UDC 517.947.42

ON THE SMOOTHNESS OF THERMAL POTENTIALS. III

A SPECIAL THERMAL SINGLE-LAYER POTENTIAL \(P(x,t)\) ON SURFACES OF TYPE

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

L. I. Kamynin

In papers [1—3] we investigated the smoothness of ordinary thermal single- and double-layer potentials with densities distributed on noncylindrical surfaces of type \(\mathcal{L}^{0,1,\frac{1+\alpha}{2}}_{1,\alpha,\alpha/2}\) (see [1]) and \(\mathcal{L}^{1,\alpha,\alpha/2}_{1,1,\frac{1+\alpha}{2}}\) (see [2]), which, in combination with the a priori estimates from [4], made it possible, in particular, to construct solutions with prescribed smoothness of the second and third boundary-value problems with conormal derivative for a parabolic equation.

In the present paper we consider the special thermal potential \(P(x,t)\), introduced by M. Pani in [7] and playing an important role in the theory of heat conduction in the solution of boundary-value problems with oblique derivative. We investigate the smoothness of the thermal potential \(P(x,t)\) as a function of the smoothness of its density, distributed on a noncylindrical surface of type \(\mathcal{L}^{0,1,\frac{1+\alpha}{2}}_{1,\alpha,\alpha/2}\) or \(\mathcal{L}^{1,\alpha,\alpha/2}_{1,1,\frac{1+\alpha}{2}}\). The results obtained are used in proving the existence of a solution of class \(H^{1,\alpha,\alpha/2}_{1,1,\frac{1+\alpha}{2}}(\overline{D_T^{B}})\) of the second and third boundary-value problems with oblique derivative for a general linear parabolic equation of second order under minimally admissible smoothness requirements on the data of the problem. The proof of the corresponding existence theorem is carried out by the method of analytic continuation with respect to a parameter, relying essentially on the \((2+\alpha)\) a priori estimate (up to the boundary) established for solutions of the second and third boundary-value problems with oblique derivative by the author jointly with V. N. Maslennikova in [5, 6].

Our article continues the author’s series of works (see [1, 2]) devoted to the properties of thermal potentials and their applications in the theory of boundary-value problems for the parabolic equation. In the papers of this series, unified notation and continuous numbering of paragraphs are used.

§ 10. A SPECIAL THERMAL SINGLE-LAYER POTENTIAL \(P(x,t)\) AND ITS PROPERTIES

On the \(n\)-dimensional hyperplane \(t=\tau\), for \((y,\tau)\in\Gamma_\tau\) (see § 1 in [1]), we introduce the unit vectors \(N=N(y,\tau)=\{N_1(y,\tau),\ldots,N_n(y,\tau)\}\)—the unit vector of the inner normal to the section \(\Gamma_\tau\); \(\nu=\nu(y,\tau)=\{\nu_1(y,\tau),\ldots,\nu_n(y,\tau)\}\)—the unit vector of the di-

direction forming an acute angle (see (10.3)) with \(N(y,\tau)\); \(\mu=\mu(y,\tau)=\{\mu_1(y,\tau),\ldots,\mu_n(y,\tau)\}\) is a unit vector lying in the two-dimensional plane spanned by the vectors \(N(y,\tau)\) and \(\nu(y,\tau)\) and orthogonal to \(\nu(y,\tau)\) (see (10.2)); \(\lambda_j=\lambda_j(y,\tau)=\{\lambda_{j1}(y,\tau),\ldots,\lambda_{jn}(y,\tau)\}\) \((j=1,2,\ldots,n-2)\) is an orthonormal system of vectors orthogonal to the vectors \(\nu\) and \(N\). Denoting the scalar product of the vectors \(a\) and \(b\) by \((a,b)\), we have

\[ (N,N)=(\nu,\nu)=(\mu,\mu)=(\lambda_j,\lambda_j)=1 \quad (j=1,2,\ldots,n-2), \tag{10.1} \]

\[ (\mu,\nu)=(\lambda_j,\nu)=(\lambda_j,N)=0, \quad (j=1,2,\ldots,n-2), \tag{10.2} \]

\[ (\nu,N)\geqslant 2d_0>0. \tag{10.3} \]

Let, for \((y,\tau)\in\Gamma_\tau\) and arbitrary \((\bar x,t)\),

\[ r_{y\bar x}=\{\bar x_1-y_1,\ldots,\bar x_n-y_n\},\qquad r(y,\bar x)=\sqrt{(r_{y\bar x},r_{y\bar x})}. \]

In the local coordinate system \(\{\xi,\tau\}\) associated with the point \((x,t)\in\Gamma_t\) (see § 1, [1]), we shall use the following notation: for \((\xi,\tau)\in\Gamma_\tau\),

\[ r_{\xi\bar x}=\{\bar x_1-\xi_1,\ldots,\bar x_{n-1}-\xi_{n-1}, \bar x_n-\psi(\xi_1,\ldots,\xi_{n-1},\tau)\} \]

and, for \((x,t)\in\Gamma_t\),

\[ r_{\xi x}=\{x_1-\xi_1,\ldots,x_{n-1}-\xi_{n-1}, \psi(x_1,\ldots,x_{n-1},t)-\psi(\xi_1,\ldots,\xi_{n-1},\tau)\}. \]

Obviously,

\[ r_{\xi\bar x}=(r_{\xi\bar x},\nu)\nu+(r_{\xi\bar x},\mu)\mu+ \sum_{j=1}^{n-2}(r_{\xi\bar x},\lambda_j)\lambda_j, \tag{10.4} \]

whence, in view of (10.1), (10.2),

\[ (r_{\xi\bar x},r_{\xi\bar x}) =(r_{\xi\bar x},\nu)^2+(r_{\xi\bar x},\mu)^2+ \sum_{j=1}^{n-2}(r_{\xi\bar x},\lambda_j)^2. \tag{10.5} \]

Recall that the domain \(\Omega_\tau=D_T^{B}\cap\{t=0\}\), by § 1 [1], has boundary section \(\Gamma_\tau\).

Condition \((P_1)\) (see § 4 in [7]). Let the surface \(\Gamma\) be such that, for \(0\leqslant \tau\leqslant T\), for every point \((y,\tau)\in\Gamma_\tau\) the direction opposite to \(\nu(y,\tau)\) has no common points with \(\overline{\Omega}_\tau\), except for \((y,\tau)\).

Remark 6. If, for \(0\leqslant \tau\leqslant T\), the domain \(\Omega_\tau\) is convex, then \(\Gamma\) satisfies condition \((P_1)\).

If \(\Gamma\) satisfies condition \((P_1)\), then, following § 4 of [7] and assuming \((y,\tau)\in\Gamma_\tau\), \((\bar x,t)\in\Omega_t\), we introduce the functions (see the notation (1.20) of § 1 in [1])

\[ P(r(y,\bar x),t-\tau) =(\nu(y,\tau),N(y,\tau))\,g_{0,n/2}(r(y,\bar x),t-\tau) +p(r(y,\bar x),t-\tau), \tag{10.6} \]

\[ p(r(y,\bar x),t-\tau) =-(\mu(y,\tau),N(y,\tau))(\mu(y,\tau),r_{y\bar x})\times G_{\frac{n+1}{2}}(r(y,\bar x),t-\tau), \tag{10.7} \]

where, in view of (10.5),

\[ G_m(r(y,\bar x),t-\tau) = g_{0,m}(r(y,\bar x),t-\tau)\times \]

\[ \times \frac{ \left\{ \Phi\!\left(\dfrac{(\nu(y,\tau),r_{y\bar x})}{2\sqrt{t-\tau}}\right) -\dfrac{V_{\bar x}^{\tau}}{2} \right\} }{ g_{0,0}\bigl((\nu(y,\tau),r_{y\bar x}),t-\tau\bigr) } = \]

\[ = g_{0,m}\left(\sqrt{(\mu(y,\tau),r_{\bar y x})^2+ \sum_{j=1}^{n-2}(\lambda_j(y,\tau),r_{\bar y x})^2},\,t-\tau\right)\times \]
\[ \times\left\{\Phi\left(\frac{(\nu(y,\tau),r_{\bar y x})}{2\sqrt{t-\tau}}\right)-\frac{\sqrt{\pi}}{2}\right\} \tag{10.8} \]

\[ \Phi(s)=\int_0^s \exp\{-z^2\}\,dz. \tag{10.9} \]

If in (10.6)—(10.8) the point \((\bar x,\bar t)\) is replaced by \((x,t)\in\Gamma_t\), then we shall denote the functions obtained by

\[ \bar P(r(y,x),\,t-\tau),\qquad \bar p(r(y,x),\,t-\tau),\qquad \bar G_m(r(y,x),\,t-\tau). \tag{10.10} \]

Condition \((P_2)\) (see § 4 in [7]). Let the surface \(\Gamma\) be such that if, for \((y,\tau)\in\Gamma_t\), \((y^*,\tau)\) is the first point of intersection of \(\Gamma_\tau\) at which the direction opposite to \(\nu(y,\tau)\) meets \(\bar\Omega_\tau\), then the inequality

\[ \inf_{(y,\tau)\in\Gamma_\tau,\ 0\le \tau\le T} r(y,y^*)=2d_2>0 \tag{10.11} \]

is satisfied.

Remark 7. If \(\Gamma\) is of type
\[ Л^{0,1,\frac{1+\alpha}{2}}_{1,\alpha,\alpha/2} \quad\text{or}\quad Л^{1,\alpha,\alpha/2}_{1,1,\frac{1+\alpha}{2}}, \]
then the condition \((P_2)\) is satisfied for \(\Gamma\).

Remark 8. If, for the surface \(\Gamma\), instead of condition \((P_1)\) condition \((P_2)\) is satisfied, then in (10.8), instead of the function

\[ \left\{\Phi\left(\frac{(\nu(y,\tau),r_{\bar y x})}{2\sqrt{t-\tau}}\right)-\frac{\sqrt{\pi}}{2}\right\}, \]

one must write (see § 4 in [7] and (10.11))

\[ \left\{\Phi\left(\frac{(\nu(y,\tau),r_{\bar y x})}{2\sqrt{t-\tau}}\right) -\Phi\left(\frac{(\nu(y,\tau),r_{\bar y x})+d_2}{2\sqrt{t-\tau}}\right)\right\}. \tag{10.12} \]

(The fundamental solution of Paneah (10.6), (10.12) for \(n\ge 3\) is written out explicitly in [8].)

Lemma 10 (§ 4 of [7]). If the surface \(\Gamma\) satisfies condition \((P_i)\) \((i=1,2)\), then

1) for \(\nu(y,\tau)=N(y,\tau)\)

\[ P(r(y,\bar x),\,t-\tau)\equiv g_{0,n/2}(r(y,\bar x),t-\tau) \quad \bigl((y,\tau)\in\Gamma_\tau,\ 0\le\tau\le T,\ (\bar x,t)\in\Omega_t\bigr); \]

2) for fixed \((y,\tau)\in\Gamma_\tau\), the functions
\[ P(r(y,\bar x),\,\bar t-\tau),\quad p(r(y,\bar x),\,\bar t-\tau) \]
satisfy, with respect to \((\bar x,\bar t)\), the heat equation (1.34) (§ 1 of [1]) at every point \((\bar x,\bar t)\in D_T^{B}\);

3) for \((y,\tau)\in\Gamma_\tau,\ 0\le\tau\le T,\ (\bar x,t)\in\Omega_t\)

\[ \frac{\partial P(r(y,\bar x),t-\tau)}{\partial \nu(y,\tau)} = \frac{\partial g_{0,n/2}(r,(y,\bar x),t-\tau)}{\partial N(y,\tau)} + \]

\[ +\begin{cases} 0, & \text{under condition }(P_1),\\[4pt] P_2(r(y,\bar x),t-\tau), & \text{under condition }(P_2), \end{cases} \]

where

\[ P_2(r(y,\bar x),t-\tau) = -\frac12\,(\mu(y,\tau),N(y,\tau))(\mu(y,\tau),r_{y\bar x})\times \]

\[ {}\times g_{0,\,\frac n2+1}\bigl(r(y,\bar x),\,t-\tau\bigr) \exp\left\{-\frac{1}{4(t-\tau)} \left[((\nu(y,\tau),r_{yx})+d_2)^2-(\nu(y,\tau),r_{yx})^2\right]\right\} \]

—is a function continuous in \(\bar D_T^{\,B}\).

From (1.1), (1.2), (1.6), and (1.7) of § 1 in [1], assuming condition (10.3) to be fulfilled for \((y,\tau)\in\Gamma_\tau\), \(0\le \tau\le T\), we obtain the following.

