \hat b_2^{(0)}(x,\xi,\tau) = -\sum_{j=1}^{2}(\xi_j-x_j) \frac{\partial \psi(\xi_1,\xi_2,\tau)}{\partial \xi_j}. \]

Taking into account that

\[ d\sigma_{\xi}(\tau)=[\bar b_0^{(0)}(\xi,\tau)]^{-1}d\xi_1\,d\xi_2, \tag{3.43} \]

we introduce the functions \(\varphi_1\) and \(\psi_1\) from (3.3) and obtain, respectively, \(\bar W_1(x,t)\), \(a\), \(\bar b_j\), and \(\hat b_j\) \((j=1,2)\). In studying the smoothness of \(\bar W_1(x,t)\) with respect to \(t\), we use representation (3.41) with \(\bar b_j\) \((j=1,2)\), which behave as do

\(b_j\) from (3.4) \((j=1,2)\), and moreover the term of the form \(b_3\) from (3.4), which depended on \(t\), is absent. Therefore one easily obtains the estimate

\[ \left|\overline{W}_1(x,t+\Delta t)-\overline{W}_1(x,t)\right|\leq (C)[\varphi]_0|\Delta t|^{\alpha/2}, \]

and also (2.5).

In studying the smoothness of \(\overline{W}_1\) with respect to \(x\) we use the representation (3.41) with \(\hat b_j\) \((j=1,2)\). By virtue of (1.14) and (1.15), for \(|y-x|=\delta\) we have

\[ |a(y;\xi,t,\tau)-a(x;\xi,t,\tau)|\leq (C)\left(\sum_{j=1}^{2}|x_j-y_j| \left|\frac{\partial \psi_1(\xi_1,\xi_2,\tau)}{\partial \xi_j}\right| +\right. \]

\[ \left. +|\psi_1(y_1,y_2,t)-\psi_1(x_1,x_2,t)|\right) \leq (C)[\delta(\rho^\alpha+(t-\tau)^{\alpha/2})+\delta^{1+\alpha}], \]

whence we obtain the estimate

\[ \left|\overline{W}_1(y,t)-\overline{W}_1(x,t)\right|\leq (C)[\varphi]_0\delta^{\alpha_0}, \]

which completes the proof of Theorem 2.

§ 4. PROOF OF THEOREMS 5–7.

We begin with the proof of Theorem 5, distinguishing the cases: a) \(\varphi\) is only continuous and b) \(\varphi\in H^{0,\alpha,\alpha/2}(\Gamma)\) and (2.8) is satisfied. We shall confine ourselves to considering only case b), referring in case a) to works [1] and [2]. Let first the point \((\bar x,t)\) lie on the inner normal \(N(x,t)\) to the surface \(\Gamma_t\) at the point \((x,t)\in\Gamma_t\), and let \(r(\bar x,x)=\delta\), where \(0<\delta<d/4\). Put (see (1.37))

\[ W(\bar x,t)-\overline{W}(x,t)=\frac12\sum_{i=1}^{3}W_i, \]

where, for \(t_1=t-d_1\),

\[ W_1=\int_{t_1}^{t}d\tau\iint_{\Gamma_\tau(x,d)} \left\{\left[\cos\bigl(N(y,\tau),r_{xy}(\tau)\bigr)\times\right.\right. \]

\[ \left.\times g_{1,\frac52}(r(\bar x,y),t-\tau) -\cos\bigl(N(y,\tau),r_{xy}(\tau)\bigr)\times\right. \]

\[ \left.\left.\times g_{1,\frac52}(r(x,y),t-\tau)\right]\varphi(y,\tau)\right\}\,d\sigma_y(\tau), \]

\[ W_2=\int_{0}^{t_1}d\tau\iint_{\Gamma_\tau}\{\}\,d\sigma_y(\tau), \qquad W_3=\int_{t_1}^{t}d\tau\iint_{\Gamma_\tau\setminus\Gamma_\tau(x,d)}\{\}\,d\sigma_y(\tau). \]

Obviously,

\[ |W_i|\leq (C)[\varphi]_0\delta^\alpha,\qquad i=2,3. \tag{4.1} \]

Pass on \(S_{dd_1}(x,t)\) to the local system \(\{\xi,\tau\}\), associated with the point \((x,t)\in\Gamma_t\) (see also (1.11)), and note that, by virtue of (3.41), (3.42),

\[ r(x,\xi)\cos\bigl(N(\xi,\tau),r_{x\xi}(\tau)\bigr)=\overline{b}_0^{(0)}(\xi,\tau)b^{(0)}, \]

\[ r(\bar x,\xi)\cos\bigl(N(\xi,\tau),r_{\bar x\xi}(\tau)\bigr)=\overline{b}_0^{(0)}(\xi,\tau)(b^{(0)}-\delta), \]

where (see (3.42) and (1.10))

\[ b^{(0)}\equiv b^{(0)}(\rho,\bar\tau)=\sum_{j=1}^{2}\overline{b}_j^{(0)}(x,\rho,t,\tau). \tag{4.2} \]

Introduce the notation (see (3.5), (3.3), and (1.10)):

\[ \psi_1 \equiv \psi_1(\rho s_1,\rho s_2,\tau),\quad \varphi_1 \equiv \varphi_1(\rho s_1,\rho s_2,\tau),\quad \varphi \equiv \varphi(\rho s_1,\rho s_2,\tau), \tag{4.3} \]

\(b=b(\rho,\tau)\) is obtained from \(b^{(0)}(\rho,\tau)\) in (4.2) by replacing \(\psi\) by \(\psi_1\) from (3.3). Put

\[ W_1=\sum_{i=1}^{5} W_{1i}(\hat\delta), \]

where (see (3.43), (3.42))

\[ W_{1i}(\hat\delta)=\int_{t_1}^{t}d\tau\int_{0}^{2\pi}d\vartheta\int_{0}^{+\infty} a_i(\hat\delta)\,g_{1,\frac52}(\rho,t-\tau)\,d\rho,\quad i=1,2,3,4 \tag{4.4} \]

\[ \left(\text{for } i=5 \text{ the integration with respect to } \rho \text{ is over the limits } \frac34 d\leq \rho\leq d\right), \]

\[ a_1(\hat\delta)=-\varphi_1(0,0,t)\,g_{1,0}(\hat\delta,t-\tau), \]

\[ a_2(\hat\delta)=(\varphi_1(0,0,t)-\varphi_1)\,g_{1,0}(\hat\delta,t-\tau), \]

\[ a_3(\hat\delta)=\hat\delta\varphi_1\bigl(g_{0,0}(\hat\delta,t-\tau)-g_{0,0}(\psi_1-\hat\delta,t-\tau)\bigr), \tag{4.5} \]

\[ a_4(\hat\delta)=\varphi_1 b\bigl(g_{0,0}(\psi_1-\hat\delta,t-\tau)-g_{0,0}(\psi_1,t-\tau)\bigr), \]

\[ a_5(\hat b)=\varphi\bigl[(b^{(0)}-\hat\delta)g_{0,0}(\psi-\hat\delta,t-\tau)-b^{(0)}g_{0,0}(\psi,t-\tau)\bigr]- \]

\[ -\varphi_1\bigl[(b-\hat\delta)g_{0,0}(\psi_1-\hat\delta,t-\tau)-b\,g_{0,0}(\psi_1,t-\tau)\bigr]. \]

Obviously,

\[ |W_{15}(\hat\delta)|\leq (C)[\varphi]_0\,\hat\delta . \tag{4.6} \]

Making the substitution \(\frac14\hat\delta^2(t-\tau)^{-1}=z\), we obtain

\[ W_{11}(\hat\delta)=-8\pi^{3/2}\varphi(0,0,t)+8\pi\varphi(0,0,t)\sum_{i=1}^{2} C_i(\hat\delta), \tag{4.7} \]

where

\[ C_1(\hat\delta)=\int_{0}^{\frac{\hat\delta^2}{4t}} z^{-\frac12}e^{-z}\,dz,\quad C_2(\hat\delta)=\int_{\frac{\hat\delta^2}{4t}}^{\frac{\hat\delta^2}{4d_1}} z^{-\frac12}e^{-z}\,dz \quad \text{for } t>d_1, \]

and \(C_2(\hat\delta)\equiv 0\)

\[ \text{for } t\leq d_1. \]

Obviously,

\[ |C_2(\hat\delta)|\leq (C)\hat\delta,\quad |C_1(\hat\delta)|\leq (C)t^{-\frac{\alpha}{2}}\hat\delta^\alpha \tag{4.8} \]

(the following easily proved inequality was used:

\[ 0\leq \int_{0}^{u} z^{-\frac12}e^{-z}\,dz\leq (C)u^{\alpha/2} \quad \text{for } 0\leq u<+\infty,\ 0<\alpha\leq 1). \tag{4.9} \]

In case b), from (4.7), (4.8), by virtue of the inequality

\[ |\varphi_1(0,0,t)|=|\varphi(0,0,t)|\leq [\varphi]_{\alpha}t^{\frac{\alpha}{2}}, \tag{4.10} \]

which follows from (1.17), (2.8), we obtain

\[ |W_{11}(\hat\delta)+8\pi^{3/2}\varphi(0,0,t)|\leq (C)[\varphi]_{\alpha}\hat\delta^\alpha . \tag{4.11} \]

Next, making the substitutions \(t-\tau=\eta,\ \frac14\rho^2\eta^{-1}=s,\ \frac14\delta^2\eta^{-1}=z\), we obtain, by virtue of (1.17), (1.23) (and also (3.27) for \(\alpha=1\)),

\[ |W_{12}(\delta)|\leq (C)[\varphi]_\alpha\,\delta \int_0^t \eta^{-\frac{3-\alpha}{2}} \exp\left\{-\frac14\delta^2\eta^{-1}\right\}\,d\eta \leq (C)[\varphi]_\alpha\,\delta^\alpha . \tag{4.12} \]

To estimate \(|W_{1i}(\delta)|\) for \(i=3,4\), by the mean value theorem (putting \(\lambda(z,\delta)\equiv \delta-z\psi_1\), see (4.3)) we obtain

\[ \delta |g_{0,0}(\delta,t-\tau)-g_{0,0}(\psi_1-\delta,t-\tau)| = \frac{\delta}{2}|\psi_1|\int_0^1 g_{1,1}(|\lambda(z,\delta)|,t-\tau)\,dz \leq \]

\[ \leq (C)|\psi_1|\int_0^1 (\delta^2+z\delta|\psi_1|)g_{0,1}(\lambda(z,\delta),t-\tau)\,dz . \tag{4.13} \]

