UDC 517.51

ON MIXED DERIVATIVES OF FUNCTIONS SUMMABLE WITH A WEIGHT

S. V. USPENSKII

In the present paper we study classes of functions that are defined in a smooth domain \(G\) and whose unmixed derivatives—generally speaking, of different orders with respect to different variables—are summable there with a certain weight that vanishes on the boundary \(\partial G\) of the domain. The main problem solved here is the description of those function spaces to which, on \(G\), the mixed derivatives of functions of the class under consideration belong. The formulation of this problem and the first results go back to S. N. Bernstein [1], who, for periodic functions \(f(x,y)\), proved that if

\[ \frac{\partial^{2} f}{\partial x^{2}},\quad \frac{\partial^{2} f}{\partial y^{2}} \in L_{2}, \]

then

\[ \frac{\partial^{2} f}{\partial x \partial y}\in L_{2}. \]

For Sobolev classes \(W_{p}^{l}(G)\) in the case when \(G=R_n\), the most general result was announced by L. N. Slobodetskii [2], who formulated the following theorem:

If \(f\in W_{p}^{l_1\cdots l_n}(R_n)\), \(1<p<\infty\), and

\[ \sum_{i=1}^{n}\frac{k_i}{l_i}\leq 1, \]

then

\[ D^{k}f\in L_p(R_n),\quad \text{where}\quad |k|=\sum_{i=1}^{n} k_i, \]

and the estimate

\[ \|D^k f\|_{L_p(R_n)} \leq c\|f\|_{W_p^l(R_n)} . \tag{1} \]

holds.

The case in which \(G\) is a domain with a curvilinear boundary turned out to be more complicated. Here one should note the work of A. P. Calderón [3], in which, for \(l_i=l\) \((i=1,\ldots,n)\), theorem (1) was extended to a broad class of domains. However, this result is rather the exception than the rule, since from examples constructed by S. M. Nikol’skii and V. P. Il’in [4] it follows that, when \(l_i\ne l_j\) \((i=1,2,\ldots,n;\ j=12,\ldots,n)\), for an arbitrary closed domain, however smooth (even analytic), theorem (1), generally speaking, does not hold. Thus, if one restricts oneself to considering cases in which embedding theorems for \(R_n\) can be transferred to \(G\), then one can pose only the question of finding, for a given class of functions, its own family of domains, completely determined by this class of functions. The most complete result of this kind was recently obtained for the spaces \(W_p^{l_1\cdots l_n}\) by O. V. Beso-

by V. P. Il’in [5, 4]. In the present paper theorem (1) is generalized to the case of functions summable with a weight, and a class of domains with smooth boundary is singled out for which this theorem holds. The paper also contains a number of results developing the author’s work [6].

Let us introduce definitions. Let \(G\) be a domain of the form

\[ \varphi(x_1,\ldots,x_{n-1}) \leq x_n \leq \infty,\qquad a_i \leq x_i \leq b_i \quad (i=12,\ldots,n-1), \tag{2} \]

where

\[ \varphi\in C^l,\qquad |D^k\varphi|<M,\qquad |k|=\sum_{i=1}^n k_i>0. \]

Let also

\[ \rho=\rho(x,\partial G)\leq c_1|x_n-\varphi|\leq c_2\rho(x,\partial G). \tag{3} \]

Define the functional space

\[ W^{l_1\ldots l_n}_{p,\alpha_1,\ldots,\alpha_n}(G)=W^l_{p,\alpha}(G) \]

as the closure of \(C^l(G)\) with respect to the norm

\[ \|f\|_{W^l_{p,\alpha}(G)} = \|f\|_{L_p(G)} + \sum_{i=1}^n \left\| \rho^{\frac{\alpha_i}{p}} \frac{\partial^{l_i}f}{\partial x_i^{l_i}} \right\|_{L_p(G)}. \tag{4} \]

We shall denote by \(R_n\) the Euclidean space of points \(x=(x_1,\ldots,x_n)\), by \(R_n^+\) the half-space

\[ x_n>0,\qquad x=(x_1,\ldots,x_{n-1})\in R_{n-1}, \]

and by \(\square^+\) the half-strip

\[ x_n>0,\qquad a_i\leq x_i\leq B_i\quad (i=1,\ldots,n-1). \]

Set also

\[ \lambda_i=l_i-\frac{\alpha_i}{p},\qquad \gamma_i=\frac{\lambda_n-\lambda_i}{l_i}+1 \quad (i=1,2,\ldots,n). \tag{5} \]

We formulate the main results of the paper.