Remark 9. If \(\Gamma\) is of type \(\mathcal L_{1,\alpha,\alpha/2}^{0,1,\frac{1+\alpha}{2}}\) or \(\mathcal L_{1,1,\frac{1+\alpha}{2}}^{1,\alpha,\alpha/2}\) and condition (10.3) is fulfilled for \((y,\tau)\in\Gamma_\tau\), \(0\le \tau\le T\), then we shall assume \(d\) and \(d_1\) so small that, together with (1.8) of § 1 in [1], the inequalities

\[ (\nu(x,t),\nu(y,\tau))\ge \frac12,\qquad (\nu(x,\tau),\nu(y,\tau))\ge \frac12, \]

\[ (\nu(y,\tau),N(x,t))\ge d_0,\qquad (\nu(y,\tau),N(x,\tau))\ge d_0 \tag{10.13} \]

hold for \((y,\tau)\in\Gamma_{dd_1}(x,t)\).

Remark 10. For simplicity of exposition, in what follows, without diminishing the generality of the arguments (cf. [7]), we shall assume that \(n=2\) and \(\Omega_\tau\) is a \(B\)-convex domain for \(0\le \tau\le T\). Then, in the local coordinate system \((\xi,\tau)\) associated with \((x,t)\in\Gamma_t\) (see § 1 in [1]), by virtue of (1.5) of § 1 in [1], (10.8), we have:

\[ (\xi,\tau)=(\xi_1,\psi(\xi_1,\tau),\tau)\in\Gamma_\tau, \]

\[ \mu(\xi_1,\tau)=\{\mu_1(\xi_1,\tau),\mu_2(\xi_1,\tau)\} \equiv\{-\nu_2(\xi_1,\tau),\nu_1(\xi_1,\tau)\}, \tag{10.14} \]

\[ G_m(r(\xi,\bar x),t-\tau) = g_{0,m}\bigl((\mu(\xi_1,\tau),r_{\xi\bar x}),t-\tau\bigr)\times \]

\[ {}\times\left\{ \Phi\left(\frac{(\nu(\xi_1,\tau),r_{\xi\bar x})}{2\sqrt{t-\tau}}\right) -\frac{\sqrt{\pi}}{2} \right\}, \tag{10.15} \]

\[ r_{\xi\bar x}=\{\bar x_1-\xi_1,\ \bar x_2-\psi(\xi_1,\tau)\} \]

and, for \((x,t)\in\Gamma_t\),

\[ r_{\xi x}=\{x_1-\xi_1,\ \psi(x_1,t)-\psi(\xi_1,\tau)\}, \tag{10.16} \]

\[ N_1(\xi_1,\tau)=\cos(N(\xi_1,\tau),\xi_1) = -\frac{\partial\psi(\xi_1,\tau)}{\partial\xi_1}\,N_2(\xi_1,\tau), \tag{10.17} \]

\[ N_2(\xi_1,\tau)=\cos(N(\xi_1,\tau),\xi_2) = \left[1+\left(\frac{\partial\psi(\xi_1,\tau)}{\partial\xi_1}\right)^2\right]^{-1/2}. \]

Lemma 11 (cf. § 4 of [7]). If \(\Gamma\) is of type \(\mathcal L_{1,\alpha,\alpha/2}^{0,1,\frac{1+\alpha}{2}}\) \((0<\alpha\le 1)\), then for \((y,\tau)\in\Gamma_\tau\), \((\bar x,t)\in\bar\Omega_t\), the estimates

\[ \left\{ \left|\frac{\partial^l P(r(y,\bar x),t-\tau)} {\partial x_i^{\,l_1}\partial x_j^{\,l_2}}\right|, \quad \left|\frac{\partial^l p(r(y,\bar x),t-\tau)} {\partial x_i^{\,l_1}\partial x_j^{\,l_2}}\right| \right\} \le \]

\[ \le (C)\,g_{0,\,\frac{n+l}{2}}\bigl(\varkappa r(y,\bar x),t-\tau\bigr) \tag{10.18} \]

\[ (i,j=1,2,\ldots,n;\quad l_1+l_2=l=0,1,2), \]

hold.

\[ \left|\frac{\partial^k G_m(r(y,\bar x),t-\tau)}{\partial \bar x_i^k}\right| \le (C)g_{0,m+\frac{k}{2}}(\chi r(y,\bar x),t-\tau) \tag{10.19} \]

\[ (i=1,2,\ldots,n;\ k=0,1), \]

where \(0<\chi<1\) is a constant depending on \(d_0\) from (10.3) and independent of \((\bar x,t),(y,\tau)\); \((C)\) (see Remark 1, § 2 of [1]) is a constant depending on \(\Gamma\), \(d_0\), and \(\nu\).

Proof. Let \(\operatorname{pr}\Omega_\tau\) be the projection of the section \(\Omega_\tau\) onto the plane \((x_1,x_2)\). If \(\operatorname{pr}\Omega_\tau \supseteq \operatorname{pr}\Omega_t\) for any \(0\le \tau\le t\le T\), then Lemma 11 follows from § 4 of [7]. Now let \((\bar x,t)\in \bar\Omega_t\), and moreover

\[ \bar x=(\bar x_1,\bar x_2)\in \operatorname{pr}\Omega_t\setminus \operatorname{pr}\Omega_\tau . \tag{10.20} \]

We carry out, for example, the proof of (10.19) for \(k=m=0\). If \((\nu(y,\tau),r_{y\bar x})>0\), then by § 4 of [7] estimate (10.19) is valid. Let

\[ (\nu(y,\tau),r_{y\bar x})<0 . \tag{10.21} \]

Introduce the points \(x=(\bar x_1,\psi(\bar x_1,t))\) and \(x^*=(\bar x_1,\psi(\bar x_1,\tau))\). By virtue of (10.20), \(\bar x\) is a point of the segment joining \(x\) and \(x^*\), and from (1.1) of § 1 [1] it follows that

\[ |r(x^*,\bar x)|\le |\psi(\bar x_1,t)-\psi(\bar x_1,\tau)| \le (C)(t-\tau)^{\frac{1+\alpha}{2}} . \tag{10.22} \]

Consider two cases: a) \((\nu(y,\tau),r_{yx^*})<0\) and b) \((\nu(y,\tau),r_{yx^*})\ge 0\). From (10.15) it follows that it is enough to estimate

\[ g_{0,0}\bigl((\mu(y,\tau),r_{y\bar x}),\,t-\tau\bigr). \tag{10.23} \]

In case a) put

\[ r_{y\bar x}=r_{yx^*}+r_{x^*\bar x} \tag{10.24} \]

and note the inequality, valid for any \(0<\varepsilon<1\),

\[ 2\left|(\mu(y,\tau),r_{yx^*})(\mu(y,\tau),r_{x^*\bar x})\right| \le \varepsilon(\mu(y,\tau),r_{yx^*})^2+ \frac{1}{\varepsilon}(\mu(y,\tau),r_{y^*x})^2 . \tag{10.25} \]

For \((y,\tau)\in \Gamma_\tau\) and \((x^*,\tau)\in \Gamma_\tau\), from condition a) by virtue of (10.3) there follows the inequality (cf. § 4 of [1])

\[ |(\mu(y,\tau),r_{yx^*})|\ge \chi_1 r(y,x^*), \tag{10.26} \]

where the constant \(\chi_1\) \((0<\chi_1<1)\) depends only on \(d_0\). Therefore, applying in (10.23) representation (10.24), we easily obtain with the aid of (10.25), (10.22), and (10.26) the inequalities

\[ g_{0,0}\bigl((\mu(y,\tau),r_{y\bar x}),\,t-\tau\bigr) \le g_{0,0}\bigl(\sqrt{1-\varepsilon}\,(\mu(y,\tau),r_{yx^*}),\,t-\tau\bigr) \exp\left\{\frac14\left(\frac1\varepsilon-1\right)(t-\tau)^{-1} (\mu(y,\tau),r_{x^*\bar x})^2\right\} \le \]

\[ \le (C)g_{0,0}\left(\chi_1\sqrt{(1-\varepsilon)(r_{yx^*},r_{yx^*})},\,t-\tau\right). \tag{10.27} \]

Using the expansion \(r_{yx^*}=r_{y\bar x}+r_{\bar x x^*}\), the inequality

\[ 2\left|(r_{yx}^{-},\, r_{xx^{*}}^{-})\right| \leq \varepsilon (r_{yx}^{-},\, r_{yx}^{-})+\frac{1}{\varepsilon}(r_{xx^{*}},\, r_{xx^{*}}) \]

and (10.22), we obtain the estimate

\[ \left|g_{0,0}\left((\mu(y,\tau),\, r_{yx}^{-}),\, t-\tau\right)\right| \leq (C) g_{0,0}(\varkappa r(y,\bar{x}),\, t-\tau), \tag{10.28} \]

from which, in case a), estimate (10.19) follows. In case b), in view of (10.21), there exists a point \(\tilde{x}\), lying on the segment joining \(x\) and \(x^{*}\), for which \((\nu(y,\tau),\, r_{y\tilde{x}}^{-})=0\), and therefore

\[ |(\mu(y,\tau),\, r_{yx}^{-})|=r(y,\tilde{x}). \tag{10.29} \]

Moreover (cf. (10.22)),

\[ |r(\bar{x},\tilde{x})|\leq (C)(t-\tau)^{\frac{1+\alpha}{2}} . \tag{10.30} \]

Using the representation \(r_{yx}^{-}=r_{y\tilde{x}}+r_{\tilde{x}x}^{-}\), we easily obtain, by means of (10.29), (10.30) (cf. the derivation of (10.28)), the estimate

\[ g_{0,0}((\mu(y,\tau),\, r_{yx}^{-}),\, t-\tau)\leq \]

\[ \leq (C) g_{0,0}\left(\sqrt{1-\varepsilon}\, r(y,\tilde{x}),\, t-\tau\right) \leq (C) g_{0,0}(\varkappa r(y,\bar{x}),\, t-\tau), \]

which completes the derivation of (10.29) also in case b).

Lemma 12. Let \(\Gamma\) be of type \(\mathcal{L}_{1,\alpha,\alpha/2}^{0,1,\frac{1+\alpha}{2}}\) or \(\mathcal{L}_{1,1,\frac{1+\alpha}{2}}^{1,\alpha,\alpha/2}\) \((0<\alpha\leq 1)\), and let in the local coordinate system \(\{\xi,\tau\}\) associated with \((x,t)\in\Gamma_{t}\), conditions (1.10), (1.11) be satisfied. Then for \(0\leq t-\tau\leq d_{1}\), \(0<\Delta t<d_{1}\),

\[ \hat{x}=(x_{1},\ldots,x_{n-1},\,\psi(x_{1},\ldots,x_{n-1},\,t+\Delta t)), \tag{10.31} \]

\[ x=(x_{1},\ldots,x_{n-1},\,\psi(x_{1},\ldots,x_{n-1},\,t)) \]

the following estimates hold:

\[ \left| \frac{\partial P\left(r(\xi,\hat{x}),\, t+\Delta t-\tau\right)} {\partial x_{j}} \right| \leq \]

\[ \leq (C)g_{0,\frac{n+1}{2}}(\varkappa r(\xi,\hat{x}),\, t+\Delta t-\tau) \quad (j=1,2,\ldots,n), \tag{10.32} \]

\[ \left| \frac{\partial}{\partial x_{i}} \left[ \frac{\partial P\left(r(\xi,\hat{x}),\, t+\Delta t-\tau\right)} {\partial x_{j}} \right] \right| \leq \]

\[ \leq (C)g_{0,\frac{n}{2}+1}(\varkappa r(\xi,\hat{x}),\, t+\Delta t-\tau) \tag{10.33} \]

\[ (i=1,2,\ldots,n-1;\ j=1,2,\ldots,n), \]

\[ \left| \frac{\partial}{\partial x_{i}}\, \bar{G}_{m}(r(\xi,\hat{x}),\, t+\Delta t-\tau) \right| \leq \]

\[ \leq (C)g_{0,m+\frac{1}{2}}(\varkappa r(\xi,\hat{x}),\, t+\Delta t-\tau) \tag{10.34} \]

\[ (i=1,2,\ldots,n-1) \]