Similarly, putting \(\lambda_1(z,\delta)\equiv z\delta-\psi_1\), we have

\[ |g_{0,0}(\psi_1-\delta,t-\tau)-g_{0,0}(\psi_1,t-\tau)|\leq \]

\[ \leq \frac{\delta}{2}\int_0^1 g_{1,1}(|\lambda_1(z,\delta)|,t-\tau)\,dz \leq \]

\[ \leq (C)\int_0^1 (z\delta^2+\delta|\psi_1|)g_{0,1}(\lambda_1(z,\delta),t-\tau)\,dz . \tag{4.14} \]

Consider, in the Lyapunov ball \(s_{3d}(x^*,\tau)\) (\(|t-\tau|\leq d_1\)), the points

\[ M_0=(0,0,z^{-1}\delta,\tau), \]

\[ M_1=(x^*,\tau)=(0,0,\psi(0,0,\tau),\tau)\in s_{3d}(x^*,\tau) \]

and (see (3.5))

\[ M_2=(\xi,\tau)=(\rho s_1,\rho s_2,\psi(\rho s_1,\rho s_2,\tau),\tau)\in \Gamma_\tau(x,d). \]

By the sine theorem,

\[ \frac{M_0M_2}{\sin(M_1M_0,M_1M_2)} = \frac{M_1M_0}{\sin(M_2M_1,M_2M_0)}, \]

whence

\[ M_0M_2\geq M_0M_2\sin(M_2M_1,M_2M_0) = M_1M_0\sin(M_1M_0,M_1M_2). \]

But the straight line \(\overline{M_1M_2}\) intersects the surface \(\Gamma_\tau(x,d)\) at the two points \(M_1\) and \(M_2\); therefore, according to Lemma 2,

\[ |(M_2M_1,M_2M_0)|\geq \omega_0>\frac{\pi}{3}, \]

whence

\[ M_0M_2\geq \frac12\,M_1M_2, \]

i.e.

\[ \sqrt{\rho^2+\bigl(\psi(\rho s_1,\rho s_2,\tau)-z^{-1}\delta\bigr)^2} \geq \frac12\,|z^{-1}\delta-\psi(0,0,\tau)|, \]

and, for \(0<z\leq 1\),

\[ \rho^2+\lambda^2(z,\delta)\geq z^2\rho^2+\lambda^2(z,\delta) \geq \frac14(\delta-z\psi(0,0,\tau))^2 . \]

Let \(t_0(z)\) be such that

\[ |z\psi_1(0,0,t_0(z))|=\frac{\delta}{2}, \tag{4.15} \]

where for \(t_1 \leq t_0(z)<\tau\leq t\), \(|z\psi_1(0,0,\tau)|<\dfrac{\delta}{2}\) (if there is no such \(t_0(z)\), we set \(t_0(z)=t_1\)). Obviously, for \(t_0(z)<\tau\leq t\) and \(\rho\leq \dfrac{\delta}{4\sqrt2}<\dfrac34 d\),

\[ \lambda^2(z,\delta)\geq \frac{\delta^2}{32}. \tag{4.16} \]

Analogously, with the aid of Lemma 2 it is shown that

\[ \rho^2+\lambda_1^2(z,\delta)\geq \frac14\,(z\delta-\psi_1(0,0,\tau))^2. \tag{4.17*} \]

Let \(t^0(z)\) be such that

\[ |\psi_1(0,0,t^0(z))|=\frac{z\delta}{2} \tag{4.17} \]

and for \(t_1\leq t^0(z)<\tau\leq t\), \(|\psi_1(0,0,\tau)|<\dfrac{z\delta}{2}\) (if there is no such \(t^0(z)\), then we set \(t^0(z)=t_1\)). Obviously, for \(t^0(z)<\tau\leq t\) and \(\rho\leq \dfrac{\delta z}{4\sqrt2}\),

\[ \lambda_1^2(z,\delta)\geq \frac{\delta^2 z^2}{32}. \tag{4.18} \]

Introduce the representations:

\[ W_{1jk}(\delta)=\sum_{i=1}^{3} W_{1jki}(\delta),\qquad j=3,4;\quad k=1,2, \]

where

\[ W_{13k1}(\delta)=\int_{0}^{1} dz\int_{t_1}^{t} d\tau\int_{0}^{2\pi} d\vartheta\int_{\delta/4\sqrt2}^{+\infty} b_k\,d\rho; \]

\[ W_{13k2}(\delta)=\int_{0}^{1} dz\int_{t_1}^{t_0(z)} d\tau\int_{0}^{2\pi} d\vartheta\int_{0}^{\delta/4\sqrt2} b_k\,d\rho; \]

\[ W_{13k3}(\delta)=\int_{0}^{1} dz\int_{t_0}^{t} d\tau\int_{0}^{2\pi} d\vartheta\int_{0}^{\delta/4\sqrt2} b_k\,d\rho. \tag{4.19} \]

\(W_{14ki}(\delta)\) is obtained from \(W_{13ki}(\delta)\) by replacing in (4.19) \(b_k\) by \(b_{k+2}\), \(\lambda(z,\delta)\) by \(\lambda_1(z,\delta)\), \(t_0(z)\) by \(t^0(z)\), and \(\dfrac{\delta}{4\sqrt2}\) by \(\dfrac{\delta z}{4\sqrt2}\); here

\[ b_k=b_{0k}\,g_{1,7/2}(\rho,t-\tau)\,g_{0,0}(\lambda(z,\delta),t-\tau), \]

\[ b_{01}=\delta^2|\psi_1|,\qquad b_{02}=z\delta|\psi_1|^2, \]

\[ b_{03}=z\delta^2 b,\qquad b_{04}=\delta|\psi_1|\,b. \]

From (4.4), (4.13), (4.14) it follows that

\[ |W_{1j}(\delta)|\leq (C)[\varphi]_\nu \sum_{k=1}^{2} W_{1jk}(\delta),\qquad j=3,4. \tag{4.20} \]

From (1.13), (1.15) it follows that

\[ |b|\leq (C)\left(\rho^{1+\alpha}+(t-\tau)^{\frac{1+\alpha}{2}}\right). \tag{4.21} \]

From (1.15), (1.23), making the substitutions \(t-\tau=\eta\), \(\dfrac{1}{8}\rho^2\eta^{-1}=s\), \(\dfrac{1}{128}\delta^2\eta^{-1}=v\), we obtain

\[ |W_{1311}(\delta)| \leq (C)\delta^2 \int_0^t \eta^{-2+\frac{\alpha}{2}} \times \]

\[ \times \exp\left\{-\frac{\delta^2\eta^{-1}}{128}\right\}\,d\eta \leq (C)\delta^\alpha \int_0^{+\infty} v^{-\alpha/2} e^{-v}\,dv \leq (C)\delta^\alpha . \]

Using (1.15), (1.23), and (4.15), we obtain, by virtue of (1.12) and the inequality

\[ \frac{1}{2}(2-\alpha)(1+\alpha)-\left(1-\frac{\alpha}{2}\right) = \frac{\alpha}{2}(2-\alpha) > 0 \]

\[ |W_{1312}(\delta)| \leq (C)\delta^2 \int_0^1 dz \int_{t_1}^{t_0(z)} (t-\tau)^{-2+\frac{\alpha}{2}}\,d\tau \leq \]

\[ \leq (C)\delta^\alpha \int_0^1 |\psi_1(0,0,t_0(z))-\psi_1(0,0,t)|^{2-\alpha} z^{2-\alpha} |(t-t_0(z))^{-1+\alpha/2} \]

\[ -(t-t_1)^{-1+\alpha/2}|\,dz \leq (C)\delta^\alpha \int_0^1 z^{2-\alpha}\,|t- \]

\[ -t_0(z)|^{\frac{(2-\alpha)(1+\alpha)}{2}-\left(1-\frac{\alpha}{2}\right)} \left|\frac{t_0(z)-t_1}{t-t_1}\right|^{1-\alpha/2} dz \leq (C)\delta^\alpha . \]

From (4.16), with the substitutions \(\dfrac{1}{8}\rho^2(t-\tau)^{-1}=s\), \(\dfrac{1}{128}\delta^2(t-\tau)^{-1}=v\), we have

\[ |W_{1313}(\delta)| \leq (C)\delta^2 \int_0^1 dz \int_{t_0(z)}^t (t-\tau)^{-2+\frac{\alpha}{2}} \exp\left\{-\frac{1}{128}\delta^2(t-\tau)^{-1}\right\}\,d\tau \leq (C)\delta^\alpha . \]

If one uses the inequality

\[ 0 \leq \int_u^{+\infty} z^{-\frac12-\alpha} e^{-z}\,dz \leq (C)u^{-\alpha} \quad \text{for } 0<u<+\infty,\quad 0<\alpha<1, \tag{4.22} \]

then, in an analogous manner, we obtain

\[ |W_{1321}(\delta)| \leq (C)\delta \int_0^t \eta^{-\frac32+\alpha} \exp\left\{-\frac{1}{128}\delta^2\eta^{-1}\right\}\,d\eta \leq (C)t^{\frac{\alpha}{2}}\delta^\alpha \]

and, in view of (4.15), (4.16),

\[ |W_{1322}(\delta)| \leq (C)\delta^\alpha \int_0^1 z^{2-\alpha}|t-t_0(z)|^{\frac{\alpha(1-\alpha)}{2}} \left|\frac{t_0(z)-t_1}{t-t_1}\right|^{\frac{1-\alpha}{2}} dz \leq (C)\delta^\alpha , \]

\[ |W_{1323}(\delta)| \leq (C)\delta \int_0^1 z\,dz \int_{t_0(z)}^t (t-\tau)^{-\frac32+\alpha} \exp\left\{-\frac{1}{128}\delta^2(t-\tau)^{-1}\right\}\,d\tau \leq (C)\delta^\alpha . \]

Finally, if one uses (4.21), (1.15), (4.17), and (4.18), then in an analogous manner we obtain, in view of (4.20)

$$ |W_{14kj}(\delta)| \leq (C)\delta^\alpha,\quad k=1,2;\ j=1,2,3;\quad |W_{1i}(\delta)| \leq (C)[\varphi]_0\delta^\alpha,\quad i=3,4. \tag{4.23} $$

From (4.1), (4.6), (4.10), (4.12), and (4.23), (2.7) and (2.9) (in case b)) follow under the condition that \((\bar x,\bar t)\) tends to \((x,t)\) along \(N(x,t)\). The use of Theorem 2 makes it possible to prove relations (2.7) and (2.9) for an arbitrary approach of the point \((\bar x,\bar t)\) to \((x,t)\in\Gamma_t\). Indeed, let \((\bar x,\bar t)\) be an arbitrary point lying inside \(D_T^k\) (\(k=v\) or \(k=n\)) sufficiently close to \(\Gamma_t(x,d)\). Let the point \((\bar x_t^-,\bar t)\in\Gamma_t(x,d)\) be the nearest to \((\bar x,\bar t)\). In addition, consider the point \((\bar x^{--},\bar t)=(x_1,x_2,\psi(x_1,x_2,\bar t),\bar t)\), \((\bar x_t^-,\bar t)\in\Gamma_t(x,d)\). By the choice of \((\bar x_t^-,\bar t)\) and estimate (1.12),

$$ r(\bar x,\bar x_t^-)\leq r(\bar x,\bar x_t^-)\leq r(\bar x_t^-,x)+r(x,\bar x) \leq L_{0,t}|\bar t-t|^{\frac{1+\alpha}{2}}+r(\bar x,x), $$

$$ r(x,\bar x_t^-)\leq r(x,\bar x)+r(\bar x,\bar x_t^-) \leq L_{0,t}|\bar t-t|^{\frac{1+\alpha}{2}}+2r(\bar x,x). \tag{4.24} $$