1. Let \(f\in W^l_{p,\alpha}(G)\), \(\lambda_n>\lambda_i\) \((i=1,2,\ldots,n)\),

\[ G=[\varphi(x)\leq x_n<\infty,\quad x=(x_1,\ldots,x_{n-1})\in R_{n-1}]. \tag{6} \]

Consider the function

\[ F(y_1,\ldots,y_n)=f(y,y_n+\varphi(y)), \tag{7} \]

which is defined on \(R_n^+\). Then if \(\alpha_n>\alpha_i>0\) \((i=1,2,\ldots,n)\), then \(F\in W^l_{p,\alpha}(R_n^+)\) and

\[ \|F\|_{W^l_{p,\alpha}(R_n^+)} \leq c\|f\|_{W^l_{p,\alpha}(G)}. \tag{8} \]

This theorem may also be formulated as follows:

If \(\lambda_n>\lambda_i\), \(\alpha_n>\alpha_i>0\) \((i=1,\ldots,n)\), then the coordinate transformation

\[ x_i=y_i\quad (i=1,2,\ldots,n-1),\qquad x_n=y_n+\varphi(y) \]

generates a mapping

\[ W^l_{p,\alpha}(G)\to W^l_{p,\alpha}(R_n^+), \]

which is a linear operator.

This proposition admits a converse, namely, if \(f \in W_{p,\alpha}^{l}(R_n^+)\), \(\lambda_n \geqslant \lambda_i\), \(\alpha_n \geqslant \alpha_i \geqslant 0\) \((i=1,2,\ldots,n)\), then the change of coordinates

\[ x_i=y_i \quad (i=1,\ldots,n-1), \qquad x_n=y_n-\varphi(y) \]

generates a mapping

\[ f(x)\to F(y)=f(y,y_n-\varphi(y)), \]

which is a linear operator

\[ W_{p,\alpha}^{l}(R_n^+) \to W_{p,\alpha}^{l}(G). \]

2. Let \(f \in W_{p,\alpha}^{l}(G)\), where \(G\) has the form (6).

Denote

\[ \gamma_i=\frac{\lambda_n-\lambda_i}{l_i}+1 \quad (i=1,2,\ldots,n), \qquad \beta=\sum_{i=1}^{n} k_i\gamma_i-\lambda_n, \]

where \(k_i\) are integers,

\[ \varkappa_i=l_i\gamma_i-\sum_{j=1}^{n} k_j\gamma_j \quad (i=1,2,\ldots,n). \]

Then, if \(\lambda_n\geqslant \lambda_i\), \(\alpha_n\geqslant \alpha_i\geqslant 0\) \((i=1,2,\ldots,n)\), \(\varkappa_i\geqslant 0\) for those \(i\) for which \(k_i\neq 0\), and \(\varkappa_n\geqslant 0\), then

\[ \left\|\rho^{\tilde{\beta}}\cdot D^k f\right\|_{L_p(G)} \leqslant c\|f\|_{W_{p,\alpha}^{l}(G)}, \]

where

\[ |k|=\sum_{i=1}^{n}k_i, \qquad \tilde{\beta}= \begin{cases} \beta & \text{if } \beta\geqslant 0,\\ 0 & \text{if } \beta<0. \end{cases} \]

§ 1. ON ESTIMATES IN \(L_p\) WITH A WEIGHT OF INTEGRALS OF POTENTIAL TYPE

Definition. Let a vector \(\alpha=(\alpha_1,\ldots,\alpha_n)\), \(\alpha_i>0\) \((i=1,2,\ldots,n)\), \(\sum_{i=1}^{n}\alpha_i=n\), be given. A single-valued function \(K(x)=K(x_1,\ldots,x_n)\) will be called an \(\alpha\)-homogeneous function of degree \(m\) if, for every \(t>0\),

\[ K(t^\alpha x)=K(t^{\alpha_1}x_1,\ldots,t^{\alpha_n}x_n)=t^mK(x). \]

Let \(f\in L_p\), \(1<p<\infty\), in the half-space \(x_n>\delta\), \(x(x_1,\ldots,x_{n-1})\in R_{n-1}\), where \(\delta\) is an arbitrary positive number, and let \(K(x,x_n)\) be homogeneous of degree \(m\geqslant -n\) and

\[ \int_{R_{n-1}} |K(x,1)|\,dx=c<\infty. \tag{9} \]

Consider the function

\[ J(x,x_n)=\int_{0}^{\infty}\int_{R_{n-1}} f(t+x,x_n+v)K(t,v)\,dv, \tag{10} \]

where the integral (10) is defined as a singular integral.