(the estimates (10.32)—(10.34) are preserved when \((\hat{x},t+\Delta t)\) is replaced by \((x,t)\),

\[ \left|\bar G_m(r(\xi,\hat x),\, t+\Delta t-\tau)-\bar G_m(r(\xi,x),\, t-\tau)\right|\le \]

\[ \le \begin{cases} \left[(\Delta t)^{\frac{1+\alpha}{2}}(t-\tau)^{-m-\frac12} +\Delta t\,(t-\tau)^{-m-1}\right]\,g_{0,0}(\varkappa\rho,\,2(t+\Delta t-\tau)),\\[6pt] \hfill \text{if } \Gamma\in \mathcal{L}^{0,1,\frac{1+\alpha}{2}}_{1,\alpha,\alpha/2},\\[10pt] \Delta t\,(t-\tau)^{-m-1}\,g_{0,0}(\varkappa\rho,\,2(t+\Delta t-\tau)),\\[6pt] \hfill \text{if } \Gamma\in \mathcal{L}^{1,\alpha,\alpha/2}_{1,1,\frac{1+\alpha}{2}}. \end{cases} \tag{10.35} \]

\[ \left| \frac{\partial \bar G_m(r(\xi,\hat x),\, t+\Delta t-\tau)}{\partial x_i} - \frac{\partial \bar G_m(r(\xi,x),\, t-\tau)}{\partial x_i} \right|\le \]

\[ \le (C)\left[(\Delta t)^{\frac{1+\alpha}{2}}(t-\tau)^{-m-\frac12} +\Delta t\,(t-\tau)^{-m-1}\right]\,g_{0,0}(\varkappa\rho,\,2(t+\Delta t-\tau)) \tag{10.36} \]

\[ (i=1,2,\ldots,n-1);\quad \Gamma\in \mathcal{L}^{1,\alpha,\alpha/2}_{1,1,\frac{1+\alpha}{2}}. \]

Further, putting

\[ x=(0,\ldots,0,\psi(0,\ldots,0,t)),\quad x^\delta=(\delta,0,\ldots,0,\psi(\delta,0,\ldots,0,t)), \tag{10.37} \]

we have, for \(\rho\ge 2\delta\), the estimates

\[ \left|\bar G_m(r(\xi,x^\delta),\,t-\tau)-\bar G_m(r(\xi,x),\,t-\tau)\right|\le \]

\[ \le (C)\delta\, g_{0,m+\frac12}\left(\frac{\varkappa\rho}{2},\,2(t-\tau)\right) \quad \text{for } \Gamma\in \mathcal{L}^{0,1,\frac{1+\alpha}{2}}_{1,\alpha,\alpha/2}, \tag{10.38} \]

\[ \left| \frac{\partial \bar G_m(r(\xi,x^\delta),\,t-\tau)}{\partial x_i} - \frac{\partial \bar G_m(r(\xi,x),\,t-\tau)}{\partial x_i} \right|\le \]

\[ \le (C)\delta\, g_{0,m+1}\left(\frac{\varkappa\rho}{2},\,2(t-\tau)\right) \quad (i=1,2,\ldots,n-1) \tag{10.39} \]

\[ \text{for } \Gamma\in \mathcal{L}^{1,\alpha,\alpha/2}_{1,1,\frac{1+\alpha}{2}}. \]

Proof. Estimates (10.32)—(10.36) follow from Lemma 11. To derive estimates (10.35), (10.36) and (10.38), (10.39), we note that, by the mean-value theorem and by virtue of remarks 9, 10 and (10.15)—(10.17), we have

\[ \bar G_m(r(\xi,\hat x),\,t+\Delta t-\tau)-\bar G_m(r(\xi,x),\,t-\tau)= \]

\[ =\int_0^1 \frac{\partial}{\partial\theta} G_m(r(\xi,\hat x(\theta)),\,t+\theta\Delta t-\tau)\,d\theta, \]

\[ \bar G_m(r(\xi,x^\delta),\,t-\tau)-\bar G_m(r(\xi,x),\,t-\tau)= \]

\[ = \int_0^1 \frac{\partial}{\partial \theta} G_m(r(\xi, x^\delta(\theta)),\, t-\tau)\,d\theta, \]

where

\[ r_{\xi x(\theta)}=\{\,x_1-\xi_1,\ \psi(x_1,t)-\psi(\xi_1,\tau) +\theta(\psi(x_1,t+\Delta t)-\psi(x_1,t))\,\}, \tag{10.40} \]

\[ r_{\xi x^\delta(\theta)}=\{\,x_1-\xi_1+\theta\delta,\ \psi(x_1,t)-\psi(\xi_1,\tau) +\theta(\psi(\delta,t)-\psi(x_1,t))\,\}. \tag{10.41} \]

Moreover (see (10.16)),

\[ \frac{\partial \overline{G}_m(r(\xi,x),\,t-\tau)}{\partial x_1} = -\frac{1}{2}\bigl(\mu(\xi,\tau),r_{\xi x}\bigr) \left(\mu(\xi,\tau),\frac{\partial r_{\xi x}}{\partial x_1}\right) \overline{G}_{m+1}(r(\xi,x),\,t-\tau) + \]

\[ +\frac{1}{2} \left(\nu(\xi_1,\tau),\frac{\partial r_{\xi x}}{\partial x_1}\right) g_{0,m+\frac12}(r(\xi,x),\,t-\tau). \tag{10.42} \]

Using Lemma 11 and estimates (1.12)—(1.15) from § 1 [1] (for \(\Gamma\in Л^{0,1,\frac{1+\alpha}{2}}_{1,\alpha,\alpha/2}\)), (6.3), (6.4), (6.9), (6.10) from § 6 [2] (for \(\Gamma\in Л^{1,\alpha,\alpha/2}_{1,1,\frac{1+\alpha}{2}}\)), by virtue of Lemma 4 and (1.24) from § 1 [1], it is now easy to obtain the estimates (10.35), (10.36), (10.38), (10.39).

Let \((x,t)\in\Gamma_t\), \((\bar x,\bar t)\notin\Gamma_t\), and let the function \(\varphi(y,\tau)\) be given on \(\Gamma\). With the aid of \(P(r(y,\bar x),\bar t-\tau)\)—the special fundamental solution (10.6)—we introduce the special thermal potentials (cf. (1.35), (1.36) from § 1 [1])

\[ P(\bar x,\bar t)=\int_0^{\bar t} d\tau \iint_{\Gamma_\tau} P(r(y,\bar x),\,\bar t-\tau)\,\varphi(y,\tau)\,d\sigma_y(\tau), \tag{10.43} \]

\[ Q(\bar x,\bar t)=\int_0^{\bar t} d\tau \iint_{\Gamma_\tau} \frac{\partial P(r(y,\bar x),\,\bar t-\tau)}{\partial \nu(x,t)} \,\varphi(y,\tau)\,d\sigma_y(\tau). \tag{10.44} \]

If \((\bar x,\bar t)\equiv(x,t)\in\Gamma_t\), then we introduce the functions \(\overline P(x,t)\) and \(\overline Q(x,t)\), obtained from (10.43) and (10.44) by replacing \((\bar x,\bar t)\) by \((x,t)\). The functions \(\overline P(x,t)\) and \(\overline Q(x,t)\) will be called the direct values (on \(\Gamma\)) of the corresponding thermal potentials.

§ 11. FORMULATION OF THE MAIN RESULTS ON THE SMOOTHNESS OF THE SPECIAL THERMAL POTENTIALS \(P(x,t)\) AND \(Q(x,t)\)

ON SURFACES OF TYPE \(Л^{0,1,\frac{1+\alpha}{2}}_{1,\alpha,\alpha/2}\) AND \(Л^{1,\alpha,\alpha/2}_{1,1,\frac{1+\alpha}{2}}\)

In what follows we shall use the notation (2.1)—(2.3) from § 2 [1]. In addition, we put, for \(0<\alpha\leq \beta\leq 1\),

\[ \alpha^*=\min(\alpha,\beta'), \tag{11.1} \]

where \(\beta'\) is any number for which \(0<\beta'<\beta\).

Theorem 14. Let \(\Gamma\) be of type \(\Pi^{0,1,\frac{1+\alpha}{2}}_{1,\alpha,\alpha/2}\) \((0<\alpha\leq 1)\), let the field of directions \(\nu\) satisfy (10.3), with \([\nu_j]_0<+\infty\) \((j=1,2,\ldots,n)\), and let the density \(\varphi\) be bounded and integrable. Then the function \(P(\bar{x},\bar{t})\) (the special thermal single-layer potential (10.43)) belongs to the class \(H^{0,\alpha_0,\frac12}(\bar{D}^{\,B}_T)\), and the Hölder constants \(H_{0,x}(P)\), \(H_{0,t}(P)\) have the form \((C)|\varphi|_0\). Moreover,

\[ |P(\bar{x},\bar{t})|\leq (C)|\varphi|_0\sqrt{\bar{t}},\qquad (\bar{x},\bar{t})\in \bar{D}^{\,B}_T . \tag{11.2} \]

Theorem 15. Let \(\Gamma\) be of type \(\Pi^{0,1,\frac{1+\alpha}{2}}_{1,\alpha,\alpha/2}\), \(\nu_j\in H^{0,\beta,\beta/2}(\Gamma)\) \((j=1,2,\ldots,n;\ 0<\alpha\leq\beta\leq 1)\), let (10.3) be fulfilled, and let the density \(\varphi\) be a bounded integrable function. Then \(\bar{Q}(x,t)\) (the direct value of the thermal potential (10.44), where \((\bar{x},\bar{t})\equiv(x,t)\in\Gamma_t\)) belongs to the class \(H^{0,\alpha^*,\alpha^*/2}(\Gamma)\) (see (11.1)), and the Hölder constants \(H_{0,x}(\bar{Q})\), \(H_{0,t}(\bar{Q})\) have the form \((C)|\varphi|_0\). Moreover,

\[ |\bar{Q}(x,t)|\leq (C)|\varphi|_0\,t^{\alpha/2},\qquad (x,t)\in\Gamma . \tag{11.3} \]

Theorem 16. Let \(\Gamma\) be of type \(\Pi^{0,1,\frac{1+\alpha}{2}}_{1,\alpha,\alpha/2}\), \(\nu_j\in H^{0,\beta,\beta/2}(\Gamma)\) \((j=1,2,\ldots,n;\ 0<\alpha\leq\beta\leq 1)\), and let (10.3) be fulfilled. Suppose that \((\bar{x},\bar{t})\equiv(x,t)\), in tending to \((x,t)\in\Gamma_t\), lies on the normal \(N(x,t)\) to \(\Gamma_t\). Then, for continuous \(\varphi\),

\[ \lim_{(\bar{x},\bar{t})\to(x,t)\in\Gamma_t} Q(\bar{x},t) = \bar{Q}(x,t)-\frac{(2\sqrt{\pi})^n}{2}\,\varphi(x,t). \tag{11.4} \]

If, moreover, \(\varphi\in H^{0,\alpha,\alpha/2}(\Gamma)\) and

\[ |\varphi(y,\tau)|\leq |\varphi|_\alpha\,\tau^{\alpha/2},\qquad (y,\tau)\in\Gamma_\tau,\quad 0\leq \tau\leq T, \tag{11.5} \]

then

\[ \left| Q(\bar{x},t)-\bar{Q}(x,t)+\frac{(2\sqrt{\pi})^n}{2}\,\varphi(x,t) \right| \leq (C)|\varphi|_\alpha\, r^{\alpha_0}(\bar{x},x) \tag{11.6} \]

and

\[ |Q(\bar{x},t)|\leq (C)|\varphi|_\alpha\,t^{\alpha/2},\qquad (\bar{x},t)\in\bar{D}^{\,B}_T . \tag{11.7} \]

Theorem 17. Let \(\Gamma\) be of type \(\Pi^{1,\alpha,\alpha/2}_{1,1,\frac{1+\alpha}{2}}\), \(\nu_j\in H^{0,1,\frac{1+\beta}{2}}_{1,\beta,\beta/2}(\Gamma)\) \((j=1,2,\ldots,n;\ 0<\alpha\leq\beta\leq 1)\), \(\varphi\in H^{0,\alpha,\alpha/2}(\Gamma)\), and let (10.3) and (11.5) be fulfilled. Then \(\bar{Q}(x,t)\in H^{0,1,\frac{1+\alpha^*}{2}}_{1,\alpha^*,\alpha^*/2}(\Gamma)\), and \(H_{1,x}(\bar{Q})\) and \(H_{0,t}(\bar{Q})\) have the form \((C)|\varphi|_\alpha\). Moreover,

\[ |\bar{Q}(x,t)|\leq (C)|\varphi|_\alpha\, t^{\frac{1+\alpha}{2}}, \tag{11.8} \]

\[ \left| \frac{\partial \bar{Q}(x,t)}{\partial x_i} \right| \leq (C)|\varphi|_\alpha\, t^{\alpha/2} \qquad (i=1,2,\ldots,n-1),\quad (x,t)\in\Gamma . \]