Obviously, \((\bar x,\bar t)\) lies on the normal \(N(\bar x_t^-,\bar t)\). The use of Theorem 2 and relation (2.9), which by what has been proved is valid for \((\bar x,\bar t)\) and \((\bar x_t^-,\bar t)\in\Gamma_{\bar t}\), gives, in view of (4.24), the inequalities

$$ |W(\bar x,\bar t)-\bar W(x,t)+2\pi\varphi(x,t)| \leq |W(\bar x,\bar t)-\bar W(\bar x_t^-,\bar t)+2\pi\varphi(\bar x_t^-,\bar t)|+ $$

$$ +|\bar W(\bar x_t^-,\bar t)-\bar W(x,t)|+ 2\pi|\varphi(\bar x_t^-,\bar t)-\varphi(x,t)| \leq $$

$$ \leq (C)|\varphi|_\alpha\bigl(r^{\alpha_0}(x,\bar x)+|t-\bar t|^{\alpha/2}\bigr), $$

which proves (2.9) for an arbitrary approach of \((\bar x,\bar t)\) to \((x,t)\in\Gamma_t\).

Theorem 5 is proved.

To prove Theorem 6, note that by the method of this paragraph (see also [1], § 36) one can prove the estimate

$$ \{|W(\bar x,\bar t)|,\ |V(\bar x,\bar t)|\}\leq (C)[\varphi]_0,\quad (\bar x,\bar t)\in D_T^k, \tag{4.25} $$

from which (2.10) follows in view of (4.10). By Theorem 5, \(W(\bar x,\bar t)\) has continuous (on \(\Gamma\)) limiting values \(W_k(x,t)\) (\(k=v,n\)) when \((\bar x,\bar t)\) tends to \((x,t)\in\Gamma_t\), remaining inside \(D_T^k\), and moreover, in view of (2.9),

$$ W_k(x,t)=\bar W(x,t)\mp 2\pi\varphi(x,t),\quad (x,t)\in\Gamma,\quad W_k\in H^{0,\alpha_0,\frac{\alpha}{2}}(\Gamma), \tag{4.26} $$

whence it follows that \(W(\bar x,\bar t)\) is a solution of the heat-conduction equation (1.34) in the open domains \(D_T^k\), continuous in \(\overline{D_T^k}\) by virtue of (2.10), (4.26), and bounded at infinity (in \(x\)) in \(D_T^n\) (see (4.25)). In addition, from (2.10) it follows that \(W(\bar x,0)=0\). Therefore Theorem 6 follows from Theorem 5 and (2.10) by virtue of the maximum principle.

We now prove Theorem 7.

Let \((x,t)\in D_T^v\), and let \((\bar x,\bar t)\) lie on the normal \(N(x,t)\) to the section \(\Gamma_t\) at the point \((x,t)\in\Gamma_t\), \(r(\bar x,x)=\delta<d/4\). In the local coordinate system \(\{\xi,\tau\}\) (see (1.10), (1.11)) let \((x,t)=(0,0,0,t)\), \((\bar x,t)=(0,0,\delta,t)\). Put

$$ V(\bar x,t)-\bar V(x,t)=W(\bar x,t)-\bar W(x,t)+V(\delta), $$

where

\[ V(\delta)=\left[V(\bar{x},t)-W(\bar{x},t)\right]-\left[\bar{V}(x,t)-\bar{W}(x,t)\right]. \]

Theorem 7 will follow from Theorem 5 and (4.25), (4.10), if we establish the estimate

\[ |V(\delta)|\leqslant (C)[\varphi]_0\delta^{\alpha^0}. \tag{4.27} \]

Obviously, it is enough to estimate

\[ V_1(\delta)=\frac{1}{2}\sum_{i=1}^{3}V_{1i}(\delta), \]

where

\[ V_{1i}(\delta)=\int_{t_1}^{t}d\tau \iint_{\Gamma_\tau(x,2\delta)} a_i\,d\sigma_\xi(\tau)\quad (i=1,2),\qquad V_{13}(\delta)=\int_{t_1}^{t}d\tau \iint_{\Gamma_\tau(x,d)\setminus \Gamma_\tau(x,2\delta)} a_3\,d\xi(\tau); \]

\[ a_i=\varphi(\xi,\tau)a_{i0}g_{1,\frac{5}{2}}\bigl(r(\bar{x},\xi),t-\tau\bigr),\quad i=1,2, \]

\[ a_{10}=\cos\bigl(r_{\bar{x}\xi}(\tau),N(x,t)\bigr)-\cos\bigl(r_{\bar{x}\xi}(\tau),N(\xi,\tau)\bigr), \]

\[ a_{20}=-\left[\cos\bigl(r_{x\xi}(\tau),N(x,t)\bigr)-\cos\bigl(r_{x\xi}(\tau),N(\xi,\tau)\bigr)\right], \]

\[ a_3=a_1+a_2=\varphi(\xi,\tau)(a_{31}+a_{32}),\quad a_{31}=(a_{311}-a_{312})g_{0,5/2}\bigl(r(\bar{x},\xi),t-\tau\bigr), \]

\[ a_{32}=a_{321}\left(g_{0,5/2}\bigl(r(\bar{x},\xi),t-\tau\bigr)-g_{0,5/2}\bigl(r(x,\xi),t-\tau\bigr)\right), \]

\[ a_{311}=r(\bar{x},\xi)\cos\bigl(r_{\bar{x}\xi}(\tau),N(\xi,\tau)\bigr) -r(x,\xi)\cos\bigl(r_{x\xi}(\tau),N(x,t)\bigr), \]

\[ a_{312}=r(\bar{x},\xi)\cos\bigl(r_{\bar{x}\xi}(\tau),N(\xi,\tau)\bigr) -r(x,\xi)\cos\bigl(r_{x\xi}(\tau),N(\xi,\tau)\bigr), \]

\[ a_{321}=-r(x,\xi)a_{20}. \]

From the vector equality \(r_{\bar{x}\xi}(\tau)=r_{\bar{x}x}(\tau)+r_{x\xi}(\tau)\) it follows that

\[ a_{311}-a_{312} =r(\bar{x},x)\left[\cos\bigl(r_{\bar{x}x}(\tau),N(x,t)\bigr) -\cos\bigl(r_{\bar{x}x}(\tau),N(\xi,\tau)\bigr)\right]. \]

From the Lyapunov conditions (see (1.13)), using polar coordinates (1.11) and the mean-value theorem, in view of (1.23) we have

\[ |a_i|\leqslant (C)[\varphi]_0\bigl(\rho^\alpha+(t-\tau)^{\alpha/2}\bigr) g_{0,2}\bigl(\rho,2(t-\tau)\bigr),\quad i=1,2, \]

\[ |a_{31}|\leqslant (C)\delta\bigl(\rho^\alpha+(t-\tau)^{\alpha/2}\bigr) g_{0,5/2}(\rho,t-\tau); \]

\[ |a_{32}|\leqslant (C)\left[\delta g_{0,\frac{5-\alpha}{2}}\bigl(\rho,2(t-\tau)\bigr) +\delta^2 g_{0,3-\frac{\alpha}{2}}(\rho,t-\tau)\right], \]

whence, with the aid of substitutions of the form \(t-\tau=\eta\), \(\frac{1}{8}\rho^2\eta^{-1}=s\), \(\frac{1}{4}\delta^2\eta^{-1}=z\) (using (3.27) for \(\alpha=1\)), we obtain

\[ |V_{1i}(\delta)|\leqslant (C)[\varphi]_0 \int_{0}^{t}\eta^{-1+\frac{\alpha}{2}} \left(1-\exp\left\{-\frac{1}{2}\delta^2\eta^{-1}\right\}\right)d\eta \leqslant (C)[\varphi]_0\delta^\alpha,\quad i=1,2, \]

\[ |V_{13}(\delta)|\leqslant (C)[\varphi]_0 \int_{0}^{t}\left[\delta\eta^{-\frac{3-\alpha}{2}}+\delta^2\eta^{-2+\alpha/2}\right] \exp\left\{-\frac{1}{4}\delta^2\eta^{-1}\right\}d\eta\leqslant \]

\[ \leqslant (C)[\varphi]_0\delta^{\alpha^0}. \]

Thus the estimate (4.27), and hence Theorem 7, are proved. The validity of Remark 3 follows easily from the inequality (cf. the notation in (4.24)), for \((x,t)\in \Gamma_t\):

\[ \left| \frac{\partial U(\bar{x},\bar{t})}{\partial N(\bar{x}_{t}^{-},\bar{t})} - \frac{\partial U(\bar{x},\bar{t})}{\partial N(x,t)} \right| \leq \sum_{j=1}^{3} \left| \cos\bigl(N(\bar{x}_{t}^{-},\bar{t}),\bar{x}_{j}\bigr) - \cos\bigl(N(x,t),\bar{x}_{j}\bigr) \right| \left| \frac{\partial U(\bar{x},\bar{t})}{\partial \bar{x}_{j}} \right|, \tag{4.28} \]

in view of (see the Lyapunov conditions (1.7))

\[ \left| \cos\bigl(N(\bar{x}_{t}^{-},\bar{t}),\bar{x}_{j}\bigr) - \cos\bigl(N(x,t),\bar{x}_{j}\bigr) \right| \leq (C)\,\bigl|\bigl(N(\bar{x}_{t}^{-},\bar{t}),N(x,t)\bigr)\bigr| \leq (C)\,[r^{\alpha}(\bar{x}_{t}^{-},x)+|\bar{t}-t|^{\alpha/2}], \]

(4.24) and the estimate (cf. [6], [7])

\[ \left| \frac{\partial U(\bar{x},\bar{t})}{\partial \bar{x}_{j}} \right| \leq (C)\,[\varphi]_{0}\,|\ln r(\bar{x}_{t}^{-},\bar{x})|, \qquad j=1,2,3. \]

The validity of Remark 4 follows from (4.28), (4.24), and Theorem 4, if one notes that, by Theorem 8, (2.15) holds.

§ 5. PROOF OF THEOREM 8.