Lemma 1. If the kernel \(K(t,v)\) satisfies (9), and is homogeneous of degree \(m>-n\), then the estimate

\[ \int_{R_n^+} x_n^\beta |J(x,x_n)|^p\,dR_n^+ \le c \int_{R_n^+} x_n^{\beta+\chi p}|f(x,x_n)|^p\,dR_n^+, \tag{11} \]

where \(\chi=\dfrac{n+m}{a_n}\), \(\beta>-1\), holds, and \(c\) does not depend on \(f\).

Proof.

\[ \int_{R_n^+} x_n^\beta |J(x,x_n)|^p\,dR_n^+ \le \]

\[ \le c\int_{R_n^+} x_n^\beta \left|\int_0^{x_n}\int_{R_{n-1}} |f(t+x,x_n+v)|\cdot |K(t,v)|\,dR_{n-1}\,dv\right|^p dR_n^+ \]

\[ +c\int_{R_n^+} x_n^\beta \left|\int_{x_n}^{\infty}\int_{R_{n-1}} |f(t+x,x_n+v)|\cdot |K(t,v)|\,dR_{n-1}\,dv\right|^p dR_n^+ . \tag{12} \]

Consider separately each term on the right-hand side of (12). We have

\[ \int_{R_n^+} x_n^\beta \left|\int_0^{x_n}\int_{R_{n-1}} |f(t+x,x_n+v)|\cdot |K(t,v)|\,dR_{n-1}\,dv\right|^p dR_n^+ \le \]

\[ \text{(generalized Minkowski inequality)} \]

\[ \le c\int_0^\infty x_n^\beta \left[\int_0^{x_n}\int_{R_{n-1}} \left[\int_{R_{n-1}} |f(t+x,x_n+v)|^p\,dx\right]^{1/p} |K(t,v)|\,dt\,dv\right]^p dx_n \le \]

\[ \text{(substitution \(x+t=x'\), homogeneity condition and (9))} \]

\[ \le c\int_0^\infty x_n^\beta \left|\int_0^{x_n} v^{-1+\chi} \left[\int_{R_{n-1}} |f(x^1,x_n+v)|^p\,dx\right]^{1/p} dv\right|^p dx_n \le \]

\[ \left(\text{substitution \(\dfrac{v}{x_n}=v_1\)}\right) \]

\[ \le c\int_0^\infty x_n^{\beta+\chi p} \left[\int_0^1 v_1^{-1+\chi} \left[\int_{R_{n-1}} |f(x,x_n(1+v_1))|^p\,dx\right]^{1/p} dv_1\right]^p dx \le \]

\[ \text{(generalized Minkowski inequality)} \]

\[ \le c\left[\int_0^1 v^{-1+\chi} \left[\int_0^\infty x_n^{\beta+\chi p} \int_{R_{n-1}} |f(x,x_n(1+v))|^p\,dx\,dx_n\right]^{1/p} dv\right]^p \le \]

\[ \text{(substitution \(x_n(v+1)=y\))} \]

\[ \le c\left[\int_0^1 \frac{v^{-1+\chi}\,dv}{(v+1)^{\frac{\alpha}{p}+\chi+\frac1p}}\right]^p \int_{R_n^+} x_n^{\beta+\chi p}|f(x,x_n)|^p\,dR_n^+ . \tag{13} \]

We also have

\[ \int_{R_n^+} x_n^\beta \left| \int_{x_n}^{\infty}\int_{R_{n-1}} |f(t+x,x_n+v)|\cdot |K(t,v)|\,dv\,dt \right|^p dR_n^+ \le \]

\[ \text{(generalized Minkowski inequality, homogeneity condition and (9))} \]

\[ \le c\int_0^\infty x_n^\beta \left[ \int_{x_n}^{\infty} \left[ \int_{R_{n-1}} |f(x,v+x_n)|^p\,dx \right]^{1/p} v^{-1+\varkappa}\,dv \right]^p dx_n \le \]

\[ \text{(substitution \(v+x_n=v_1\))} \]

\[ \le c\int_0^\infty x_n^\beta \left| \int_{x_n}^{\infty} v_1^{-1+\varkappa} \left[ \int_{R_{n-1}} |f(x,v_1)|^p\,dx \right]^{1/p} dv_1 \right|^p dx_n \le \]

\[ \text{(Hardy inequality)} \]

\[ \le c\int_{R_n^+} x_n^{\beta+p\varkappa}|f(x,x_n)|^p\,dR_n^+ . \tag{14} \]

From (13) and (14) we obtain estimate (11).