Theorem 18. Let \(\Gamma\) be of type \(\Pi^{0,1,\frac{1+\alpha}{2}}_{1,\alpha,\alpha/2}\), \(\nu_j\in H^{0,\beta,\beta/2}(\Gamma)\) \((j=1,2,\ldots,n;\` <!-- source-page: 010 --> \(0<\alpha\leqslant\beta<1\)), \(\varphi\in H^{0,\alpha,\alpha/2}(\Gamma)\), and suppose (10.3) and (11.5) are satisfied. Then

\[ P\in H_{1,\alpha',\alpha'/2}^{0,1,\frac{1-\alpha'}2}(\overline{D_T^{\mathrm{B}}}), \]

where \(H_{0,t}(P)\), \(H_{1,x}(P)\), and \(H_{1,t}(P)\) have the form \((C)|\varphi|_\alpha\), and

\[ |P(\bar x,t)|\leqslant (C)|\varphi|_\alpha t^{\frac{1+\alpha}{2}}, \]

\[ \left|\frac{\partial P(\bar x,t)}{\partial x_j}\right| \leqslant (C)|\varphi|_\alpha t^{\alpha/2} \quad (j=1,2,\ldots,n),\qquad (\bar x,t)\in \overline{D_T^{\mathrm{B}}}. \tag{11.9} \]

Theorem 19. Let \(\Gamma\) be of type \(\Lambda_{1,1,\frac{1+\alpha}{2}}^{1,\alpha,\alpha/2}\), \(v_j\in H_{1,\beta,\beta/2}^{0,1,\frac{1+\beta}{2}}(\Gamma)\) \((j=1,2,\ldots,n;\)

\(0<\alpha\leqslant\beta\leqslant 1)\), \(\varphi\in H_{1,\alpha,\alpha/2}^{0,1,\frac{1+\alpha}{2}}(\Gamma)\), and suppose (11.5) is satisfied and

\[ |\varphi(y,\tau)|\leqslant |\varphi|_{1+\alpha}\,\tau^{\frac{1+\alpha}{2}}, \qquad 0\leqslant \tau\leqslant T, \]

\[ \left|\frac{\partial \varphi(y,\tau)}{\partial y_i}\right| \leqslant |\varphi|_{1+\alpha}\,\tau^{\alpha/2} \quad (i=1,2,\ldots,n-1),\qquad (y,\tau)\in\Gamma_\tau . \tag{11.10} \]

Then

\[ P(\bar x,t)\in H_{1,1,\frac{1+\alpha}{2}}^{1,\alpha',\alpha'/2}(\overline{D_T^{\mathrm{B}}}), \]

where \(H_{1,x}(P)\), \(H_{1,t}(P)\), and \(H_{0,t}(P)\) have the form \((C)|\varphi|_{1+\alpha}\). Moreover,

\[ \left|\frac{\partial P(\bar x,t)}{\partial x_j}\right| \leqslant (C)|\varphi|_{1+\alpha}t^{\frac{1+\alpha}{2}} \quad (j,k=1,2,\ldots,n),\qquad (\bar x,t)\in \overline{D_T^{\mathrm{B}}}, \]

\[ \left\{ \left|\frac{\partial^2P(\bar x,t)}{\partial x_j\,\partial x_k}\right|, \ \left|\frac{\partial P(\bar x,t)}{\partial t}\right| \right\} \leqslant (C)|\varphi|_{1+\alpha}t^{\alpha/2}. \tag{11.11} \]

Remark 11. If \(\Gamma\) is of type \(\Lambda_{1,1,\frac{1+\beta}{2}}^{1,\beta,\beta/2}\) and \(\beta\) is any number for which \(0<\alpha<\beta\leqslant 1\), then, with the remaining conditions of Theorem 19 retained,

\[ P\in H_{1,1,\frac{1+\alpha}{2}}^{1,\alpha,\alpha/2}(\overline{D_T^{\mathrm{B}}}). \]

Consider the following boundary-value problem with oblique derivative for the heat equation:

\[ \sum_{i=1}^{n}\frac{\partial^2 u}{\partial x_i^2} -\frac{\partial u}{\partial t} =f_1(\bar x,t), \qquad (\bar x,t)\in D_T^{\mathrm{B}}, \tag{11.12} \]

\[ u(\bar x,0)=f_2(\bar x), \qquad \bar x\in\Omega_0, \tag{11.13} \]

\[ \frac{\partial u(x,t)}{\partial \nu(x,t)} +b(x,t)u(x,t)=f_3(x,t), \qquad (x,t)\in\Gamma . \tag{11.14} \]

(The problem (11.12)—(11.14) with \(b(x,t)\equiv 0\) is the second boundary-value problem, and otherwise the third boundary-value problem with oblique derivative.)

Theorem 20. Let \(\Gamma\) be of type \(\Lambda_{1,1,\frac{1+\beta}{2}}^{1,\beta,\beta/2}\), \(v_j\in H_{1,\beta,\beta/2}^{0,1,\frac{1+\beta}{2}}(\Gamma)\) \((0<\alpha<\beta\leqslant 1,\ \beta\) arbitrary), and suppose (10.3) holds. Let

\[ f_1(\bar x,t)\in H^{0,\alpha,\alpha/2}(\bar D_T^{\mathrm{B}}),\quad f_2(\bar x)\in H_{1,1,\frac{1+\alpha}{2}}^{1,\alpha,\alpha/2}(\bar\Omega_0),\quad b(x,t),\, f_3(x,t)\in H_{1,\alpha,\alpha/2}^{0,1,\frac{1+\alpha}{2}}(\Gamma), \]

moreover \(f_i\) \((i=2,3)\) are compatible, by virtue of equation (11.12), on the edge \(\Omega_0\cap\Gamma\). Then there exists a solution of problem (11.12)—(11.14) from the class
\[ H_{1,1,\frac{1+\alpha}{2}}^{1,\alpha,\alpha/2}(\bar D_T^{\mathrm{B}}). \]

Consider the boundary-value problem with an oblique derivative for the linear parabolic equation of second order

\[ \sum_{i,j=1}^{n} a_{ij}(\bar x,t)\frac{\partial^2 u}{\partial x_i\partial x_j} +\sum_{i=1}^{n} b_i(\bar x,t)\frac{\partial u}{\partial x_i} +c(\bar x,t)u- \]

\[ -\frac{\partial u}{\partial t}=f_1(\bar x,t), \]

\[ (\bar x,t)\in D_T^{\mathrm{B}} \tag{11.15} \]

with the initial condition (11.13) and the boundary condition (11.14). The \((2+\alpha)\) a priori estimate of the solution of the second and third boundary-value problems (11.15), (11.13), (11.14) in the closed domain \(\bar D_T^{\mathrm{B}}\), obtained by the author jointly with V. N. Maslennikova in [5, 6], makes it possible, with the aid of theorem 20, to prove by the classical method of continuation with respect to a parameter the following theorem on the existence of a solution of problem (11.15), (11.13), (11.14).

Theorem 21. Let \(\Gamma\) be of type
\[ \Lambda_{1,1,\frac{1+\beta}{2}}^{1,\beta,\beta/2} \]
\((0<\alpha<\beta<1,\ \beta\ \text{arbitrary})\), and suppose that (10.3) holds. Let equation (11.15) be of parabolic type in \(\bar D_T^{\mathrm{B}}\), and let the coefficients \(a_{ij}, b_i, c\) belong to the class
\[ H^{0,\alpha,\alpha/2}(\bar D_T^{\mathrm{B}}). \]
Let \(f_i\) \((i=1,2,3)\) satisfy the conditions of theorem 20 and be compatible, by virtue of (11.15), on the edge \(\Omega_0\cap\Gamma\). Then there exists a solution of problem (11.15), (11.13), (11.14) from the class
\[ H_{1,1,\frac{1+\alpha}{2}}^{1,\alpha,\alpha/2}(\bar D_T^{\mathrm{B}}). \]

The proof of theorems 14—20 is given in §§ 12—16.

§ 12. PROOF OF THEOREMS 14, 15, AND 16

Under the assumptions of theorem 14, continuity of \(P(\bar x,t)\) in \(\bar D_T^{\mathrm{B}}\) was proved in [7]; therefore there exist direct values \(\bar P(x,t)=(P(x,t))_{\mathrm{B}}\) (for the notation see § 5 [1]). With the aid of Lemmas 11, 12 it is easy to prove (cf. § 3 of [1]) that
\[ \bar P(x,t)\in H^{0,\alpha_0,\frac12}(\Gamma). \]
But then the assertions of theorem 14 follow from the maximum principle.

For the proof of theorem 15 put (see (10.44), where \((\bar x,t)\equiv(x,t)\in\Gamma_t\)),

\[ \bar Q(x,t)=\sum_{i=1}^{2}\bar Q_i(x,t), \]

where

\[ \bar Q_1(x,t)=\int_{0}^{t}d\tau\iint_{\Gamma_\tau}\varphi(y,\tau) \left[ \frac{\overline{\partial P(r(y,x),\,t-\tau)}}{\partial\nu(x,t)} -\right. \]

\[ \left. -\frac{\overline{\partial P(r(y,x),\,t-\tau)}}{\partial\nu(y,\tau)} \right]\,d\sigma_y(\tau), \]

\[ \overline Q_2(x,t)=\int_0^t d\tau \iint_{\Gamma_\tau}\varphi(y,\tau)\, \frac{\overline{\partial P(r(y,x),\,t-\tau)}}{\partial \nu(y,\tau)}\,d\sigma_y(\tau). \]

By Lemma 10 (see Remark 10, §10) and Theorem 2 of §2 [1],

\[ \overline Q_2(x,t)\in H^{0,\alpha^0,\alpha/2}(\Gamma). \tag{12.1} \]

We pass to the local coordinate system \(\{\xi,\tau\}\) associated with the point \((x,t)\in\Gamma_t\) (see §1 of [1], in particular (1.10), (1.11)), and use the functions \(\varphi_0(\xi_1,\tau)\), \(\psi_1(\xi_1,\tau)\), \(\varphi_1(\xi_1,\tau)\) from (3.3), §3 [1], and \(\nu_j^{(1)}(\xi_1,\tau)=\nu_j(\xi_1,\tau)\omega(\rho)\).

Remark 12. Everywhere in the sequel, unless the contrary is specially stated, by \(\varphi\), \(\psi\), \(\nu_j\) we shall mean the smoothed functions \(\varphi_1\), \(\psi_1\), \(\nu_j^{(1)}\) with the indices omitted.

By Lemma 1 of §1 [1], Lemma 11 of §10, Remark 10 of §10, and the considerations of §3 [1], we see that it is enough to investigate the function

\[ \overline Q_{11}(x,t)\equiv \overline Q_{11}(x_1,t) =\int_{t_1}^t d\tau\int_{-\infty}^{+\infty} \varphi(\xi_1,\tau)\,\overline q_1(r(\xi,x),\,t-\tau)\,d\xi_1, \tag{12.2} \]

where (see (10.6), (10.7), (10.14)—(10.16)) \(\overline q_1(r(\xi,\overline x),\,t-\tau)\) is the direct value of the function

\[ q_1(r(\xi,\overline x),\,t-\tau) =\sum_{j=1}^2[\nu_j(x_1,t)-\nu_j(\xi_1,\tau)]\, \frac{\partial P(r(\xi,\overline x),\,t-\tau)}{\partial x_i} = \]

\[ =-\frac12\Bigl[(\nu(\xi_1,\tau),N(\xi_1,\tau))(\nu(x_1,t)-\nu(\xi_1,\tau),r_{\xi\overline x}) \]

\[ +(\nu(\xi_1,\tau),\nu(x_1,t)-\nu(\xi_1,\tau))(\mu(\xi_1,\tau),r_{\xi\overline x})\Bigr]\, g_{0,2}(r(\xi,\overline x),\,t-\tau) \]

\[ -(\nu(x_1,t)-\nu(\xi_1,\tau),\mu(\xi_1,\tau)) \left[1-\frac12(t-\tau)^{-1}(\mu(\xi_1,\tau),r_{\xi\overline x})^2\right] G_{3/2}(r(\xi,\overline x),\,t-\tau). \tag{12.3} \]

Making the substitution

\[ \xi_1-x_1=\pm \rho\quad \text{for}\quad \pm(\xi_1-x_1)>0, \tag{12.4} \]

from the estimates (10.32) and (1.17) (for \(\nu_j\) with \(\alpha=\beta\)) of §1 [1] we easily obtain the inequality

\[ |\overline Q_{11}(x,t)|\le \]

\[ \le (C)[\varphi]_0\int_{t_1}^t (t-\tau)^{\frac{\beta-3}{2}}\,d\tau \int_0^{+\infty} g_{0,0}(x\rho,\,2(t-\tau))\,d\rho \le (C)[\varphi]_0\,t^{\beta/2}, \tag{12.5} \]

from which, in view of (2.5) of §2 [1], estimate (11.3) follows. In view of (12.5), when studying the smoothness of the function (12.2) with respect to \(t\), one may restrict oneself (cf. §3 of [1]) to the case

\[ 0<\Delta t<t. \tag{12.6} \]