The function \(U(\bar{x},\bar{t})\) in the open domains \(D_T^k\) \((k=e,\ i)\) satisfies the heat equation (1.34), is continuous in \(\overline{D}_T^k\) by Theorem 1 (and has continuous limiting values \(U_k(x,t)\equiv \bar{U}(x,t)\) on \(\Gamma\)), and is bounded in \(\overline{D}_T^k\) (see (2.4)). Moreover, from (2.4), (2.8), in view of (4.10), it follows that \(U(\bar{x},\bar{t})\) satisfies the zero initial condition at \(\bar{t}=0\), and the estimate holds

\[ |U(\bar{x},\Delta t)| \leq (C)[\varphi]_{\alpha}|\Delta t|^{\frac{1+\alpha}{2}}, \qquad (\bar{x},\Delta t)\in D_T. \tag{5.1} \]

By the maximum principle, the estimate

\[ |U(\bar{x},t+\Delta t)-U(\bar{x},t)| \leq (C)[\varphi]_{\alpha}|\Delta t|^{\frac{1+\alpha^{\circ}}{2}}, \tag{5.2} \]

\[ (\bar{x},t+\Delta t),\;(\bar{x},t)\in \overline{D}_T^k, \qquad k=e,\ i \]

will be proved if one establishes on \(\Gamma\) the estimate

\[ |\bar{U}(x,t+\Delta t)-\bar{U}(x,t)| \leq (C)[\varphi]_{\alpha}|\Delta t|^{\frac{1+\alpha^{\circ}}{2}}, \tag{5.3} \]

\[ (x,t+\Delta t),\;(x,t)\in \Gamma. \]

In view of the Lyapunov conditions (1.12) and the considerations of § 3, it is sufficient to consider the function (see (4.3))

\[ \bar{U}_{1}(x,t)=\sum_{i=1}^{3}\bar{U}_{1i}(x,t), \]

where (see (1.10), (1.11))

\[ \overline{U}_{11}(x,t)=4\pi\int_{t_1}^{t}(t-\tau)^{-1/2}\varphi_1(0,0,\tau)\,d\tau, \]

\[ \overline{U}_{12}(x,t)=\int_{t_1}^{t}\varphi_1(0,0,\tau)\bigl(1-g^{(0)}(\psi_1;0,t)\bigr)\,d\tau \int_{0}^{2\pi}d\vartheta\int_{0}^{+\infty}g_{1,3/2}(\rho,t-\tau)\,d\rho, \tag{5.4} \]

\[ \overline{U}_{13}(x,t)=\int_{t_1}^{t}d\tau\int_{0}^{2\pi}d\vartheta\int_{0}^{+\infty} \bigl[\varphi_1 g(\psi_1;0,\rho,t) \]

\[ -\varphi_1(0,0,\tau)g^{(0)}(\psi_1;0,t)\bigr]g_{1,3/2}(\rho,t-\tau)\,d\rho . \]

The function \(\varphi_1(0,0,\tau)\) satisfies the Hölder condition in \(\tau\) with exponent \(\alpha/2\), and
\[ |\varphi_1(0,0,\tau)|\le [\varphi]_\alpha \tau^{\alpha/2} \]
(cf. (2.8)); therefore, by Lemma 3,

\[ \left|\overline{U}_{11}(x,t+\Delta t)-\overline{U}_{11}(x,t)\right| \le (C)[\varphi]_\alpha|\Delta t|^{\frac{1+\alpha}{2}} . \tag{5.5} \]

In view of (5.1), (3.8) may be regarded as fulfilled. Splitting in (5.4), for \(\overline{U}_{12}(x,t)\), the interval of integration with respect to \(\rho\) into \((0,\sqrt{\Delta t})\) and \((\sqrt{\Delta t},+\infty)\), after simple transformations one may write

\[ \overline{U}_{12}(x,t+\Delta t)-\overline{U}_{12}(x,t)=\sum_{i=1}^{5}\overline{U}_{12i}, \]

where

\[ \overline{U}_{121}=\int_{t-\Delta t}^{t+\Delta t}a^{(1)}\,d\tau,\qquad \overline{U}_{122}=-\int_{t-\Delta t}^{t}a^{(2)}\,d\tau,\qquad \overline{U}_{12i}=\int_{t_1}^{t-\Delta t}a^{(i)}\,d\tau,\quad i=3,4,5, \]

\[ a^{(i)}=4\pi\varphi_1(0,0,\tau)a_0^{(i)},\quad i=1,2;\qquad a_0^{(1)}=1-g^{(0)}(\psi_1;0,t+\Delta t), \]

\[ a_0^{(2)}=1-g^{(0)}(\psi_1;0,t), \]

\[ a^{(3)}=a^{(1)}\bigl(1-g_{0,0}(\sqrt{\Delta t},t+\Delta t-\tau)\bigr),\qquad a^{(4)}=a^{(2)}\bigl(1-g_{0,0}(\sqrt{\Delta t},t-\tau)\bigr), \]

\[ a^{(5)}=a^{(1)}g_{0,0}(\sqrt{\Delta t},t+\Delta t-\tau) -a^{(2)}g_{0,0}(\sqrt{\Delta t},t-\tau). \]

Use of (1.12) gives

\[ |a^{(2)}|\le (C)[\varphi]_0(t-\tau)^{-3/2} |\psi_1(0,0,t)-\psi_1(0,0,\tau)|^2 \le (C)[\varphi]_0(t-\tau)^{\alpha-\frac12}, \tag{5.6} \]

\[ |a^{(1)}|\le (C)[\varphi]_0(t+\Delta t-\tau)^{\alpha-\frac12}, \]

whence

\[ |\overline{U}_{12i}|\le (C)[\varphi]_0|\Delta t|^{\frac12+\alpha},\qquad i=1,2. \tag{5.7} \]

Use of (5.6), (3.8), and the substitution \(z=\dfrac14\Delta t(t+\Delta t-\tau)^{-1}\) gives

\[ |\overline{U}_{123}| \leq (C)[\varphi]_0 \int_0^{t-\Delta t} (t+\Delta t-\tau)^{-\frac12+\alpha} \bigl(1-g_{0,0}(\sqrt{\Delta t},\,t+\Delta t-\tau)\bigr)\,d\tau \leq \]

\[ \leq (C)[\varphi]_0(\Delta t)^{\frac12+\alpha} \int_{\frac{\Delta t}{8t}}^{\frac18} z^{-\frac32-\alpha}(1-e^{-z})\,dz \leq (C)[\varphi]_0|\Delta t|^{\frac{1+\alpha}{2}} . \tag{5.8} \]

(the inequality was used

\[ 0 \leq \int_u^1 z^{-\frac32-\alpha}(1-e^{-z})\,dz \leq (C)u^{-\frac{\alpha}{2}} \quad \text{for } \quad 0<u\leq 1,\quad 0<\alpha\leq 1). \]

Similarly,

\[ |\overline{U}_{124}| \leq (C)[\varphi]_0|\Delta t|^{\frac{1+\alpha}{2}} . \tag{5.9} \]

By the mean-value theorem, in view of (5.6), (1.23), (3.8), and the inequality

\[ (t+\Delta t-\tau)^\alpha \leq (t+z\Delta t-\tau)^\alpha+(1-z)^\alpha(\Delta t)^\alpha, \quad 0\leq z\leq 1, \]

we have

\[ |a^{(5)}| \leq (C)[\varphi]_0 \left\{ \int_0^1 \left[ \Delta t(t+z\Delta t-\tau)^{-\frac32+\alpha} +\right.\right. \]

\[ \left. +(\Delta t)^{1+\alpha}(1-z)^\alpha (t+z\Delta t-\tau)^{-3/2} \right] g_{0,0}(\sqrt{\Delta t},\,t+z\Delta t-\tau)\,dz + \]

\[ \left. +\left[ \Delta t(t-\tau)^{-1+\alpha} +(\Delta t)^{\frac{1+\alpha}{2}}(t-\tau)^{\frac{\alpha-1}{2}} \right] g_{0,\frac12}(\sqrt{\Delta t},\,t-\tau) \right\}, \]

whence, with the help of substitutions of the form

\[ \frac14\,\Delta t\,(t+z\Delta t-\tau)^{-1}=u, \]

we obtain, in view of (3.8),

\[ |\overline{U}_{125}| \leq (C)[\varphi]_0|\Delta t|^{\frac{1+\alpha}{2}} \left[ (\Delta t)^{\frac{\alpha}{2}-\frac14} \int_{\frac{\Delta t}{8t}}^1 \bigl(u^{-\frac12-\alpha}+u^{-\frac12}\bigr)e^{-u}\,du + \right. \]

\[ \left. +(\Delta t)^{\frac{\alpha}{2}} \right] \leq (C)[\varphi]_0|\Delta t|^{\frac{1+\alpha}{2}}, \tag{5.10} \]

since for \(0<u\leq 1,\ 0<\alpha\leq 1\)

\[ 0 \leq \int_u^1 u^{-\frac12-\alpha}e^{-u}\,du \leq (C)u^{-\alpha/2}. \]

From (5.7)—(5.11) we have, under (3.8),

\[ |\overline{U}_{12}(x,t+\Delta t)-\overline{U}_{12}(x,t)| \leq (C)[\varphi]_0|\Delta t|^{\frac{1+\alpha}{2}} . \tag{5.11} \]

Put

\[ \overline{U}_{13}(x,t+\Delta t)-\overline{U}_{13}(x,t) =\sum_{i=1}^4 \overline{U}_{13i}. \]

where

\[ \overline U_{131}= \int_{t-\Delta t}^{t+\Delta t}d\tau\int_0^{2\pi}d\vartheta\int_0^{+\infty}b^{(1)}\,d\rho,\qquad \overline U_{132}= -\int_{t-\Delta t}^{t}d\tau\int_0^{2\pi}d\vartheta\int_0^{+\infty}b^{(2)}\,d\rho, \tag{5.12} \]

\[ \overline U_{13j}= \int_{t_1}^{t-\Delta t}d\tau\int_0^{2\pi}d\vartheta\int_0^{+\infty}b^{(j)}\,d\rho,\qquad j=3,4, \]

where (see (4.3))

\[ b^{(2)}\equiv b_0(t)\equiv [\varphi_1 g(\psi_1;0,\rho,t)-\varphi_1(0,0,\tau)g^{(0)}(\psi_1;0,t)]g_{1,3/2}(\rho,t-\tau); \]

\[ b^{(1)}\equiv b_0(t+\Delta t), \]

\[ b^{(3)}=(g_{1,3/2}(\rho,t+\Delta t-\tau)-g_{1,3/2}(\rho,t-\tau))\times \]

\[ \times[(\varphi_1-\varphi_1(0,0,\tau))g(\psi_1;0,\rho,t+\Delta t)+\varphi_1(0,0,\tau)]\times \]

\[ \times[g(\psi_1;0,t+\Delta t)-g^{(0)}(\psi_1;0,t+\Delta t)]; \]

\[ b^{(4)}=g_{1,3/2}(\rho,t-\tau)\{\varphi_1[g(\psi_1;0,\rho,t+\Delta t)-g(\psi_1;0,\rho,t)]- \]

\[ -\varphi_1(0,0,\tau)[g^{(0)}(\psi_1;0,t+\Delta t)-g^{(0)}(\psi_1;0,t)]\}. \]