The following lemma generalizes to the case of functions summable with a weight the corresponding results of [7].

Lemma 2. Let

1) \(K(t,v)\) be \(\alpha\)-homogeneous of degree \(-n\) and \(n\ge 2\);

2)
\[ \int_{S^+} K(x)\sum_{i=1}^n \alpha_i x_i^2\,dS=0, \]
where \(dS\) is an element of the surface of the unit sphere \(S\), \(S^+\) is the unit hemisphere;

3)
\[ |K(x)-K(y)|\le \omega(|x-y|),\qquad x,y\in S^+,\qquad \int_0^1 \frac{\omega(t)}{t}\,dt<\infty; \]

4)
\[ \int_{R_{n-1}} |K(x,1)|\,dx=c<\infty . \]

Then the estimate holds
\[ \int_{R_n^+} x_n^\beta |J(x,x_n)|^p\,dR_n^+ \le c\int_{R_n^+} x_n^\beta |f(x,x_n)|^p\,dR_n^+, \tag{15} \]
where \(\beta\ge 0,\quad 1<p<\infty\).

Proof.

\[ \int_{R_n^+} x_n^\beta |J(x,x_n)|^p\,dR_n^+ \le \]

\[ \le c\int_{R_n^+} \left| \int_0^\infty\int_{R_{n-1}} (x_n+v)^{\beta/p} f(t+x,x_n+v)K(t,v)\,dt\,dv \right|^p dR_n^+ \]

\[ +\,c\int_{R_n^+} \left| \int_0^\infty\int_{R_{n-1}} \left[(x_n+v)^{\beta/p}-x_n^{\beta/p}\right] f(x+t, \right. \]

\[ \left. x_n+v)K(t,v)\,dt\,dv \right|^p dR_n^+ . \tag{16} \]

Consider each term separately. For \(\beta=0\) the estimate is trivial. Let \(\beta>0\). Then

\[ \begin{aligned} &\int_{R_n^+}\left|\int_0^\infty \int_{R_{n-1}} \left[(x_n+v)^{\beta/p}-x_n^{\beta/p}\right] f(t+x,\right.\\ &\left.\qquad x_n+v)K(t,v)\,dt\,dv\right|^p dR_n^+ \leq c\int_{R_n^+} x_n^{\beta-\varepsilon}\times\\ &\times \left|\int_0^{x_n}\int_{R_{n-1}} v^{\varepsilon/p}|f(t+x,x_n+v)|\cdot |K(t,v)|\,dt\,dv\right|^p dR_n^+ +\\ &\quad +\int_{R_n^+}\left|\int_{x_n}^{\infty}\int_{R_{n-1}} v^{\beta/p}|f(t+x,x_n+v)|\cdot |K(t,v)|\,dt\,dv\right|^p dR_n^+ . \end{aligned} \]

Putting \(\beta_1=\beta-\varepsilon,\ \chi_1=\dfrac{\varepsilon}{p}\) and \(\beta_2=0,\ \chi_2=\dfrac{\beta}{p}\), and taking into account the estimates (13) and (14), we obtain

\[ \int_{R_n^+}\left|\int_0^\infty \int_{R_{n-1}} \left[(x_n+v)^{\beta/p}-x_n^{\beta/p}\right] f(t+x,x_n+v)K(t,v)\,dt\,dv\right|^p dR_n^+ \leq \]

\[ \leq c\int_{R_n^+} x_n^\beta |f(x,x_n)|^p\,dR_n^+ . \tag{17} \]

In considering the first term on the right-hand side of (16), we use a generalization of the Zygmund—Calderón theorem on the boundedness in \(L_p\) of a singular operator, which was recently obtained by O. V. Besov and P. I. Lizorkin [7].