Put, as usual (cf. the decomposition (3.9) of §3 [1]) (see also (10.31)),

\[ Q_{11}(x_1,t+\Delta t)-\overline Q_{11}(x_1,t) \equiv \sum_{i=1}^3 \overline Q_{11i}(\Delta t)= \]

\[ \begin{aligned} \equiv {}& \int_{t-\Delta t}^{t+\Delta t} d\tau \int_{-\infty}^{+\infty} \varphi(\xi_1,\tau)\bar q_1(r(\xi,\hat x),t+\Delta t-\tau)\,d\xi_1 +\\ &+(-1)\int_{t-\Delta t}^{t} d\tau \int_{-\infty}^{+\infty} \varphi(\xi_1,\tau)\bar q_1(r(\xi,x),t-\tau)\,d\xi_1 +\\ &+\int_{t_1}^{t-\Delta t} d\tau \int_{-\infty}^{+\infty} \varphi(\xi_1,\tau) \left[\bar q_1(r(\xi,\hat x),t+\Delta t-\tau)-\right.\\ &\hspace{3.7cm}\left.-\bar q_1(r(\xi,x),t-\tau)\right]\,d\xi_1 . \end{aligned} \tag{12.7} \]

From (12.5) follows the estimate
\[ |\bar Q_{11i}(\Delta t)|\leq (C)[\varphi]_0|\Delta t|^{\beta/2} \quad \text{for } i=1,2 . \]

The use of the mean-value theorem, by virtue of Lemma 4 and (1.24), (1.17) (for \(\nu_j\) with \(a=\beta\)) from § 1 [1], as well as (10.35), gives the following inequality:
\[ \left|\bar q_1(r(\xi,\hat x),t+\Delta t-\tau)-\bar q_1(r(\xi,x),t-\tau)\right|\leq \]
\[ \leq (C)\left[(\Delta t)^{\beta/2}(t-\tau)^{-3/2}+ (\Delta t)^{\frac{1+\alpha}{2}}(t-\tau)^{-2+\frac{\beta}{2}} +\Delta t\,(t-\tau)^{\frac{\beta-5}{2}}\right] g_{0,0}\bigl(\varkappa\rho,\,2(t+\Delta t-\tau)\bigr), \]
whence, using the substitutions (12.4) and
\[ t-\tau=\eta,\qquad s=\frac{\varkappa^2\rho^2}{8\eta}, \tag{12.8} \]
we obtain
\[ |\bar Q_{113}(\Delta t)|\leq \]
\[ \leq (C)[\varphi]_0\int_{\Delta t}^{t} \left[(\Delta t)^{\beta/2}\eta^{-1} +(\Delta t)^{\frac{1+\alpha}{2}}\eta^{\frac{\beta-3}{2}} +\Delta t\,\eta^{-2+\frac{\beta}{2}}\right]\,d\eta \int_{0}^{+\infty}s^{-1/2}e^{-s}\,ds\leq \]
\[ \leq (C)[\varphi]_0|\Delta t|^{\beta'/2}, \]
whence
\[ |\bar Q_{11}(x_1,t+\Delta t)-\bar Q_{11}(x_1,t)| \leq (C)[\varphi]_0|\Delta t|^{\beta'/2}. \tag{12.9} \]

Let us now consider (see (1.10), § 1 of [1], and (10.37)) (cf. the decomposition (3.16) from § 3 [1])
\[ \bar Q_{11}(\delta,t)-\bar Q_{11}(0,t)=\sum_{i=1}^{3}\bar Q_{11}^{(i)}(\delta)\equiv \]
\[ \equiv \int_{t_1}^{t} d\tau \int_{|\xi_1-x_1|\leq 2\delta} \varphi(\xi_1,\tau)\bar q_1(r(\xi,x^\delta),t-\tau)\,d\xi_1+ \]
\[ +(-1)\int_{t_1}^{t}d\tau \int_{|\xi_1-x_1|\leq 2\delta} \varphi(\xi_1,\tau)\bar q_1(r(\xi,x),t-\tau)\,d\xi_1+ \]

\[ + \int_{t_1}^{t} d\tau \int_{|\xi_1-x_1|\geqslant 2\delta} \varphi(\xi_1,\tau)\left[\bar q_1(r(\xi,x^\delta),t-\tau)-\bar q_1(r(\xi,x),t-\tau)\right]\,d\xi_1. \tag{12.10} \]

Making the substitutions (12.4) and

\[ \xi_1-x_1=\pm \rho_1 \quad \text{for } \quad \pm(\xi_1-x_1)>0, \tag{12.11} \]

we obtain, in view of (1.17) (for \(v_j\) with \(\alpha=\beta\)) from § 1 [1] and Lemma 12 from § 10, with the aid of substitutions of the form (12.8) and \(\left(c=\dfrac{9}{8}\varkappa^2\right)\),

\[ c\delta^2\eta^{-1}=z, \tag{12.12} \]

\[ \left|\bar Q_{11}^{(i)}(\delta)\right| \leqslant (C)[\varphi]_0 \int_{0}^{t}\eta^{\frac{\beta-3}{2}}\,d\eta \int_{0}^{3\delta}\exp\left\{-\frac{c\rho_1^2}{9\eta}\right\}\,d\rho_1 \leqslant \]

\[ \leqslant (C)[\varphi]_0 \int_{0}^{t}\eta^{-1+\beta/2}\,d\eta \int_{0}^{c\delta^2\eta^{-1}} s^{-1/2}e^{-s}\,ds \leqslant \]

\[ \leqslant (C)[\varphi]_0\delta^\beta \int_{c\delta^2t^{-1}}^{+\infty} z^{-\frac{1+\beta}{2}}\,dz \int_{0}^{z}s^{-1/2}e^{-s}\,ds \leqslant \]

\[ \leqslant (C)[\varphi]_0\delta^{\beta_0} \quad \text{for } i=1. \]

An analogous estimate also holds for \(i=2\). Since, when \(|\xi_1-x_1|\geqslant 2\delta\), \(\rho_1\geqslant \delta\), we have

\[ \frac12\rho \leqslant (1-\lambda)\rho+\lambda\rho_1 \leqslant \frac32\rho \quad \text{for any } 0\leqslant \lambda\leqslant 1, \tag{12.13} \]

then, by Lemma 12 from § 10 and (3.25) from § 3 [1], we have

\[ \left|\bar q_1(r(\xi,x^\delta),t-\tau)-\bar q_1(r(\xi,x),t-\tau)\right| \leqslant \]

\[ \leqslant (C)\left[ \delta^\beta g_{0,3/2}\left(\frac{\varkappa}{2}\rho,2(t-\tau)\right) +\delta\, g_{0,2-\beta/2}\left(\frac{\varkappa}{2}\rho,2(t-\tau)\right) \right], \]

whence (cf. the derivation of (3.26) from § 3 [1]), with the aid of substitutions of the form (12.4), (12.8), (12.12),

\[ \left|\bar Q_{11}^{(3)}(\delta)\right|\leqslant \]

\[ \leqslant (C)[\varphi]_0\delta^\beta \int_{c\delta^2t^{-1}}^{+\infty} \left[z^{-1}+z^{-\frac{1+\beta}{2}}\right]\,dz \int_{z}^{+\infty}s^{-1/2}e^{-s}\,ds \leqslant (C)[\varphi]_0\delta^{\beta'}, \]

therefore

\[ \left|\bar Q_{11}(\delta,t)-\bar Q_{11}(0,t)\right| \leqslant (C)[\varphi]_0\delta^{\beta'}. \tag{12.14} \]

From (12.1), (12.9), (12.14) follows the assertion
\[ \bar Q(x,t)\in H^{0,\beta',\beta'/2}(\Gamma), \]
which completes the proof of Theorem 15.

The proof of formula (11.4) of Theorem 16 for continuous \(\varphi\) is carried out as in [7].

Now let \(\varphi \in H^{0,\alpha,\alpha/2}(\Gamma)\). In view of (12.1) and Theorems 5, 6 of § 2 [1], it suffices to estimate the expression

\[ \left| Q_{11}(\bar{x}, t) - \bar{Q}_{11}(x,t) \right| \]

for

\[ x_1=\bar{x}_1=0,\qquad \bar{x}_2=\psi(x_1,t)=0,\qquad x_2=\delta+\psi(x_1,t) \tag{12.15} \]

(see (1.10) from § 1 [1]), where \(\bar{Q}_{11}(x,t)\) is obtained from (12.2) by replacing \(\bar{q}_1\) by \(q_1\) from (12.3). But, by the mean-value theorem,

\[ \frac{\partial P(r(\xi,\bar{x}),t-\tau)}{\partial x_j} - \frac{\partial P(r(\xi,x),t-\tau)}{\partial x_j} = \int_0^1 \frac{\partial}{\partial \theta} \left[ \frac{\partial P(r(\xi,\bar{x}(\theta)),t-\tau)}{\partial x_j} \right]\,d\theta, \tag{12.16} \]

where

\[ r_{\xi\bar{x}(\theta)} = \{x_1-\xi_1,\ \theta\delta+\psi(x_1,t)-\psi(\xi_1,\tau)\} \]

and

\[ \frac{\partial r_{\xi\bar{x}(\theta)}}{\partial\theta}=\{0,\delta\}. \tag{12.17} \]

With the help of Lemma 11 from § 10, Lemma 4 from § 1 [1], and (1.17) (for \(\nu_j\) with \(\alpha=\beta\)) of § 1 [1], from (12.3), (12.16), (12.17) we have

\[ \left| q_1(r(\xi,\bar{x}),t-\tau)-\bar{q}_1(r(\xi,x),t-\tau)\right| \leq \]

\[ \leq (C)\int_0^1 q_1(\theta,\rho,t-\tau)\,d\theta = \]

\[ = (C)\delta\,\sigma_{0,2-\frac{\beta}{2}}(x\rho,2(t-\tau)) \int_0^1 \sigma_{0,0}\bigl(|\theta\delta-\psi(\xi_1,\tau)|,2(t-\tau)\bigr)\,d\theta. \]

For what follows we shall use the method of deriving estimates (4.23) from § 4 [1]. Choose (cf. (4.17*), (4.17), (4.18) from § 4 [1]) \(t(\theta)\) so that

\[ |\psi(0,t(\theta))|=\frac12\,\theta\delta \tag{12.18} \]

and, for \(t_1\leq t(\theta)<\tau\leq t\), the inequalities hold

\[ |\psi(0,\tau)|\leq \frac12\,\theta\delta \quad\text{and}\quad |\theta\delta-\psi(\xi_1,\tau)|\geq k\theta\delta \]

for \(\rho<k\theta\delta,\quad k=2^{-5/2}\).

\[ \tag{12.19} \]

Using decompositions of the form (4.19) from § 4 [1], we obtain

\[ \left| Q_{11}(\bar{x},t)-\bar{Q}_{11}(x,t)\right| \leq (C)\|\varphi\|_0\sum_{i=1}^{3} Q^{(i)}(\delta), \]

where, in view of (12.4),

\[ Q^{(1)}(\delta)=\int_0^1 d\theta\int_{t_1}^{t}d\tau\int_{k\theta\delta}^{+\infty}q_1(\theta,\rho,t-\tau)\,d\rho, \]

\[ Q^{(2)}(\delta)=\int_0^1 d\theta\int_{t(\theta)}^{t}d\tau\int_0^{k\theta\delta}q_1(\theta,\rho,t-\tau)\,d\rho, \]

\[ Q^{(3)}(\delta)=\int_0^1 d\theta\int_{t_1}^{t(\theta)}d\tau\int_0^{k\theta\delta}q_1(\theta,\rho,t-\tau)\,d\rho. \]

By means of changes of variables of the form (12.8), (12.12), we have

\[ Q^{(1)}(\delta)\leqslant (C)\delta\int_0^1d\theta\int_0^t\eta^{-2+\frac{\beta}{2}}\,d\eta \int_{k\theta\delta}^{+\infty}g_{0,0}(\chi,\rho,2(t-\tau))\,d\rho\leqslant \]

\[ \leqslant (C)\delta\int_0^1d\theta\int_0^t\eta^{\frac{\beta-3}{2}}\,d\eta \int_{c\theta^2\delta^2\eta^{-1}}^{+\infty}s^{-\frac12}e^{-s}\,ds \leqslant (C)\delta^\beta. \]

Using (12.19), we have

\[ Q^{(22)}(\delta)\leqslant \]

\[ \leqslant (C)\delta\int_0^1d\theta\int_{t(\theta)}^t (t-\tau)^{-2+\frac{\beta}{2}}g_{0,0}(c\theta\delta,t-\tau)\,d\tau \int_0^{k\theta\delta}g_{0,0}(\chi\rho,2(t-\tau))\,d\rho\leqslant \]

\[ \leqslant (C)\delta^\beta\int_0^1\theta^{\beta-1}\,d\theta \int_{4^{-1}(c\theta\delta)^2t^{-1}} z^{-\frac{1+\beta}{2}}e^{-z}\,dz \leqslant (C)\delta^\beta. \]

Finally, using (12.18), (12.15) and
\[ |\psi(0,t(\theta))-\psi(0,t)|\leqslant (C)|t-t(\theta)|^{\frac{1+\alpha}{2}}, \]
we obtain

\[ Q^{(3)}(\delta)\leqslant \]

\[ \leqslant (C)\delta\int_0^1d\theta\int_{t_1}^{t(\theta)} (t-\tau)^{-2+\frac{\beta}{2}}\,d\tau \int_0^{k\theta\delta}g_{0,0}(\chi\rho,2(t-\tau))\,d\rho\leqslant \]

\[ \leqslant (C)\delta\int_0^1d\theta\int_{t_1}^{t(\theta)} (t-\tau)^{\frac{\beta-3}{2}}\,d\tau\leqslant \]

\[ \leqslant (C)\delta^\beta\int_0^1\theta^{\beta-1} \left|\psi(0,t(\theta))-\psi(0,t)\right|^{1-\beta} \left|(t-t(\theta))^{\frac{\beta-1}{2}}-\right. \]

\[ \left.-(t-t_1)^{\frac{\beta-1}{2}}\right|\,d\theta \leqslant (C)\delta^\beta, \]

whence

\[ |\overline{Q}_{11}(\bar{x},t)-\overline{Q}_{11}(x,t)| \leqslant (C)|\varphi|_0\delta^\beta, \tag{12.20} \]

which completes the proof of estimate (11.6).