From (1.17), (1.27), (3.8), with the aid of (1.23), we easily obtain

\[ |\overline U_{13i}|\leq (C)|\varphi|_\alpha|\Delta t|^{\frac{1+\alpha}{2}},\qquad i=1,2. \tag{5.13} \]

From the estimates (1.24), (1.17), (3.8), (1.27), (1.25), (1.26) we have

\[ |b^{(3)}|\leq (C)|\varphi|_\alpha (t-\tau)^{-5/2}g_{1,-\alpha/2}(\rho,2(t+\Delta t-\tau))\Delta t, \]

\[ |b^{(4)}|\leq (C)[\varphi]_0 g_{1,3/2}(\rho,t-\tau) [\Delta t(\rho^{1+\alpha}+(t-\tau)^{\frac{1+\alpha}{2}})^2(t-\tau)^{-2}+ \]

\[ +(\Delta t)^{\frac{1+\alpha}{2}}(t+\Delta t-\tau)^{-1} (\rho^{1+\alpha}+(t+\Delta t-\tau)^{\frac{1+\alpha}{2}})+ \]

\[ +\Delta t(t-\tau)^{-1+\alpha}+(\Delta t)^{\frac{1+\alpha}{2}}(t-\tau)^{\frac{\alpha-1}{2}}], \]

whence, with the aid of (1.24) and (3.8), we obtain

\[ |\overline U_{133}|\leq (C)|\varphi|_\alpha \int_0^{t-\Delta t}\left[\Delta t(t-\tau)^{\frac{\alpha-3}{2}} +(\Delta t)^{\frac{1+\alpha}{2}}(t-\tau)^{-1+\alpha/2}\right]d\tau\leq \]

\[ \leq (C)|\varphi|_\alpha|\Delta t|^{\frac{1+\alpha}{2}}. \tag{5.14} \]

From (5.13), (5.14) we have

\[ |\overline U_{13}(x,t+\Delta t)-\overline U_{13}(x,t)| \leq (C)|\varphi|_\alpha \Delta t^{\frac{1+\alpha^\circ}{2}}. \tag{5.15} \]

From (5.5), (5.11), (5.15) follows (5.3), and hence also (5.2). The functions

\[ \frac{\partial U(x,t)}{\partial x_j}\qquad (j=1,2,3) \]

in the open domains \(D_T^k\) (\(k=b,h\)) satisfy

satisfy equation (1.34) and (as is easily verified by the methods of [1], [2]) are bounded at infinity (in \(x\)). To prove the assertion that

\[ \frac{\partial U}{\partial x_j}\in H^{0,\alpha',\alpha'/2}(\overline{D_T^k}),\quad k=\mathrm{в},\mathrm{н};\quad j=1,2,3, \tag{5.16} \]

it is therefore sufficient, by virtue of the maximum principle, to show that, under the conditions of Theorem 8, the following assertions hold: 1) when the point \((\bar x,\bar t)\), remaining inside \(D_T^k\) \((k=\mathrm{в},\mathrm{н})\), approaches a point \((x,t)\in\Gamma\), the limiting values

\[ \left(\frac{\partial U(x,t)}{\partial x_j}\right)_k\quad (k=\mathrm{в},\mathrm{н}); \]

exist;

\[ \text{2) }\quad \left(\frac{\partial U}{\partial x_j}\right)_k\in H^{0,\alpha',\alpha'/2}(\Gamma); \tag{5.17} \]

\[ \text{3) }\quad \left| \frac{\partial U(\bar x,\bar t)}{\partial \bar x_j} - \left(\frac{\partial U(x,t)}{\partial x_j}\right)_k \right| \le (C)|\varphi|_{\alpha}\, r^{\alpha}(\bar x,x) \tag{5.18} \]

as \((\bar x,\bar t)\) approaches \((x,t)\in\Gamma_t\) along the normal \(N(x,t)\);

\[ \text{4) }\quad \left|\frac{\partial U(\bar x,\bar t)}{\partial \bar x_j}\right| \le (C)|\varphi|_{\alpha}\, t^{\alpha/2},\quad (\bar x,\bar t)\in D_T^k . \tag{5.19} \]

Let \((x,t)\in\Gamma_t\) and \((\bar x,\bar t)\in D_T^{\mathrm{в}}\) (for definiteness), with \((\bar x,\bar t)\) lying on the normal \(N(x,t)\) to the section \(\Gamma_t\) at the point \((x,t)\) and \(r(\bar x,x)=\delta<d/2\). We pass to the local coordinates \(\{\xi,\tau\}\) associated with \((x,t)\in\Gamma_t\) (see § 1, (1.10), (1.11)). Note that, by virtue of the choice of the axes \(O\xi_j\) \((j=1,2,3)\) (which we shall also denote by \(Ox_j\)), \(\partial/\partial x_3=\partial/\partial N(x,t)\), and, by Theorem 7,

\[ \lim_{(\bar x,t)\to(x,t)} \frac{\partial U(\bar x,t)}{\partial x_3} = \overline V(x,t)-2\pi^{3/2}\varphi(x_1,x_2,t) = \left(\frac{\partial U(x,t)}{\partial x_3}\right)_{\mathrm{в}}, \tag{5.20} \]

\[ \left| \frac{\partial U(\bar x,t)}{\partial x_3} - \left(\frac{\partial U(x,t)}{\partial x_3}\right)_{\mathrm{в}} \right| \le (C)|\varphi|_{\alpha}\, r^{\alpha}(\bar x,x), \tag{5.21} \]

\[ \left\{ \left|\frac{\partial U(\bar x,t)}{\partial x_3}\right|, \left|\left(\frac{\partial U(x,t)}{\partial x_3}\right)_{\mathrm{в}}\right| \right\} \le (C)|\varphi|_{\alpha}\, t^{\alpha/2}, \]

\[ (\bar x,t)\in D_T^{\mathrm{в}},\quad (x,t)\in\Gamma, \tag{5.22} \]

and, by virtue of Theorem 4,

\[ \left(\frac{\partial U(x,t)}{\partial x_3}\right)_{\mathrm{в}} \in H^{0,\alpha^0,\alpha/2}(\Gamma). \tag{5.23} \]

We now consider differentiation \(\partial/\partial x_k\) \((k=1,2)\) in the direction \(Ox_k\) \((k=1,2)\) tangent to \(\Gamma_t\) at the point \((x,t)\in\Gamma_t\), and set (see (3.3), where \(r_1(\bar x,\xi)\) is obtained from \(r(\bar x,\xi)\) (1.19) by replacing \(\psi\) by \(\psi_1\)):

\[ \frac{\partial U(\bar x,t)}{\partial x_k}=\sum_{i=1}^4 U_i^{(k)}(\bar x,t),\qquad k=1,2, \]

where

\[ U_1^{(k)}(\bar x,t)=\int_{t_1}^{t} d\tau \int_{-\infty}^{+\infty} d\xi_1 \int_{-\infty}^{+\infty} \varphi_1(\xi_1,\xi_2,\tau)\, \frac{\partial}{\partial x_k} g_{0,\,3/2}(r_1(\bar x,\xi),t-\tau)\,d\xi_2, \tag{5.24} \]

\[ U_2^{(k)}(\bar x,t)=\frac12\int_{t_1}^{t} d\tau \iint_{\Gamma_\tau(x,d)} \{g_{1,\,5/2}(r(\bar x,\xi),t-\tau)\, \cos(r_{\bar x\xi}(\tau),x_k)\varphi(\xi,\tau)\}_{\bar x}\, d\sigma_\xi(\tau)-U_1^{(k)}(\bar x,t), \]

\[ U_3^{(k)}(\bar x,t)=\int_{t_1}^{t} d\tau \iint_{\Gamma_\tau\setminus \Gamma_\tau(x,d)} \{\ \}_{\bar x}\,d\sigma_\xi(\tau), \]

\[ U_4^{(k)}(\bar x,t)=\int_{0}^{t_1} d\tau \iint_{\Gamma_\tau} \{\ \}_{\bar x}\,d\sigma_\xi(\tau). \]

It can be shown that, by virtue of (2.8), (4.10),

\[ |U_j^{(k)}(\bar x,t)|\le (C)|\varphi|_\alpha\,t^{\alpha/2},\qquad (\bar x,t)\in D_T^{\mathrm{v}},\quad j=2,3,4, \tag{5.25} \]

\[ |U_j^{(k)}(\bar x,t)-U_j^{(k)}(x,t)| \le (C)[\varphi]_0\,r^\alpha(\bar x,x),\qquad j=2,3,4. \]

Let us note that the integral in (5.24) exists for \(\delta>0\). For this we use Lemma 2 and the relations (4.17\({}^{**}\)), (4.17), (4.18), where \(z=1\); \(t^0(1)=t^0(\delta)\). Then, by the method used to obtain the estimates (4.19), (4.23), it is easy to obtain the absolute convergence of the integral in (5.24) for \(\delta>0\). Let us now observe that, in view of \(\bar x_i=x_i=0\) \((i=1,2)\), \(\bar x_3=\delta\),

\[ \varphi_1(\xi_1,\xi_2,\tau)\frac{\partial}{\partial x_k} g_{0,\,3/2}(r_1(\bar x,\xi),t-\tau)= \]

\[ =-\varphi_1(x_1,x_2,t)\frac{\partial}{\partial \xi_k} g_{0,\,3/2}(r_1(\bar x,\xi),t-\tau) -\frac12\sum_{j=1}^2 a_{kj}, \tag{5.26} \]

where

\[ a_{k1}=[\varphi_1(x_1,x_2,t)-\varphi_1(\xi_1,\xi_2,\tau)] (\xi_k-x_k)g_{0,\,5/2}(r_1(\bar x,\xi),t-\tau), \]

\[ a_{k2}=\varphi_1(x_1,x_2,t)(\psi_1(\xi_1,\xi_2,\tau)-\delta) \frac{\partial \psi_1(\xi_1,\xi_2,\tau)}{\partial \xi_k} g_{0,\,5/2}(r_1(\bar x,\xi),t-\tau). \]

Therefore, from (5.24), (5.26) we easily obtain

\[ U_1^{(k)}(\bar x,t)=-\frac12\sum_{j=1}^2 U_{1j}^{(k)}(\bar x,t), \tag{5.27} \]

where \(U_{1j}^{(k)}(\bar x,t)\) is obtained from \(U_1^{(k)}(\bar x,t)\) by replacing the integrand in (5.24) by \(a_{kj}\) from (5.26). By virtue of the estimates (1.17), (1.14), passing to polar coordinates (1.11), we easily obtain the estimates

\[ |U_{1j}^{(k)}(\bar x,t)|\le (C)|\varphi|_\alpha\,t^{\alpha/2},\qquad (\bar x,t)\in D_T^{\mathrm{v}},\quad j=1,2. \tag{5.28} \]