Besov—Lizorkin Theorem. If conditions 1), 2), 3) of Lemma 1 are satisfied, then the operator \(K_\varepsilon\), defined by the formula

\[ K_\varepsilon f=\int_{\sum_{i=1}^{n} \frac{y_i^2}{\varepsilon^{2\alpha_i}}>1\cap R_n^+} f(y+x)K(y)\,dy, \]

is bounded in \(L_p\), i.e.

\[ \|K_\varepsilon f\|_{L_p}\leq c\|f\|_{L_p}\, ^*). \]

Applying this theorem, we shall have

\[ \int_{R_n^+}\left|\int_0^\infty \int_{R_{n-1}} (x_n+v)^{\beta/p} f(t+x,x_n+v)K(t,v)\,dt\,dv\right|^p dR_n^+ \leq \]

\[ \leq c\int_{R_n^+} x_n^\beta |f(x,x_n)|^p\,dR_n^+ . \tag{18} \]

From (17) and (18) we obtain the estimate (15).

\[ \rule{3cm}{0.4pt} \]

\(^*)\) O. V. Besov and P. I. Lizorkin formulated the theorem for the case of the whole space; however, for \(n\geq 2\) the proof is also valid for \(R_n^+\).

§ 2. MAIN RESULTS

Theorem 1. Let \(f \in W^{l}_{p,\alpha}(G)\), where \(G\) is of the form (2). We shall also assume that \(f\) admits an extension, with preservation of the class, to the domain \(G_{\varepsilon}\):

\[ G_{\varepsilon}=\bigl(\psi(x)\leq x_n<\infty,\quad a_i-\varepsilon\leq x_i\leq b_i+\varepsilon,\quad i=1,\ldots,n-1\bigr), \]

where \(\psi(x)\in C^l\) and \(\psi(x)=\varphi(x)\) for \(x_i\in [a_i,b_i]\).
Consider the function \(F(y,y_n)=f(y,y_n+\varphi(y))\), which is defined on the half-strip \(\square^{+}\):

\[ \square^{+}=(x_n\geq 0,\ a_i\leq x_i\leq b_i,\quad i=12,\ldots,n-1). \]

Then, if \(\lambda_n\geq \lambda_i,\ \alpha_n\geq \alpha_i\geq 0\) \((i=1,\ldots,n)\), we have \(F\in W^{l}_{p,\alpha}(\square^{+})\) and

\[ \|F\|_{W^{l}_{p,\alpha}(\square^{+})}\leq c\|f\|_{W^{l}_{p,\alpha}(G_{\varepsilon})}. \tag{19} \]

For the proof of this theorem we shall use some results obtained by the author in [8] and [9].
Let

\[ K_0(x,u)=\prod_{i=1}^{n-1}\frac{u^{\gamma_i}}{x_i^2+u^{2\gamma_i}}, \tag{20} \]

\[ K(x,u)=\sum_{i=0}^{N} c_i u^i \frac{\partial^i K_0(x,u)}{\partial u^i}, \tag{21} \]

where the \(c_i\) satisfy Lemma 1 of [8], with \(c_0=1\). Then \(K(x,u)\) has the following properties:

\[ \frac{\partial}{\partial u}K(x,u)=\sum_{i=1}^{n-1}\frac{\partial^{l_i}}{dx_i^{l_i}}\Phi_i(x,u), \tag{22} \]

\[ \Phi_i=\sum_{\substack{j=s\\ \alpha_i\geq l_i}}^{m}\ \sum_{\sum \alpha_i=j} c_{\alpha}u^{\sum_{i=1}^{n-1}\alpha_i\gamma_i-1} \frac{\partial^{j-l_i}K_0(x,u)} {\partial x_1^{\alpha_1}\cdots \partial x_i^{\alpha_i-l_i}\cdots \partial x_{n-1}^{\alpha_{n-1}}}, \tag{23} \]

\(c_{\alpha}\) are certain constants independent of \(x,u\). As is easy to see, the \(\Phi_i\) have the estimate

\[ \left|\frac{\partial^{m+s}\Phi_i}{\partial x_j^s\partial u^m}\right| \leq c u^{-1-m+l_i\gamma_i-s\gamma_j}K_0(x,u). \tag{24} \]

We shall assume, without loss of generality, that \(f(x,x_n)\equiv 0\) outside the strip
\[ \psi(x)\leq x_n\leq 1+\psi(x). \]

Define a function \(\eta(x_1,\ldots,x_{n-1})\), which belongs to \(C^{2l}\), is equal to one on \(G\), and is finite outside the strip

\[ \square_{\varepsilon}=(a_i-\varepsilon\leq x_i\leq b_i+\varepsilon,\quad i=1,\ldots,n-1). \]

Such a function can always be constructed.
Let also

\[ f_1(x_1,\ldots,x_n)= \begin{cases} \eta f, & x\in G_{\varepsilon},\\ 0, & x\notin G_{\varepsilon}. \end{cases} \]

Since the proof in the \(n\)-dimensional case is no different from the 2-dimensional one, we shall restrict ourselves to the case \(n=2\). Put \(\gamma=\dfrac{\lambda_2-\lambda_1}{l_1}+1\). Then, for points \((x,y)\in G\), the following representation holds:

\[ f(x,y+\varphi(x))=k\int_0^\infty\int_{-\infty}^{\infty} f_1(t,y+\psi(t))\frac{\partial K(x-t,v)}{\partial v}\,dt\,dv, \tag{25} \]

where \(k\) is a normalizing factor with the appropriate sign.