With the aid of estimate (1.17) (for \(\nu_j\) with \(\alpha=\beta\)) from § 1 [1], (2.10) of Theorem 6 from § 2 [1], and Lemma 11 from § 10, it is easy to prove the inequality

\[ |Q(\bar{x},t)|\leq (C)|\varphi|_0,\qquad (\bar{x},t)\in \bar{D}^{\,B}_T, \]

whence, in view of (11.5), (11.7) follows, which completes the proof of Theorem 16.

§ 13. PROOF OF THEOREM 17

By virtue of (12.1) and Theorem 10 from § 7 [2], it suffices to prove (see (12.2)) that

\[ \bar{Q}_{11}(x_1,t)\in H^{0,1,\frac{1+\alpha^*}{2}}_{1,\alpha^*,\frac{\alpha^*}{2}},\qquad |\bar{Q}_{11}(x_1,t)|\leq (C)|\varphi|_\alpha t^{\frac{1+\alpha}{2}}, \]

\[ \left|\frac{\partial \bar{Q}_{11}(x_1,t)}{\partial x_1}\right| \leq (C)|\varphi|_\alpha t^{\alpha/2}. \tag{13.1} \]

By virtue of Remarks 9, 10 from § 10,

\[ \nu_2(x_1,\tau)=-\mu_1(x_1,\tau)\geq d_0>0 \quad\text{when } t-\tau\leq d_1. \tag{13.2} \]

We note the equality

\[ \begin{aligned} \nu(x_1,t)-\nu(\xi_1,\tau) ={}&\nu(x_1,t)-\nu(x_1,\tau) -(\xi_1-x_1)\frac{\partial \nu(x_1,\tau)}{\partial x_1}+{}\\ &+(\xi_1-x_1)\int_0^1 \left( \frac{\partial \nu(x_1,\tau)}{\partial x_1} - \frac{\partial \nu\bigl(x_1+\theta(\xi_1-x_1),\tau\bigr)}{\partial x_1} \right)\,d\theta . \end{aligned} \tag{13.3} \]

Let us set, taking into account (see (1.10) in § 1 [1]),

\[ x_1=0,\qquad x_2=\psi(x_1,t)=0, \tag{13.4} \]

\[ \bar{Q}_{11}(x_1,t)= \sum_{i=1}^{5}\left[ \bar{Q}^{(0)}_{11i}(x_1,t)+\bar{Q}^{(1)}_{11i}(x_1,t) \right], \tag{13.5} \]

where

\[ \bar{Q}^{(1)}_{11i}(x_1,t)=\bar{Q}_{11i}(x_1,t)-\bar{Q}^{(0)}_{11i}(x_1,t) \]

and (see (1.21) in § 1 [1])

\[ \begin{aligned} \bar{Q}_{111}(x_1,t) ={}&\frac{1}{2}\int_{t_1}^{t} d\tau \int_{-\infty}^{+\infty} \Bigl\{ \varphi(\xi_1,\tau) \Bigl[ (\nu(\xi_1,\tau),N(\xi_1,\tau)) \bigl(\nu_1(0,t)-{}\\ &\qquad\qquad -\nu_1(\xi_1,\tau)\bigr) +\mu_1(\xi_1,\tau) \bigl(\nu(\xi_1,\tau),\nu(0,t)-\nu(\xi_1,\tau)\bigr) \Bigr] \Bigr\}_1 \times{}\\ &\qquad\qquad\times g_{1,2}(\xi_1,t-\tau)\,g(\psi;0,t)\,d\xi_1 . \end{aligned} \tag{13.6} \]

\[ \begin{aligned} \bar{Q}_{112}(x_1,t) ={}&-\frac{1}{2}\int_{t_1}^{t} d\tau \int_{-\infty}^{+\infty} \Bigl\{ \varphi(\xi_1,\tau) \Bigl[ (\nu(\xi_1,\tau),N(\xi_1,\tau)) \bigl(\nu_2(0,t)-{}\\ &\qquad\qquad -\nu_2(\xi_1,\tau)\bigr) +\mu_2(\xi_1,\tau) \bigl(\nu(\xi_1,\tau),\nu(0,t)-\nu(\xi_1,\tau)\bigr) \Bigr] \bigl(\psi(0,t)-{}\\ &\qquad\qquad -\psi(\xi_1,\tau)\bigr) \Bigr\}_2 \,g_{0,2}(\xi_1,t-\tau)\,g(\psi;0,t)\,d\xi_1 . \end{aligned} \tag{13.7} \]

\[ \overline Q_{113}(x,t)=-\int_{t_1}^{t}d\tau\int_{-\infty}^{+\infty} \{\varphi(\xi_1,\tau)(\mu(\xi_1,\tau),\,\nu(0,t)-\nu(\xi_1,\tau))\}_3 \times \]
\[ \times G_{3/2}(r(\xi,x),\,t-\tau)d\xi_1, \tag{13.8} \]

\[ \overline Q_{114}(x,t)=-\frac12\int_{t_1}^{t}d\tau\int_{-\infty}^{+\infty} \{\varphi(\xi_1,\tau)\mu_1^2(\xi_1,\tau)\times \]
\[ \times(\mu(\xi_1,\tau),\,\nu(0,t)-\nu(\xi_1,\tau))\}_4 \,\xi_1^2 G_{5/2}(r(\xi,x),\,t-\tau)d\xi_1, \tag{13.9} \]

\[ \overline Q_{115}(x,t)=\frac12\int_{t_1}^{t}d\tau\int_{-\infty}^{+\infty} \{\varphi(\xi_1,\tau)\mu_2^2(\xi_1,\tau)(\mu(\xi_1,\tau),\,\nu(0,t)-\nu(\xi_1,\tau))\times \]
\[ \times(\psi(0,t)-\psi(\xi_1,\tau))^2\}_5 G_{5/2}(r(\xi,x),\,t-\tau)d\xi_1, \tag{13.10} \]

\[ \overline Q_{11j}^{(0)}(x_1,t)=\sum_{i=1}^{2}\overline Q_{11j}^{(0i)}(x_1,t)\quad (j=1,2,\ldots,5), \tag{13.11} \]

where \(\overline Q_{11j}^{(01)}(x_1,t)\) is obtained from \(\overline Q_{11j}(x_1,t)\), first, by replacing, in the representations (13.6), (13.7) (for \(j=1,2\)), \(g(\psi;0,t)\) by \(1\), and by replacing, in the representations (13.8)—(13.10) (for \(j=3,4,5\)), \(G_m\) by the expression

\[ G_{0,m}(\xi_1,t-\tau)=g_{0,m}(\nu_2(0,\tau)^\varepsilon\xi_1,t-\tau) \left\{\Phi\left(-\frac{\nu_1(0,\tau)\xi_1}{2\sqrt{t-\tau}}\right)-\frac{\sqrt\pi}{2}\right\} \tag{13.12} \]

and, secondly, by the formal replacement in \(\{\ \}_j\), in the representations (13.6)—(13.10), of the argument \(\xi_1\) by \(0\). To obtain \(\overline Q_{11j}^{(02)}(x_1,t)\) \((j=1,2,\ldots,5)\), one must, in the expressions introduced above for \(\overline Q_{11j}^{(01)}(x_1,t)\), formally replace expressions of the form \(\nu_i(0,t)-\nu_i(0,\tau)\) \((i=1,2)\) and \(\nu(0,t)-\nu(0,\tau)\) by \(\dfrac{\partial\nu_i(0,\tau)}{\partial x_1}\) \((i=1,2)\) and \(\dfrac{\partial\nu(0,\tau)}{\partial x_1}\), respectively, multiplying \(\{\ \}_j\) in this case by \(-\xi_1\).

If, in the expressions obtained, one uses the substitutions (12.4) and (see (13.2))
\(\rho=2\nu_2^{-1}(0,\tau)\sqrt{t-\tau}\sqrt z\), then we easily obtain

\[ \overline Q_{111}^{(01)}(x_1,t)=\overline Q_{112}^{(02)}(x_1,t)\equiv0. \tag{13.13} \]

\[ \overline Q_{111}^{(02)}(x_1,t)=-2\sqrt\pi\int_{t_1}^{t}(t-\tau)^{-1/2}\varphi(0,\tau) \left[(\nu(0,\tau),N(0,\tau))\times\right. \]
\[ \left.\times\frac{\partial\nu_1(0,\tau)}{\partial x_1} +\mu_1(0,\tau)\left(\nu(0,\tau),\frac{\partial\nu(0,\tau)}{\partial x_1}\right)\right]d\tau, \tag{13.14} \]

\[ \overline Q_{112}^{(01)}(x_1,t)=-\sqrt\pi\int_{t_1}^{t}(t-\tau)^{-3/2}\varphi(0,\tau) \left[(\nu(0,\tau),N(0,\tau))(\nu_2(0,t)-\right. \]
\[ \left.-\nu_2(0,\tau))+\mu_2(0,\tau)(\nu(0,\tau),\nu(0,t)-\nu(0,\tau))\right]\times \]
\[ \times(\psi(0,t)-\psi(0,\tau))\,d\tau, \tag{13.15} \]

\[ \bar Q_{113}^{(01)}(x_1,t) = \pi \int_{t_1}^{t} (t-\tau)^{-1}\varphi(0,\tau)v_2^{-1}(0,\tau) \bigl(\mu(0,\tau),\,v(0,t)- \]
\[ {}-v(0,\tau)\bigr)\,d\tau, \tag{13.16} \]

\[ \bar Q_{113}^{(02)}(x_1,t) = -4\int_{t_1}^{t} (t-\tau)^{-\frac12}\varphi(0,\tau)v_2^{-2}(0,\tau)\times \]
\[ {}\times \left(\mu(0,\tau),\,\frac{\partial v(0,\tau)}{\partial x_1}\right) \Phi_0(\tau)\,d\tau, \tag{13.17} \]

\[ \bar Q_{114}^{(01)}(x_1,t) = -\pi\int_{t_1}^{t} (t-\tau)^{-1}\varphi(0,\tau)v_2^{-3}(0,\tau)\times \]
\[ {}\times \bigl(\mu(0,\tau),\,v(0,t)-v(0,\tau)\bigr)\Phi_1(\tau)\,d\tau, \tag{13.18} \]

\[ \bar Q_{114}^{(02)}(x_1,t) = 8\int_{t_1}^{t} (t-\tau)^{-\frac12}\varphi(0,\tau)v_2^{-4}(0,\tau)\mu_1^2(0,\tau)\times \]
\[ {}\times \left(\mu(0,\tau),\,\frac{\partial v(0,\tau)}{\partial x_1}\right) \Phi_1(\tau)\,d\tau, \tag{13.19} \]

\[ \bar Q_{115}^{(01)}(x_1,t) = -\frac{\pi}{2}\int_{t_1}^{t} (t-\tau)^{-2}\varphi(0,\tau)v_2^{-1}(0,\tau)\mu_2^2(0,\tau)\times \]
\[ {}\times \bigl(\mu(0,\tau),\,v(0,t)-v(0,\tau)\bigr) \bigl(\psi(0,t)-\psi(0,\tau)\bigr)^2\,d\tau, \tag{13.20} \]

\[ \bar Q_{115}^{(02)}(x_1,t) = 2\int_{t_1}^{t} (t-\tau)^{-\frac32}\varphi(0,\tau)v_2^{-2}(0,\tau)\mu_2^2(0,\tau)\times \]
\[ {}\times \left(\mu(0,\tau),\,\frac{\partial v(0,\tau)}{\partial x_1}\right) \bigl(\psi(0,t)-\psi(0,\tau)\bigr)^2\Phi_0(\tau)\,d\tau, \tag{13.21} \]

where

\[ \Phi_l(\tau) = \int_{0}^{+\infty} z^l\Phi\left(-\frac{v_1(0,\tau)\sqrt z}{v_2(0,\tau)}\right)\,dz. \tag{13.22} \]