By the mean-value theorem we have (4.14) (see (4.3)), whence, in view of (1.17), by means of the device used in deriving estimate (4.23), we obtain

\[ \left|U_{11}^{(k)}(\bar{x},t)-\overline{U}_{11}^{(k)}(x,t)\right| \le (C)|\varphi|_{\alpha}\delta^{\alpha}. \tag{5.29} \]

Put

\[ U_{12}^{(k)}(\bar{x},t)-\overline{U}_{12}^{(k)}(x,t) = \sum_{j=1}^{3} U_{12j}^{(k)}(\delta), \]

where (cf. (4.4) for \(i=2,3,4\)) \(U_{12j}^{(k)}(\delta)\) is obtained from \(U_1^{(k)}(\bar{x},t)\) by replacing the integrand in (5.24) by \(a_j^{(k)}\) \((j=1,2,3)\),

\[ a_1^{(k)} = \delta\, g_{0,\,5/2}(\delta,t-\tau)\, \varphi_1(x_1,x_2,t)\, \frac{\partial\psi_1(\xi_1,\xi_2,\tau)}{\partial \xi_k}\, g_{0,\,0}(|x-\xi|,t-\tau), \]

\[ a_2^{(k)} = \delta\varphi_1(x_1,x_2,t)\, \frac{\partial\psi_1(\xi_1,\xi_2,\tau)}{\partial \xi_k} \,[g_{0,\,0}(\psi_1-\delta,t-\tau)- g_{0,\,0}(\delta,t-\tau)]\, g_{0,\,5/2}(|x-\xi|,t-\tau), \]

\[ a_3^{(k)} = \varphi_1(x_1,x_2,t)\, \frac{\partial\psi_1(\xi_1,\xi_2,\tau)}{\partial \xi_k}\, \psi_1(\xi_1,\xi_2,\tau)\times \]

\[ {}\times [g_{0,\,0}(\psi_1-\delta,t-\tau)- g_{0,\,0}(\psi_1,t-\tau)]\, g_{0,\,5/2}(|x-\xi|,t-\tau). \]

Passing to polar coordinates (1.11), taking (1.14) into account, and using the substitutions
\(\frac18\rho^2(t-\tau)^{-1}=s\), \(\frac18\delta^2(t-\tau)^{-1}=u\), we obtain, in view of (1.23) and (3.27) (for \(\alpha=1\)),

\[ |U_{121}^{(k)}(\delta)| \le (C)|\varphi|_0\delta \int_{0}^{t} (t-\tau)^{-\frac{3-\alpha}{2}} \times \]

\[ {}\times \exp\left\{-\frac{\delta^2}{8}(t-\tau)^{-1}\right\}\,d\tau \le (C)|\varphi|_0\delta^{\alpha^0}. \tag{5.30} \]

The estimate of \(|U_{12i}^{(k)}(\delta)|\) for \(i=2,3\) is carried out in the same way as the estimate of \(|W_{1i+1}(\delta)|\) (see the derivation of (4.23)), if one uses (1.14). Therefore

\[ |U_{12i}^{(k)}(\delta)|\le (C)|\varphi|_0\delta^{\alpha^0},\quad i=2,3. \tag{5.31} \]

From (5.25), (5.27), (5.28)—(5.30)

\[ \left|\frac{\partial U(\bar{x},t)}{\partial x_k}\right| \le (C)|\varphi|_{\alpha}\,t^{\alpha/2}, \quad (\bar{x},t)\in D_T^{\mathrm{B}},\quad k=1,2, \tag{5.32} \]

\[ \left| \frac{\partial U(\bar{x},t)}{\partial x_k} - \left(\frac{\partial U(x,t)}{\partial x_k}\right)_{\mathrm{B}} \right| \le (C)|\varphi|_{\alpha}\delta^{\alpha^0}, \quad k=1,2, \tag{5.33} \]

where

\[ \left(\frac{\partial U(x,t)}{\partial x_k}\right)_{\mathrm{B}} = \sum_{i=2}^{4}\overline{U}_{i}^{(k)}(x,t) + \sum_{j=1}^{2}\overline{U}_{1j}^{(k)}(x,t), \quad k=1,2, \]

and \(Ox_k\) is a tangential direction to \(\Gamma_t\) at the point \((x,t)\). From (5.21), (5.33) follows (5.18), while from (5.22), (5.32) follows (5.19). Put in the inte-

integral representations for \(U_1^{(k)}(\bar x,t)\) (from (5.27)) and \(U_{1i}^{(k)}(\bar x,t)\) \((i=1,2)\), \(\bar x_3=0=\psi_1(x_1,x_2,t)\), and denote the resulting expressions by \(\overline{U}_{1(x,t)}^{(k)}\), \(\overline{U}_{1i}^{(k)}(x,t)\), \(i=1,2\), respectively. From (5.28), which is also valid for \(\overline{U}_{1i}^{(k)}(x,t)\), we have, for \(t\ge |\Delta t|\),

\[ \left|\overline{U}_{1i}^{(k)}(x,t+\Delta t)-\overline{U}_{1i}^{(k)}(x,t)\right| \le (C)|\varphi|_\alpha |\Delta t|^{\alpha/2}. \tag{5.34} \]

In what follows we shall assume that (3.8) is fulfilled. Put

\[ \overline{U}_{1i}^{(k)}(x,t+\Delta t)-\overline{U}_{1i}^{(k)}(x,t) = \sum_{j=1}^{3}\overline{U}_{1ij}^{(k)}(\Delta t), \qquad i=1,2, \]

where \(\overline{U}_{1ij}^{(k)}(\Delta t)\) are obtained from \(\overline{U}_{13j}\) by replacing, in (5.12), \(b^{(j)}\) by \(b_{ij}^{(k)}\) \((j=1,2,3)\), respectively, and moreover (see (4.3), (3.5))

\[ b_{12}^{(k)} \equiv b_{01}^{(k)}(t)\equiv (\varphi_1(0,0,t)-\varphi_1)\times \]

\[ \times g(\psi_1;0,\rho,t)g_{2,\,5/2}(\rho,t-\tau)s_k; \qquad b_{11}^{(k)} \equiv b_{01}^{(k)}(t+\Delta t), \]

\[ b_{22}^{(k)} \equiv b_{02}^{(k)}(t)\equiv \varphi_1(0,0,t)(\psi_1-\psi_1(0,0,t))\times \]

\[ \times \frac{\partial\psi_1(\rho s_1,\rho s_2,\tau)}{\partial\xi_k}\, g(\psi_1;0,\rho,t)g_{1,\,5/2}(\rho,t-\tau), \]

\[ b_{21}^{(k)}\equiv b_{02}^{(k)}(t+\Delta t), \qquad b_{i3}^{(k)}\equiv b_{i2}^{(k)}-b_{i1}^{(k)}. \]

From the estimates (1.17), (1.14), (1.15), (3.8) we obtain, by virtue of (1.23),

\[ |b_{01}^{(k)}(t+l_i\Delta t)| \le (C)|\varphi|_\alpha g_{2,\,\frac{5-\alpha}{2}}(\rho,2(t+l_i\Delta t-\tau)), \]

\[ |b_{02}^{(k)}(t+l_i\Delta t)| \le (C)|\varphi|_0 g_{1,\,2-\alpha}(\rho,2(t+l_i\Delta t-\tau)), \qquad l_1=0,\quad l_2=1, \]

whence

\[ \left|\overline{U}_{1ij}^{(k)}(\Delta t)\right| \le (C)|\varphi|_\alpha(\Delta t)^{\alpha/2} \quad \text{for } i=1,2;\quad j=1,2. \]

From (1.17), (1.25), (1.24), (3.8) we have

\[ |b_{13}^{(k)}| \le (C)|\varphi|_\alpha\{(\Delta t)^{\alpha/2} g_{2,\,5/2}(\rho,t+\Delta t-\tau)+ \]

\[ +\Delta t(t-\tau)^{-7/2}g_{2,\,-\alpha/2}(\rho,2(t+\Delta t-\tau))+ \]

\[ +\left[\Delta t\left(\rho^{1+\alpha}+(t-\tau)^{\frac{1+\alpha}{2}}\right)^2(t-\tau)^{-2} +(\Delta t)^{\frac{1+\alpha}{2}}\left(\rho^{1+\alpha}(t-\tau)^{-1}+\right. \]

\[ \left. +(t-\tau)^{\frac{\alpha-1}{2}}\right)\right] g_{2,\,\frac{5-\alpha}{2}}(\rho,2(t-\tau))\}. \]

From (1.12), (1.14), (1.15), (1.25), (1.24), (3.8), by virtue of (1.23), we have

\[ |b_{23}^{(k)}| \le (C)|\varphi|_\alpha\{(\Delta t)^{\alpha/2} g_{1,\,2-\alpha}(\rho,2(t+\Delta t-\tau)) +(\Delta t)^{\frac{1+\alpha}{2}}g_{1,\,\frac{5-\alpha}{2}}\times \]

\[ \times(\rho,2(t+\Delta t-\tau)) +\Delta t\left((t-\tau)^{-7/2}g_{3+2\alpha,\,0}(\rho,2(t+\Delta t-\tau))+\right. \]

\[ \left.+g_{1,\,3-2\alpha}(\rho,2(t-\tau))\right)\}, \]

whence, by virtue of (1.23) and (3.8),

\[ \left|\overline{U}^{(k)}_{113}(\Delta t)\right| \leq (C)|\varphi|_{\alpha}(\Delta t)^{\alpha'/2},\qquad \left|\overline{U}^{(k)}_{123}(\Delta t)\right| \leq (C)|\varphi|_{\alpha}|\Delta t|^{\alpha/2}. \]

Thus, finally we have

\[ \left| \left(\frac{\partial U(x,t+\Delta t)}{\partial x_k}\right)_{\mathrm{v}} - \left(\frac{\partial U(x,t)}{\partial x_k}\right)_{\mathrm{v}} \right| \leq (C)|\varphi|_{\alpha}|\Delta t|^{\alpha'/2}, \quad k=1,2. \tag{5.35} \]