To prove (25) it suffices to note that, for \((x,y)\in G\),

\[ \lim_{v\to 0}|k|\int_{-\infty}^{\infty} f_1(t,y+\psi(t))K(x-t,v)\,dt=f(x,y+\varphi(x)), \]

\[ |k|\int_{\varepsilon}^{\infty}\int_{-\infty}^{\infty} f_1(t,y+\psi(t))\frac{\partial K(x-t,v)}{\partial v}\,dt\,dv= \]

\[ =-|k|\int_{-\infty}^{\infty} f_1(t,y+\psi(t))K(x-t,\varepsilon)\,dt. \]

Using Lemma 2 [9], we shall have

\[ \begin{aligned} f(x,y+\varphi(x))={}& \sum_{i=1}^{l_2}\alpha_i\int_0^\infty\int_{-\infty}^{\infty} f_1[t,(2^i+1)v+y+\psi(t)] \frac{\partial K(x-t,v)}{\partial v}\,dv\,dt \\ &+\sum_{i=1}^{l_2}\beta_i\int_0^\infty\int_{-\infty}^{\infty}\int_v^{(2^i+1)v} [(2^i+1)v-u]^{l_2-1}D_y^{l_2}f_1[t,u+y+\psi(t)]\,du \\ &\qquad\qquad\times \frac{\partial K(x-t,v)}{\partial v}\,dv\,dt +\beta_0\int_0^\infty\int_{-\infty}^{\infty}\int_0^v u^{l_2-1}D_y^{l_2}f_1[t,u+y+\psi(t)]\,du \\ &\qquad\qquad\times \frac{\partial K(x-t,v)}{\partial v}\,dv\,dt . \end{aligned} \tag{26} \]

Consider separately each term in (26). For \(\varkappa=l_2-l_1\gamma>0\) we have

\[ J_1=\int_{R_2^+}y^{\alpha_1} \left| \int_0^\infty\int_{-\infty}^{\infty}\int_v^{(2^i+1)v} [(2^i+1)v-u]^{l_2-1}D_y^{l_2}f_1[t+x,u+y+\psi(t)]\,du\, \frac{\partial^{l_1+1}K}{dv\,dt^{l_1}}\,dv\,dt \right|^p dR_2^+\le \]

\[ \text{(putting \(\varepsilon<\varkappa\) and taking into account (24))} \]

\[ \le C\int_{R_2^+}y^{\alpha_1} \left| \int_0^\infty\int_{-\infty}^{\infty}v^{-1+\varepsilon} \int_v^{(2^i+1)v} [(2^i+1)v-u]^{\varkappa-1-\varepsilon} |D_y^{l_2}f_1|K_0(t,v)\,dt\,dv \right|^p dR_2^+\le \]

\[ \leqslant c \int_{R_2^+} y^{\alpha_1}\left|\int_0^\infty \int_{-\infty}^\infty K_0\left[t,\,(2^i+1)^{-1}v\right]\sigma^{-1+\varepsilon}\times\right. \]
\[ \left.\times \int_{(v+y)(2^i+1)^{-1}}^{v+y} \bigl[(v+y)-u\bigr]^{\chi-1-\varepsilon} \left|D_y^{l_2} f_1[x+t,\,u+\psi(x+t)]\right|\,du\,dt\,dv\right|^p dR_2^+ . \]