From estimates (11.5), (6.9), (6.10) of § 6 [2] and the condition

\[ v_j\in H_{1,\beta,\beta/2}^{0,\,1,\,\frac{1+\beta}{2}}(\Gamma) \tag{13.23} \]

we have

\[ \left|\bar Q_{11j}^{(0)}(x_1,t)\right| \leqslant (C)|\varphi|_{\alpha}\,t^{\frac{1+\alpha}{2}} \qquad (j=1,2,\ldots,5). \tag{13.24} \]

Moreover, from (13.22) it follows that

\[ \left|\Phi_l(\tau+\Delta\tau)-\Phi_l(\tau)\right| \leqslant (C)|\Delta\tau|^{\frac{1+\beta}{2}} \qquad (l=0,1). \tag{13.25} \]

From the estimates (1.17) of § 1 [1], (11.5), and the conditions (13.23) and (13.25), by virtue of Lemma 3 of § 1 [1] we have

\[ \left|\bar Q_{11j}^{(02)}(x_1,t+\Delta t)-\bar Q_{11j}^{(02)}(x_1,t)\right| \leq (C)|\varphi|_\alpha |\Delta t|^{\frac{1+\alpha^0}{2}} \]

\[ (j=1,\,3,\,4). \]

In the following estimates one may, in view of (13.24), restrict oneself to the case (12.6). We introduce the usual expansions (cf. (8.12) in § 8 of [2] or (12.7))

\[ \bar Q_{11j}^{(0k)}(x_1,t+\Delta t)-\bar Q_{11j}^{(0k)}(x_1,t) = \sum_{i=1}^{3}\bar Q_{11j}^{(0k)(i)}(\Delta t) \]

\[ (\text{for } k=2;\ j\ne 1,\,3,\,4), \]

from (13.24) we have

\[ \left|\bar Q_{11j}^{(0k)(i)}(\Delta t)\right| \leq (C)|\varphi|_\alpha |\Delta t|^{\frac{1+\alpha}{2}} \qquad (i=1,\,2). \]

With the aid of condition (13.23) and estimate (6.9) of § 6 [2], in view of (12.6), we obtain without difficulty

\[ \left|\bar Q_{11j}^{(0k)(3)}(\Delta t)\right| \leq (C)|\varphi|_0 |\Delta t|^{\frac{1+\beta'}{2}}, \]

whence

\[ \left|\bar Q_{11j}^{(0)}(x_1,t+\Delta t)-\bar Q_{11j}^{(0)}(x_1,t)\right| \leq (C)|\varphi|_\alpha |\Delta t|^{\frac{1+\alpha^*}{2}} \tag{13.26} \]

\[ (j=1,\,2,\ldots,\,5). \]

The use of conditions (13.23), (1.17) of § 1 [1], and the inequalities (see (13.3), (13.4))

\[ \left| [v_j(x_1,t)-v_j(\xi,\tau)] - \left[ v_j(x_1,t)-v_j(x_1,\tau)-(\xi_1-x_1)\frac{\partial v_j(x_1,\tau)}{\partial x_1} \right] \right| \leq \]

\[ \leq (C)\rho^{1+\beta} \tag{13.27} \]

and (see (6.9), (6.10) in § 6 [2] and (1.21) in § [1])

\[ |1-g(\psi;x,t)|\leq (C)(t-\tau)^{-1}(\rho^2+t-\tau)^2 \tag{13.28} \]

gives the estimate (see (13.5)—(13.7), (13.11), (13.13)—(13.15))

\[ \left|\bar Q_{11j}^{(1)}(x_1,t)\right| \leq (C)|\varphi|_\alpha t^{\frac{1+\alpha}{2}} \qquad (j=1,\,2). \tag{13.29} \]

Lemma 13. If \(\Gamma\) is of type \(\Lambda_{1,1,\frac{1+\alpha}{2}}^{1,\alpha,\alpha/2}\) and \(v_j\in H_{1,1,\frac{1+\beta}{2}}^{0,\beta,\beta/2}(\Gamma)\) \((0<\alpha\leq \beta\leq 1)\), then (see (10.15), (13.12)) for (10.31), (13.4)

\[ |A_m| \equiv \left|G_m(r(\xi,\hat x),t+\Delta t-\tau)-G_{0,m}(\xi_1,t+\Delta t-\tau)\right| \leq \]

\[ \leq (C)g_{0,m+\frac12}(\chi\rho,\,2(t+\Delta t-\tau)) \]

(the estimate is preserved when \((\hat x,t+\Delta t)\) is replaced by \((x,t)\)),

\[ |B_m| \equiv \left| G_m(r(\xi,\hat x),t+\Delta t-\tau)-G_{0,m}(\xi_1,t+\Delta t-\tau) \right| - \]

\[ -\left|G_m(r(\xi,x),\,t-\tau)-G_{0,m}(\xi_1,\,t-\tau)\right|\leq \]

\[ \leq (C)\Delta t\,(t-\tau)^{-m-\frac12}g_{0,0}\bigl(\varkappa\rho,\,2(t+\Delta t-\tau)\bigr). \]

The proof of Lemma 13 is carried out by the method used in the derivation of (10.19) of Lemma 11, if one uses the representations

\[ A_m=\int_0^1 \frac{\partial}{\partial\theta}A_m(1,\theta)\,d\theta,\qquad B_m=\int_0^1 \frac{\partial}{\partial z}\left[\int_0^1 \frac{\partial}{\partial\theta}A_m(z,\theta)\,d\theta\right]dz, \]

where

\[ A_m(z,\theta)=(t+z\Delta t-\tau)^{-m}\times \]

\[ \times \exp\left\{-\frac{\left[v_2(\theta\xi_1,\tau)\xi_1+\theta v_1(\theta\xi_1,\tau)\bigl(\psi(0,t+z\Delta t)-\psi(\xi_1,\tau)\bigr)\right]^2}{4(t+z\Delta t-\tau)}\right\}\times \]

\[ \times\left\{\Phi\left(\frac{-v_1(\theta\xi_1,\tau)\xi_1+\theta v_2(\theta\xi_1,\tau)\bigl(\psi(0,t+z\Delta t)-\psi(\xi_1,\tau)\bigr)} {2\sqrt{t+z\Delta t-\tau}}\right)-\frac{\sqrt{\pi}}{2}\right\}. \]

From Lemmas 11, 13, the estimates (13.27) and (6.9), (6.10) of § 6 [2], and the conditions (13.23) and \(\varphi\in H^{0,\alpha,\alpha/2}(\Gamma)\), we obtain (see (13.5), (13.8)—(13.10), (13.16)—(13.21))

\[ \left|\overline Q_{11j}^{(1)}(x_1,t)\right|\leq (C)|\varphi|_\alpha\,t^{\frac{1+\alpha}{2}} \qquad (j=3,4,5). \tag{13.30} \]

The estimates (13.29), (13.30) make it possible, in the subsequent estimates, to restrict ourselves to the case (12.6). We introduce the decomposition (cf. (12.7))

\[ \overline Q_{11j}^{(1)}(x_1,t+\Delta t)-\overline Q_{11j}^{(1)}(x_1,t) =\sum_{i=1}^3 \overline Q_{11ji}^{(1)}(\Delta t). \]

It follows from (13.29), (13.30) that

\[ \left|\overline Q_{11ji}^{(1)}(\Delta t)\right|\leq (C)|\varphi|_\alpha|\Delta t|^{\frac{1+\alpha}{2}} \qquad (i=1,2). \]

With the aid of Lemma (7) from § 6 [2], Lemmas 12 and 13, condition (13.23), the estimates (6.9), (6.10) from § 6 [2], and (1.23), (1.24) from § 1 [1], by the methods of [2] (cf. the derivation of (8.25) in § 8 [2]) we obtain, in view of (12.6), the estimate

\[ \left|\overline Q_{11j3}^{(1)}(\Delta t)\right| \leq (C)|\varphi|_\alpha|\Delta t|^{\frac{1+\beta'}{2}} \qquad \text{for } j=1,2,\ldots,5, \]

whence

\[ \left|\overline Q_{11i}^{(1)}(x_1,t+\Delta t)-\overline Q_{11i}^{(1)}(x_1,t)\right| \leq (C)|\varphi|_\alpha|\Delta t|^{\frac{1+\alpha^*}{2}} \tag{13.31} \]

\[ (i=1,2,\ldots,5). \]

From (13.26), (13.31) follows the estimate

\[ \left|\overline Q_{11}(x_1,t+\Delta t)-\overline Q_{11}(x_1,t)\right| \leq (C)|\varphi|_\alpha|\Delta t|^{\frac{1+\alpha^*}{2}}. \tag{13.32} \]

Now put

\[ \overline Q_{11}(x_1,t)=\sum_{i=2}^3 \overline Q_{11}^{(i)}(x_1,t)\equiv \]

\[ \equiv \sum_{i=2}^3 \int_{t_1}^{t} d\tau\int_{-\infty}^{+\infty} \varphi(\xi_1,\tau)\,\overline q_i(x_1,t;\xi_1,\tau)\,d\xi_1, \tag{13.33} \]

where (cf. (12.3))

\[ \overline q_i(x_1,t;\xi_1,\tau)= \sum_{j=1}^{2}\frac{\overline{\partial P(r(\xi,x),\,t-\tau)}}{\partial x_j} \begin{cases} \bigl(v_j(x_1,t)-v_j(x_1,\tau)\bigr), & \text{for } i=2,\\ \bigl(v_j(x_1,\tau)-v_j(\xi_1,\tau)\bigr), & \text{for } i=3. \end{cases} \tag{13.34} \]

With the aid of the estimates (10.32), (10.33) of Lemma 12 and condition (13.23), it is easily shown that

\[ \frac{\partial \overline Q_{11}^{(2)}(x_1,t)}{\partial x_1} = \sum_{i=1}^{2}\overline Q_{11}^{(2i)}(x_1,t) = \sum_{i=1}^{2}\int_{t_1}^{t}d\tau\int_{-\infty}^{+\infty} \varphi(\xi_1,\tau)\,\overline q_2^{(i)}(x_1,t;\xi_1,\tau)\,d\xi_1, \]

where

\[ \overline q_2^{(1)}(x_1,t;\xi_1,\tau) = \sum_{j=1}^{2} \left( \frac{\partial v_j(x_1,t)}{\partial x_1} - \frac{\partial v_j(x_1,\tau)}{\partial x_1} \right) \frac{\overline{\partial P(r(\xi,x),\,t-\tau)}}{\partial x_j}, \]

\[ \overline q_2^{(2)}(x_1,t;\xi_1,\tau) = \sum_{j=1}^{2} \bigl(v_j(x_1,t)-v_j(x_1,\tau)\bigr) \frac{\partial}{\partial x_1} \left[ \frac{\overline{\partial P(r(\xi,x),\,t-\tau)}}{\partial x_j} \right], \]

moreover (cf. the proof of Theorem 15 in § 12)

\[ \overline Q_{11}^{(21)}(x_1,t)\in H^{0,\alpha^*,\,\frac{\alpha^*}{2}} \tag{13.35} \]

and

\[ \left|\overline Q_{11}^{(2i)}(x_1,t)\right|\le (C)|\varphi|_0\,t^{\frac{\alpha}{2}} \qquad (i=1,2). \tag{13.36} \]