We now consider two points in the local system \(\{\xi,\tau\}\) (see (1.10), (1.11))
\[ (x,t)=(x_1,x_2,x_3,t)=(0,0,0,t) \]
and
\[ (x^{\delta},t)=(x_1^{\delta},x_2^{\delta},x_3^{\delta},t) =(0,\delta,\psi_1(0,\delta,t),t), \]
lying on \(\Gamma_t(x,d)\), and set

\[ \overline{U}^{(k)}_{11}(x^{\delta},t)-\overline{U}^{(k)}_{11}(x,t) = \sum_{i=1}^{3}\overline{U}^{(k)}_{11i}(\delta), \]

where \(\overline{U}^{(k)}_{11i}(\delta)\) is obtained from \(V^{(i)}_{1i}\) \((j=1,2,3)\) by replacing, in (3.16), \(a_i(y,\xi,t)\), \(a_i(x,\xi,t)\), \(a_i(y,\xi,t)-a_i(x,\xi,t)\) by \(a^{(k)}_3(x^{\delta})\), \(a^{(k)}_3(x)\), \(a^{(k)}_3(x^{\delta})-a^{(k)}_3(x)\), respectively, with

\[ a^{(k)}_3(x)\equiv \bigl(\varphi_1(x_1,x_2,t)-\varphi_1(\xi_1,\xi_2,\tau)\bigr) g(\psi_1;x,t)\, g_{0,\,5/2}(|x-\xi|,t-\tau)(\xi_k-x_k), \tag{5.36} \]

and \(a^{(k)}_3(x^{\delta})\) is obtained from (5.36) by replacing \(x\) by \(x^{\delta}\). From the estimates (1.17) we have (see (3.18), (3.19))

\[ \left|a^{(k)}_3(x^{\delta})\right| \leq (C)|\varphi|_{\alpha} \left( \rho_1^{1+\alpha}g_{1,\,5/2}(\rho_1,t-\tau) + \rho_1 g_{1,\,(5-\alpha)/2}(\rho_1,t-\tau) \right), \]

whence passage to polar coordinates (3.19) gives, by virtue of (1.23) (cf. the derivation of (3.17), (3.20)),

\[ \left|\overline{U}^{(k)}_{11i}(\delta)\right| \leq (C)|\varphi|_{\alpha}\delta^{\alpha}, \qquad i=1,2. \]

For \(\rho\geq 2\delta\) we have (3.21); therefore, by virtue of (1.17), from the mean-value theorem, (1.23) and (3.21), with the aid of (3.27), we obtain

\[ \left|\overline{U}^{(k)}_{113}(\delta)\right| \leq (C)|\varphi|_{\alpha} \int_{0}^{t} \left[ \delta^{\alpha}(t-\tau)^{-1} + \delta (t-\tau)^{-(3-\alpha)/2} \right] \exp\left\{-\frac{\delta^2}{2(t-\tau)}\right\}\,d\tau \leq \]

\[ \leq (C)|\varphi|_{\alpha}\delta^{\alpha} \int_{\delta^2/2t}^{+\infty} \left[ z^{-1}+z^{-(1+\alpha)/2} \right]e^{-z}\,dz \leq (C)|\varphi|_{\alpha}\delta^{\alpha'}. \]

Thus,

\[ \left|\overline{U}^{(k)}_{11}(x^{\delta},t)-\overline{U}^{(k)}_{11}(x,t)\right| \leq (C)|\varphi|_{\alpha}\delta^{\alpha'}. \tag{5.37} \]

Now, alongside the local system \(\{\xi,\tau\}\) associated with the point \((x,t)\in\Gamma_t\), introduce a local system \(\{\zeta,\tau\}\) associated with the point \((x^{\delta},t)\in\Gamma_t\), placing the origin of the system \(\{\zeta,\tau\}\) at the point \((x^{\delta},0)\), directing the axis \(O\zeta_3\) along the normal \(N(x^{\delta},t)\), and directing the axis \(O\zeta_1\) parallel to the axis \(O\xi_1\). Denote
\[ \varphi=(N(x,t),N(x^{\delta},t))=(O\xi_2,O\zeta_2). \]
Then

\[ \begin{cases} \xi_1=\zeta_1,\\ \xi_2=\delta+\zeta_2\cos\varphi-\zeta_3\sin\varphi, \end{cases} \qquad \begin{cases} \zeta_1=\xi_1,\\ \zeta_2=(\xi_2-\delta)\cos\varphi+ \bigl(\xi_3-\psi_1(0,\delta,t)\bigr)\sin\varphi, \end{cases} \tag{5.38} \]

\[ \xi_3=\psi_1(0,\hat\delta,t)+\zeta_2\sin\varphi+\zeta_3\cos\varphi, \]
\[ \zeta_3=-(\xi_2-\hat\delta)\cos\varphi+(\xi_3-\psi_1(0,\hat\delta,t))\sin\varphi . \tag{5.39} \]

The equations of the surface \(\Gamma_\tau(x,d)\) in the systems \(\{\xi,\tau\}\) and \(\{\zeta,\tau\}\) will be

\[ \xi_3=\psi_1(\xi_1,\xi_2,\tau) \quad\text{and}\quad \zeta_3=\overline{\psi}_1(\zeta_1,\zeta_2,\tau) \tag{5.40} \]

respectively, and moreover (cf. (1.10)) \(\psi_1(0,0,t)=\overline{\psi}_1(0,0,t)=0\). Therefore, from (5.39), (5.40) we have

\[ \overline{\psi}_1(\zeta_1,\zeta_2,\tau) =-(\xi_2-\hat\delta)\sin\varphi+ \bigl(\psi_1(\xi_1,\xi_2,\tau)-\psi_1(0,\hat\delta,t)\bigr)\cos\varphi . \tag{5.41} \]

We note that

\[ \frac{\partial \overline{\psi}_1(\zeta_1,\zeta_2,\tau)}{\partial \zeta_1} = a\,\frac{\partial \psi_1(\xi_1,\xi_2,\tau)}{\partial \xi_1}; \qquad \frac{\partial \overline{\psi}_1(\zeta_1,\zeta_2,\tau)}{\partial \zeta_2} = \]
\[ = a\left[ \frac{\partial \psi_1(\xi_1,\xi_2,\tau)}{\partial \xi_2}\cos\varphi-\sin\varphi \right], \qquad a=\left[ \cos\varphi+ \frac{\partial \psi_1(\xi_1,\xi_2,\tau)}{\partial \xi_2}\sin\varphi \right]^{-1}. \tag{5.42} \]

By virtue of Lyapunov’s condition (1.7),

\[ |\sin\varphi|\le L_1\hat\delta^{\alpha},\qquad \cos\varphi>\frac12 . \tag{5.43} \]

Therefore, for sufficiently small \(\hat\delta\) we have

\[ 0<\hat\delta^{\alpha}\le \left[ 4L_1\sup_{\Gamma}\left|\frac{\partial\psi_1}{\partial\xi_i}\right| \right]^{-1}, \qquad 0<a<4. \tag{5.44} \]

In addition (see (3.18)), from (5.39), (5.40) we have

\[ \overline{\rho}^{\,2}+ \bigl(\overline{\psi}_1(\zeta_1,\zeta_2,\tau)-\overline{\psi}_1(0,0,t)\bigr)^2 = \rho_1^2+ \bigl(\psi_1(\xi_1,\xi_2,\tau)-\psi_1(0,\hat\delta,t)\bigr)^2, \tag{5.45} \]

where

\[ \overline{\rho}^{\,2}=\zeta_1^2+\zeta_2^2 . \tag{5.46} \]

Set, by virtue of (5.38) (where \(\xi_3\) and \(\zeta_3\) have been replaced with the aid of (5.40)),

\[ \overline{\varphi}_1(\zeta_1,\zeta_2,\tau) \equiv \varphi_1\bigl(\xi_1,(\xi_2-\hat\delta)\cos\varphi+ (\psi_1(\xi_1,\xi_2,\tau)-\psi_1(0,\hat\delta,t))\sin\varphi,\tau\bigr) \equiv \]
\[ \equiv \varphi_1(\xi_1,\xi_2,\tau), \qquad \varphi_1(\xi_1,\xi_2,\tau) \equiv \varphi_1\bigl(\zeta_1,\hat\delta+\zeta_2\cos\varphi- \overline{\psi}_1(\zeta_1,\zeta_2,\tau)\sin\varphi,\tau\bigr) \equiv \overline{\varphi}_1(\zeta_1,\zeta_2,\tau). \]

Whence, by virtue of (1.17),

\[ \overline{\varphi}_1(0,0,t)=\varphi_1(0,\hat\delta,t), \qquad |\overline{\varphi}_1(0,0,t)-\varphi_1(0,0,t)|\le [\varphi]_{\alpha}\hat\delta^{\alpha}. \tag{5.47} \]

Consider the point
\[ Q=(\xi_1,\xi_2,\psi_1(\xi_1,\xi_2,\tau),\tau) =(\zeta_1,\zeta_2,\overline{\psi}_1(\zeta_1,\zeta_2,\tau),\tau), \]
for which \((\xi_1,\xi_2)\) and \((\zeta_1,\zeta_2)\) are connected by the relations (5.38), (5.40). From (5.38), (5.40), for the point \(Q\) we have

\[ |\overline{\rho}-\rho| \le |\xi_2|\,|1-\cos\varphi| +\hat\delta|\cos\varphi| + |\psi_1(\xi_1,\xi_2,\tau)-\psi_1(0,\hat\delta,t)|\,|\sin\varphi|. \tag{5.48} \]

If we put (see (1.11)) \(\rho=d/2\), then from (5.48) we have
\[ |\overline{\rho}-\rho|\le C_1\hat\delta^{\alpha}, \]
where \(C_1\) does not depend on \(\hat\delta\). For \(\hat\delta\) so small that
\(\hat\delta^\alpha\le d/(4C_1)\), we have

\[ \frac d4\le \frac d2-C_1\hat\delta^{\alpha} =\rho-C_1\hat\delta^\alpha \le \overline{\rho} \le \rho+C_1\hat\delta^\alpha =\frac d2+C_1\hat\delta^\alpha \le \frac34\,d . \tag{5.49} \]

Put

\[ R_0=\frac{d}{2},\quad R_1=\frac{d}{2}+C_1\delta^\alpha,\quad R_2=\frac{d}{2}-C_1\delta^\alpha . \tag{5.50} \]

We introduce the transformation \((\xi_1,\xi_2)\leftrightarrow(\zeta_1,\zeta_2)\) by means of (5.38), (5.40) (taking (5.41) into account). Obviously, from (5.44) we have

\[ \left|\frac{D(\xi_1,\xi_2)}{D(\zeta_1,\zeta_2)}\right|=a^{-1}>\frac14 . \tag{5.51} \]