The kernel \(K_0(t,\sigma)\sigma^{-1+\varepsilon}\) is homogeneous of degree
\[ \alpha=\left(\frac{2\gamma}{1+\gamma},\ \frac{2}{1+\gamma}\right) \]
of degree \(m=-2+2\varepsilon/(1+\gamma)\). Then, by Lemma 1,
\[ J_1 \leqslant c \int_{R_2^+} y^{\alpha_1+\varepsilon p} \left|\int_{y(2^i+1)^{-1}}^y (y-u)^{\chi-1-\varepsilon} \left|D_y^{l_2} f_1(x,\,u+\psi(x))\right|\,du\right|^p dR_2^+ \leqslant \]
\[ \leqslant c \int_{R_2^+} y^{\alpha_1+\varepsilon p} \left|\int_{y(2^i+1)^{-1}}^y u^{\chi-1-\varepsilon} \left|D_y^{l_2} f_1(x,\,u+\psi(x))\right|\,du\right|^p dR_2^+ \leqslant \]
\[ \leqslant c \int_{R_2^+} y^{\alpha_1+\varepsilon p} \left|\int_y^\infty u^{\chi-1-\varepsilon} \left|D_y^{l_2} f_1(x,\,u+\psi(x))\right|\,du\right|^p dR_2^+ \leqslant \]
\[ \text{(Hardy’s inequality)} \]
\[ \leqslant c \int_{R_2^+} y^{\alpha_1+\chi p}|\eta(x)|^p\cdot \left|D^{l_2} f(x,\,y+\psi(x))\right|^p\,dy\,dx \leqslant \]
\[ \leqslant c\,\|f\|_{W_{p,\alpha}^{\,l}(G_\varepsilon)}, \tag{27} \]
since
\[ \alpha_1+\chi p=\alpha_1+l_2p-l_1\gamma p=\alpha_2, \tag{28} \]
\[ J_2=\int_{R_2^+} y^{\alpha_1}\left| \int_0^\infty \int_{-\infty}^\infty \int_0^v u^{l_2-1}D_y^{l_2} f_1(t+x,\,u+y+\psi(t+x))\,du\, \frac{\partial^{l_1+1}K}{\partial v\,\partial t^{l_1}}\,dv\,dt \right|^p dR_2^+ \leqslant \]
\[ \text{(integration by parts and estimate (24))} \]
\[ \leqslant c \int_{R_2^+} y^{\alpha_1}\left| \lim_{v\to 0} v^{-l_1\gamma}\int_0^v u^{l_2-1} \int_{-\infty}^\infty \left|\Delta_t D_y^{l_2} f_1(x,\,u+y+\psi(x))\right| K_0(t,v)\,dt\,du\right|^p dR_2^+ \]
\[ + \int_{R_2^+} y^{\alpha_1}\left| \int_0^\infty \int_{-\infty}^\infty v^{\chi-1}\left|D_y^{l_2} f_1(t+x,\,v+y+\psi(t+x))\right| K_0(t,v)\,dv\,dt\right|^p dR_2^+ \leqslant \]
\[ \text{(from (27), Lemma 1 and (28))} \]
\[ \leqslant c\,\|f\|_{W_{p,\alpha}^{\,l}(G_\varepsilon)} . \tag{29} \]

Consider the case \(\varkappa=l_2-l_1\gamma=0\). Denoting by

\[ \Phi_k=v^{l_2-k}\int_v^\infty \cdots \int_{s_1}^\infty \frac{\partial^{l_1+1}K(t,s_0)}{\partial t^{l_1}\partial s_0}\,ds_0\cdots ds_{k-1} \]

and integrating by parts, we shall have

\[ J_1+J_2\leq c\sum_{k=1}^{l_2}\int_{R_2^+} y^{\alpha_1}\left|\int_0^\infty\int_{-\infty}^{\infty} D_y^{l_2}f_1(t+x,v+y+ \right. \]

\[ \left. +\psi(x+t))\Phi_k(t,v)\,dt\,dv\right|^p dR_2^+ . \]

We shall show that the kernel \(\Phi_k\) satisfies all the conditions of Lemma 2. By virtue of estimate (24),

\[ \int_{R_{n-1}^{0}}\left|\Phi_k(t,1)\right|\,dt<\infty . \]

From the form of the kernel \(\Phi_k(t,v)\) it follows that it is \(\alpha=\left(\dfrac{2\gamma}{1+\gamma},\,\dfrac{2}{1+\gamma}\right)\)-homogeneous of degree \(m=-2\).

Condition 2 of Lemma 2 is also fulfilled, since the kernel \(\Phi_k\) is \(\alpha\)-homogeneous and is a partial derivative with respect to \(t\) of some kernel \(\overline{\Phi}_k\), which vanishes at infinity. From the construction of the kernel \(K(t,v)\) it follows that condition 3 is also fulfilled. Then, applying Lemma 2, we obtain

\[ J_1+J_2\leq c\int_{R_2^+} y^{\alpha_1}|\eta(x)|^p|D_y^{l_2}f(x,y+\psi(x))|^p\,dR_n^+ \leq c\|f\|_{W_{p,\alpha(\sigma_\varepsilon)}^{\,l}}, \tag{30} \]

since from the condition \(l_2-l_1\gamma=0\) it follows that \(\alpha_1=\alpha_2\).