With the aid of the mean-value theorem, Lemmas 11 and 12, condition (13.23), the estimates (6.3), (6.9), (6.10) from § 6 [2], and (1.23), (1.24) from § 1 [1], we obtain, by virtue of (12.6),

\[ \left| \overline q_2^{(2)}(x_1,t+\Delta t;\xi_1,\tau) - \overline q_2^{(2)}(x_1,t;\xi_1,\tau) \right| \le \]

\[ \le (C)\left[ (\Delta t)^{\frac{\alpha}{2}}(t-\tau)^{\frac{\beta-3}{2}} + (\Delta t)^{\frac{1+\beta}{2}}(t-\tau)^{-2} + \Delta t\,(t-\tau)^{\frac{\beta-5}{2}} \right] g_{0,0}\bigl(\varkappa\rho,\,2(t+\Delta t-\tau)\bigr), \]

whence, using (13.36), (12.6), and expansions of the form (12.7), we obtain the estimate

\[ \left| \overline Q_{11}^{(22)}(x_1,t+\Delta t) - \overline Q_{11}^{(22)}(x_1,t) \right| \le (C)|\varphi|_0(\Delta t)^{\frac{\alpha}{2}}. \tag{13.37} \]

Further, from condition (13.23) we have

\[ \left| [v_j(\delta,t)-v_j(\delta,\tau)] - [v_j(0,t)-v_j(0,\tau)] \right| = \]

\[ = \left| \delta\int_{0}^{1} \left[ \frac{\partial v_j(z\delta,t)}{\partial x_1} - \frac{\partial v_j(z\delta,\tau)}{\partial x_1} \right]dz \right| \le (C)\delta(t-\tau)^{\frac{\beta}{2}}, \]

whence, with the aid of the mean-value theorem and Lemma 12, we have, provided (12.13) is satisfied,

\[ \left| \bar q_{2}^{(2)}(x_1,\ t+\Delta t;\ \xi_1,\tau) - \bar q_{2}^{(2)}(x_1,\ t;\ \xi_1,\tau) \right| \le (C)\,\delta\, g_{0,\,2-\frac{\beta}{2}}(\chi \rho,\ 2(t-\tau)). \]

This inequality, with the aid of an expansion of the form (12.10), makes it possible to obtain the estimate

\[ \left| \bar Q_{11}^{(22)}(\delta,t)-\bar Q_{11}^{(22)}(0,t) \right| \le (C)\,|\varphi|_0\,\delta^{\alpha^\circ}. \tag{13.38} \]

From (13.33), (13.35), (13.37), (13.38) it follows that

\[ \frac{\partial \bar Q_{11}^{(22)}(x_1,t)}{\partial x_1} \in H^{0,\alpha^*,\,\frac{\alpha^*}{2}}. \tag{13.39} \]

Set (see (13.33), (13.34); cf. also [10], Supplement IV)

\[ \delta^{-1} \left[ \bar Q_{11}^{(3)}(x_1+\delta,t)-\bar Q_{11}^{(3)}(x_1,t) \right] = \sum_{i=1}^{2} q_i(\delta), \]

where

\[ q_1(\delta) = \delta^{-1} \int_{t_1}^{t}\varphi(x_1,\tau)\,d\tau \int_{0}^{+\infty} d\rho \int_{0}^{1} \frac{\partial}{\partial z}\, \hat q_3(x_1+z\delta,\ t;\ \rho,\tau)\,dz, \]

\[ q_2(\delta) = \delta^{-1} \int_{t_1}^{t} d\tau \int_{-\infty}^{+\infty} [\varphi(\xi_1,\tau)-\varphi(x_1,\tau)]\,d\xi_1 \times \]

\[ \times \int_{0}^{1} \frac{\partial}{\partial z}\, \bar q_3(x_1+z\delta,\ t;\ \xi_1,\tau)\,dz \]

and (see (13.34) and (12.4))

\[ \hat q_3(x_1+z\delta,\ t;\ \rho,\tau) \equiv \sum_{i=1}^{2} \bar q_3(x_1+z\delta,\ t;\ x_1+z\delta+s_i\rho,\tau), \qquad s_i=(-1)^{i+1}. \]

It is easy to see that, under the hypotheses of Theorem 17,

\[ \lim_{\delta\to 0} q_1(\delta) = q_1(x_1,t) \equiv \]

\[ \equiv \int_{t_1}^{t}\varphi_1(x_1,\tau)\,d\tau \int_{0}^{+\infty} \frac{\partial}{\partial x_1}\, \hat q_3(x_1,\ t;\ \rho,\tau)\,d\rho, \]

\[ \lim_{\delta\to 0} q_2(\delta) = q_2(x_2,t) \equiv \]

\[ \equiv \int_{t_1}^{t} d\tau \int_{-\infty}^{+\infty} [\varphi(\xi_1,\tau)-\varphi(x_1,\tau)]\, \frac{\partial}{\partial x_1}\, \bar q_3(x_1,\ t;\ \xi_1,\tau)\,d\xi_1, \]

i.e.,

\[ \frac{\partial \bar Q_{11}^{(3)}(x_1,t)}{\partial x_1} = \sum_{i=1}^{2} q_i(x_1,t). \tag{13.40} \]

We emphasize in particular that \(\hat q_3(x_1,t;\rho,\tau)\) is the sum, over \(i=1,2\), of the expressions obtained from (13.34) (see (12.3)) by replacing \(v(x_1,\tau)-v(\xi_1,\tau)\) by \(v(x_1,\tau)-v(x_1+s_i\rho,\tau)\); \(r_\xi\) by \(r_{x_1}(s_i\rho)\); \(g_{0,2}(r(\xi,x),t-\tau)\) by \(g_{0,2}(\rho,t-\tau)g(\psi;x_1,s_i\rho,t)\); \(G_{3/2}(r(\xi,x),t-\tau)\) by \(\hat G_{3/2}(x_1,t;s_i\rho,\tau)\), and \(\xi_1\) by \(x_1+s_i\rho\), if we introduce the notation

\[ r_{x_1}(s_i\rho)=\{s_i\rho,\ \psi(x_1,t)-\psi(x_1+s_i\rho,\tau)\}, \]

\[ \frac{\partial}{\partial x_1}r_{x_1}(s_i\rho) = \left\{0,\ \frac{\partial\psi(x_1,t)}{\partial x_1} - \frac{\partial\psi(x_1+s_i\rho,\tau)}{\partial x_1}\right\}, \tag{13.41} \]

\[ g(\psi;x_1,s_i\rho,t)\equiv g_{0,0}\bigl(\psi(x_1,t)-\psi(x_1+s_i\rho,\tau),\,t-\tau\bigr) \]

(cf. (1.21), § 1),

\[ \hat G_m(x_1,t;s_i\rho,\tau) = g_{0,m}\bigl((\mu(x_1+s_i\rho,\tau),\,r_{x_1}(s_i\rho)),\,t-\tau\bigr) \times \]

\[ \times \left\{ \Phi\left( \frac{(v(x_1+s_i\rho,\tau),\,r_{x_1}(s_i\rho))} {2\sqrt{t-\tau}} \right) -\frac{\sqrt{\pi}}{2} \right\}. \tag{13.42} \]

From (13.41), (13.42) it follows (cf. (10.42)):

\[ \frac{\partial \hat G_m(x_1,t;s_i\rho,\tau)}{\partial x_1} = \]

\[ = -\frac12(\mu(x_1+s_i\rho,\tau),\,r_{x_1}(s_i\rho))\, \hat G_{m+1}(x_1,t;s_i\rho,\tau) \times \]

\[ \times \frac{\partial}{\partial x_1} (\mu(x_1+s_i\rho,\tau),\,r_{x_1}(s_i\rho)) + \]

\[ +\frac12 g_{0,m+\frac12}(\rho,t-\tau)\, g(\psi;x_1,s_i\rho,t)\, \frac{\partial}{\partial x_1} (v(x_1+s_i\rho,\tau),\,r_{x_1}(s_i\rho)). \tag{13.43} \]

From the conditions of Theorem 17, with the aid of (13.41)—(13.43), the mean-value theorem, Lemma 4 of § 1 [1], Lemma 7 of § 6 [2], Lemmas 11, 12, and (6.3), (6.4), (6.9), (6.10) from § 6 [2], it is not difficult to obtain the estimates (cf. the proof of Lemma 12), for \(0\le t-\tau\le d_1\), \(0\le \Delta t\le d_1\), \(l_1=0\), \(l_2=1\):

\[ \left\{ \begin{array}{c} \left| \dfrac{\partial^p\hat G_m(x_1,t+l_j\Delta t;s_i\rho,\tau)} {\partial x_1^p} \right| \\[1.2em] \left| \dfrac{\partial}{\partial x_1} \bigl[ g_{0,m}(\rho,t-\tau)\, g(\psi;x_1,s_i\rho,t+l_j\Delta t) \bigr] \right| \end{array} \right\} \le \]

\[ \le (C)\,g_{0,m}(\varkappa\rho,\,2(t+l_j\Delta t-\tau)), \qquad p=0,1, \tag{13.44} \]

provided (12.6) is fulfilled,

\[ \left\{ \begin{array}{c} \left| \dfrac{\partial^p\hat G_m(x_1,t+\Delta t;s_i\rho,\tau)} {\partial x_1^p} - \dfrac{\partial^p\hat G_m(x_1,t;s_i\rho,\tau)} {\partial x_1^p} \right| \\[1.2em] \left| \dfrac{\partial}{\partial x_1} \bigl[ g_{0,m}(\rho,t+\Delta t-\tau)\, g(\psi;x_1,s_i\rho,t+\Delta t) \bigr] - \right. \\[0.8em] \left. \qquad -\frac{\partial}{\partial x_1} \bigl[ g_{0,m}(\rho,t-\tau)\, g(\psi;x_1,s_i\rho,t) \bigr] \right| \end{array} \right\} \le \]

\[ \leqslant (C)\left[(\Delta t)^{\frac{1+\alpha}{2}}(t-\tau)^{-m-\frac12} +\Delta t(t-\tau)^{-m-1}\right]g_{0,0}\bigl(x\rho,\,2(t+\Delta t-\tau)\bigr), \qquad p=0,1, \tag{13.45} \]

and, when (12.13) is satisfied,

\[ \left\{ \left| \frac{\partial^p \widehat G_m(\delta,t;\,s_i\rho,\tau)}{\partial x_1^p} - \frac{\partial^p \widehat G_m(0,t;\,s_i\rho,\tau)}{\partial x_1^p} \right| \right. \]

\[ \left. {}+ g_{0,m}(\rho,t-\tau) \left| \frac{\partial^p g(\psi;\,\delta,\,s_i\rho,t)}{\partial x_1^p} - \frac{\partial^p g(\psi;\,0,\,s_i\rho,t)}{\partial x_1^p} \right| \right\} \leqslant \]

\[ \leqslant (C)\,\delta\,g_{0,m+\frac12}\bigl(x\rho,\,2(t-\tau)\bigr), \qquad p=0,1. \tag{13.46} \]

From condition (13.23), the estimates (1.17) of § 1 [1], (6.3)—(6.5) of § 6 [2], (13.44), and (10.33), it follows that

\[ |q_1(x_1,t)|\leqslant (C)|\varphi|_0\,t^{\frac{\beta}{2}}, \]

\[ |q_2(x_1,t)|\leqslant (C)|\varphi|_\alpha\,t^{\frac{\alpha}{2}}. \tag{13.47} \]

Restricting ourselves, in view of (13.47), to condition (12.6) and using expansions of the form (12.7), we obtain, with the aid of (10.35), (10.36), (13.45) (cf. the derivation of (12.9)),

\[ |q_1(x_1,t+\Delta t)-q_1(x_1,t)| \leqslant (C)|\varphi|_0|\Delta t|^{\frac{\beta}{2}}, \]

\[ |q_2(x_1,t+\Delta t)-q_2(x_1,t)| \leqslant (C)|\varphi|_\alpha|\Delta t|^{\frac{\alpha}{2}}. \tag{13.48} \]

Next, with the aid of the estimates (10.37), (10.38), (13.46), (1.17) of § 1 [1], (6.3)—(6.5) of § 6 [2], and condition (13.23), applying expansions of the form (12.10), we obtain, by virtue of (12.13) (cf. the derivation of (8.49), (8.63) in § 8 of [2], and also the derivation of (12.14)),

\[ |q_1(\delta,t)-q_1(0,t)|\leqslant (C)|\varphi|_0\,\delta^{\beta'}, \]

\[ |q_2(\delta,t)-q_2(0,t)|\leqslant (C)|\varphi|_\alpha\,\delta^{\alpha'}. \tag{13.49} \]

From (13.40), (13.47)—(13.49) it follows that

\[ \frac{\partial \overline Q_{11}^{(3)}(x_1,t)}{\partial x_1} \in H^{0,\alpha',\,\frac{\alpha}{2}}. \tag{13.50} \]

From (13.32), (13.39), (13.50), (13.5), (13.24), (13.29), (13.30), (13.26), (13.47), (7.3) of Theorem 10 of § 7 [2], it follows that (13.1) holds, which completes the proof of Theorem 17.

(To be continued)

Received by the editors
July 2, 1965

Moscow State University
named after M. V. Lomonosov

1. Differential Equations No. 10.