From (5.49), (5.50) it follows that the disk \(\overline K_{R_1}=\{\rho\le R_1\}\) in the \((\zeta_1,\zeta_2)\)-plane is transformed, under (5.38), (5.40), into a figure \(\Lambda_{R_1}\) containing in itself the disk \(K_{R_0}=\{\rho\le R_0\}\) in the \((\xi_1,\xi_2)\)-plane. On the other hand, the figure \(\Lambda_{R_1}-K_{R_0}\), under the inverse transformation (5.38), (5.40), is transformed in the \((\zeta_1,\zeta_2)\)-plane into a figure contained in the circular ring \(\overline K_{R_2R_1}=\{R_2\le\rho\le R_1\}\), with area not exceeding, by virtue of (5.51), \(C_2\delta^\alpha\), where \(C_2\) does not depend on \(\delta\) (cf. [3], supplement). Put

\[ \overline U_{12}^{(k)}(x^\delta,t) =\int_{t_1}^{t}d\tau\iint_{\rho\le R_1} \overline a_4^{(k)}(x^\delta)\,d\zeta_1\,d\zeta_2 +\int_{t_1}^{t}d\tau\iint_{\rho>R_1} \overline a_4^{(k)}(x^\delta)\,d\zeta_1\,d\zeta_2, \tag{5.52} \]

\[ \overline U_{12}^{(k)}(x,t) =\int_{t_1}^{t}d\tau\iint_{\rho\le R_0} a_4^{(k)}(x)\,d\xi_1\,d\xi_2 +\int_{t_1}^{t}d\tau\iint_{\rho>R_0} a_4^{(k)}(x)\,d\xi_1\,d\xi_2, \tag{5.53} \]

where

\[ \overline a_4^{(k)}(x^\delta) =\overline\varphi_1(0,0,t)\bigl(\overline\psi_1(\zeta_1,\zeta_2,\tau)-\overline\psi_1(0,0,t)\bigr) \frac{\partial\overline\psi_1(\zeta_1,\zeta_2,\tau)}{\partial\zeta_k} \times \]

\[ \times g_{0,\,5/2}(r_1(x^\delta,\zeta),t-\tau), \]

\[ a_4^{(k)}(x) =\varphi_1(0,0,t)\bigl(\psi_1(\xi_1,\xi_2,\tau)-\psi_1(0,0,t)\bigr) \frac{\partial\psi_1(\xi_1,\xi_2,\tau)}{\partial\xi_k} \times \tag{5.54} \]

\[ \times g_{0,\,5/2}(r_1(x,\xi),t-\tau). \]

When in (5.52) the domain of integration \(\overline K_{R_1}\) is replaced by \(K_{R_0}\) (with passage, by virtue of (5.38), (5.40), from \((\zeta_1,\zeta_2)\) to \((\xi_1,\xi_2)\)), both terms in (5.52) change by no more than \((C)\delta^\alpha\); moreover, the second terms in (5.52), (5.53) will also differ from one another by \((C)\delta^\alpha\). Therefore it is enough to consider

\[ \overline U_{12d}^{(k)}(x^\delta,t) =\int_{t_1}^{t}d\tau\iint_{\rho\le R_0} \overline a_4^{(k)}(x^\delta)\,d\zeta_1\,d\zeta_2;\quad \overline U_{12d}^{(k)}(x,t) =\int_{t_1}^{t}d\tau\iint_{\rho\le R_0} a_4^{(k)}(x)\,d\xi_1\,d\xi_2. \]

In what follows we shall assume \(0<\alpha<1\) (for \(\alpha=1\) the computations are simplified). Put

\[ \overline U_{12d}^{(k)}(x^\delta,t)-\overline U_{12d}^{(k)}(x,t) =\sum_{j=1}^{3}\overline U_{dj}^{(k)}(\delta), \]

where

\[ \overline U_{d1}^{(k)}(\delta) =\int_{t_1}^{t}d\tau\iint_{\rho\le 2\delta} \overline a_4^{(k)}(x^\delta)\,d\zeta_1\,d\zeta_2,\quad \overline U_{d2}^{(k)}(\delta) =-\int_{t_1}^{t}d\tau\iint_{\rho\le 2\delta} a_4^{(k)}(x)\,d\xi_1\,d\xi_2, \]

\[ \overline U_{d3}^{(k)}(\delta) =\int_{t_1}^{t}d\tau\iint_{2\delta\le\rho\le d/2} \overline a_4^{(k)}(x^\delta)\,d\zeta_1\,d\zeta_2 -\int_{t_1}^{t}d\tau\iint_{2\delta\le\rho\le d/2} a_4^{(k)}(x)\,d\xi_1\,d\xi_2. \]

For \(\rho \leqslant 2\delta\), from (5.48), by virtue of (1.15), it is easy to obtain that

\[ \bar{\rho} \leqslant 2c\left(\delta+\delta^\alpha (t-\tau)^{\frac{1+\alpha}{2}}\right), \]

where the constant \(c\) does not depend on \(\delta\). Taking into account the estimates (1.15), (1.14) (for \(\psi_1\)), we obtain, passing to polar coordinates (see (5.45)),

\[ \zeta_1=\bar{\rho}\cos\vartheta,\quad \zeta_2=\bar{\rho}\sin\vartheta, \]

\[ \left|\bar{U}_{d_1}^{(k)}(\delta)\right| \leqslant (C)[\varphi]_0 \int_0^t d\tau \int_0^{2c\left(\delta+\delta^\alpha (t-\tau)^{\frac{1+\alpha}{2}}\right)} g_{1,\,2-\alpha}(\bar{\rho},\,2(t-\tau))\,d\bar{\rho} \leqslant \]

\[ \leqslant (C)[\varphi]_0 \int_0^t (t-\tau)^{\alpha-1} \left[ 1-\exp\{-c^2\delta^{2\alpha}(t-\tau)^\alpha\} \exp\left\{-\frac{c^2\delta^{1+\alpha}}{(t-\tau)^{\frac{1-\alpha}{2}}}\right\} \times \exp\left\{-\frac{c^2\delta^2}{t-\tau}\right\} \right]d\tau . \]

Obviously,

\[ 0\leqslant \int_0^t (t-\tau)^{-1+\alpha} \left[1-\exp\{-c^2\delta^{2\alpha}(t-\tau)^\alpha\}\right]d\tau \leqslant (C)\delta^{2\alpha}. \tag{5.55} \]

Making the substitutions \(t-\tau=z^{\frac{2}{1-\alpha}}\), \(c^2\delta^{1+\alpha}z^{-1}=u\), we easily obtain

\[ 0\leqslant \int_0^t (t-\tau)^{-1+\alpha} \left[ 1-\exp\left\{-c^2\delta^{1+\alpha}(t-\tau)^{\frac{\alpha-1}{2}}\right\} \right]d\tau \leqslant (C)\int_0^{t^{\frac{1-\alpha}{2}}} z^{-1+\frac{\alpha}{1-\alpha}} \times \]

\[ \times\left(1-\exp\{-c^2\delta^{1+\alpha}z^{-1}\}\right)dz \leqslant (C)t^{\frac{\alpha(1+3\alpha)}{2(1+\alpha)}} \int_0^{t^{\frac{1-\alpha}{2}}} z^{-1+\frac{\alpha}{1+\alpha}} \times \]

\[ \times\left(1-\exp\{-c^2\delta^{1+\alpha}z^{-1}\}\right)dz \leqslant (C)t^{\frac{\alpha(1+3\alpha)}{2(1+\alpha)}}\delta^\alpha \int_{c^2\delta^{1+\alpha}t^{\frac{\alpha-1}{2}}}^{+\infty} u^{-1-\frac{\alpha}{1+\alpha}} \times \]

\[ \times(1-e^{-u})\,du \leqslant (C)\delta^\alpha . \tag{5.56} \]

Finally, by means of the substitution \(c^2\delta^2(t-\tau)^{-1}=z\), we obtain

\[ 0\leqslant \int_0^t (t-\tau)^{-1+\alpha} \left[1-\exp\{-c^2\delta^2(t-\tau)^{-1}\}\right]d\tau \leqslant (C)\delta^\alpha . \tag{5.57} \]

From (5.55)—(5.57) (for \(i=1\)) and (1.15), (1.14) (for \(\psi_1\)), (1.10) (for \(i=2\)), we have

\[ \left|\bar{U}_{d_i}^{(k)}(\delta)\right| \leqslant (C)[\varphi]_0\delta^\alpha,\quad i=1,\ 2. \tag{5.58} \]

For \(\rho \geqslant 2\delta\), from (5.48) we have

\[ \bar{\rho}\leqslant (C)\left(\rho+(t-\tau)^{\frac{1+\alpha}{2}}\right). \tag{5.59} \]

Note (see (3.18)) that from (5.39)—(5.42), (5.45), (5.51), by virtue of (3.21) and (1.15), we have

\[ \begin{gathered} \left|\overline{\psi}_{1}(\zeta_{1},\zeta_{2},\tau)-\psi_{1}(\xi_{1},\xi_{2},\tau)\right| \leq (C)\left(\rho_{1}\delta^{\alpha}+\delta^{1+\alpha}\right),\\ \left| \frac{\partial \overline{\psi}_{1}(\zeta_{1},\zeta_{2},\tau)}{\partial \zeta_{k}} - \frac{\partial \psi_{1}(\xi_{1},\xi_{2},\tau)}{\partial \xi_{k}} \right| \leq (C)\delta^{\alpha}, \end{gathered} \tag{5.60} \]

\[ \begin{gathered} \left|g_{0,0}\left(r_{1}(x^{\circ},\xi),t-\tau\right) - g_{0,0}\left(r_{1}(x,\xi),t-\tau\right)\right| \leq\\ \leq (C)\left[\delta+\delta^{1+\alpha}\right]\, g_{0,\frac12}\left(\rho,2(t-\tau)\right). \end{gathered} \]

We now pass from \((\zeta_{1},\zeta_{2})\) to \((\xi_{1},\xi_{2})\) (see (5.38), (5.40)) and introduce polar coordinates (1.11). Then, by virtue of (5.47), (5.59), (5.60), (1.14) for \(\overline{\psi}_{1}\) and (3.21), using the substitution \(\frac18\delta^{2}(t-\tau)^{-1}=z\), we obtain

\[ \left|\overline{U}_{d3}^{(k)}(\delta)\right| \leq (C)|\varphi|_{\alpha} \int_{0}^{t}d\tau \int_{2\delta}^{+\infty} \left[ \delta^{\alpha}\rho^{1+2\alpha} + \delta^{\alpha}\rho^{1+\alpha} + \delta^{1+\alpha}\rho^{\alpha} + \delta\rho^{1+2\alpha}(t-\tau)^{-\frac12} \right] \times \]

\[ \times\, g_{1,5/2}\left(\rho,8(t-\tau)\right)\,d\rho \leq (C)|\varphi|_{\alpha}\delta^{\alpha}. \tag{5.61} \]

From (5.58), (5.61) we have

\[ \left|\overline{U}_{12}^{(k)}(x^{\circ},t)-\overline{U}_{12}^{(k)}(x,t)\right| \leq (C)|\varphi|_{\alpha}\delta^{\alpha},\quad k=1,2. \tag{5.62} \]

From (5.23), (5.37), (5.62), following the scheme of the proof of Lyapunov’s theorem on the first derivatives of the simple-layer potential, from [3] (Appendix I, pp. 349—356), one easily obtains (see also (5.35)) relation (5.17), which completes the proof of Theorem 8.

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Received by the editors
February 13, 1965

Moscow State University
named after M. V. Lomonosov