Consider the remaining terms in (26). Put

\[ \widetilde{D}_t^{\,k}f=\left.\frac{\partial f}{\partial x}(x,y+\psi(t))\right|_{x=t}. \]

Then we shall have

\[ J_3=\int_{R_2^+}y^{\alpha_1}\left|D_x^{l_1}\int_0^\infty\int_{-\infty}^{\infty} f_1(t,(2^i+1)v+y+\psi(t)) \frac{\partial K(x-t,v)}{\partial v}\,dt\,dv\right|^p dR_2^+\leq \]

\[ \text{(taking into account the properties of the kernel (22))} \]

\[ \leq c\sum_{k=0}^{l_1}\sum_{s_1=0}^{k}\sum_{s_2=0}^{l-k} \int_{R_2^+}y^{\alpha_1}\left| \int_0^\infty\int_{a-\varepsilon}^{b+\varepsilon} p_{k,s_1,s_2}(t)\,\widetilde{D}_t^{\,l-k-s_1}D_y^{k-s_2}f \times \right. \]

\[ \left. \times\frac{\partial^l\Phi(t-x,v)}{\partial t^l}\,dt\,dv \right|^p dR_2^+, \tag{31} \]

where \(p_{k,s_1,s_2}(t)\) are certain polynomials in \(\psi\) and its derivatives, which vanish at the endpoints \(a-\varepsilon\) and \(b+\varepsilon\). Transferring in each term the differentiation with respect to \(t\) from the kernel \(\Phi\) to the function \(f\) and back with respect to \(v\) and \(t\) so that in each

term in (31) is the non-mixed derivative either \(\widetilde D_t^{\,l_1}f\) or \(D_y^k f\), where \(0\leq k\leq l_2\), we obtain

\[ \begin{aligned} J_3 \leq &\sum_{j=0}^{l_1-1}\sum_{s=0}^{l_2}\sum_{k=0}^{l} \int_{R_2^+} y^{\alpha_1} \left| \int_0^\infty \int_{a-\varepsilon}^{b+\varepsilon} p_{k,s}(t)D_y^s f D_t^{\,l-k+l_1-l_2-j}D_v^j \Phi(t-x,v)\,dvdt \right|^p dR_2^+ \\ &+ \sum_{j=0}^{l_2-1}\sum_{k=0}^{l} \int_{R_2^+} y^{\alpha_1} \left| \int_0^\infty \int_{a-\varepsilon}^{b+\varepsilon} p_k(t)\widetilde D_t^{\,l_1}f D_t^{\,l-k-j}D_v^j\Phi(t-x,v)\,dvdt \right|^p dR_2^+ . \tag{32} \end{aligned} \]

From the construction of the kernel \(\Phi(t,v)\) it follows that

\[ \Phi_{k,j}=D_t^{\,l-k+l_1-l_2-j}D_v^j\Phi(t,v) \]

is \(\alpha=\left(\dfrac{2\gamma}{1+\gamma},\dfrac{2}{1+\gamma}\right)\)-homogeneous of degree

\[ m=\frac{2\gamma}{1+\gamma}(l_2-l_1+k+j)-\frac{2j}{1+\gamma}-2\geq -2, \]

and the kernel

\[ \widetilde\Phi_{k,j}=D_t^{\,l-k-j}D_v^j\Phi(t,v) \]

is of degree

\[ m=\frac{2\gamma}{1+\gamma}(k+j)-\frac{2j}{1+\gamma}-2\geq -2. \]

It is not hard to see that the functions \(\Phi_{k,j}\), \(\widetilde\Phi_{k,j}\) also satisfy the remaining conditions of Lemma 2. Denote by

\[ f_{k,s}= \begin{cases} p_{k,s}(t)D_y^s f(t,v+y+\psi(t)), & t\in [a-\varepsilon,b+\varepsilon],\\ 0, & t\notin [a-\varepsilon,b+\varepsilon], \end{cases} \]

\[ \widetilde f_k= \begin{cases} p_k(t)D^{l_1} f(t,v+y+\psi(t)), & t\in [a-\varepsilon,b+\varepsilon],\\ 0, & t\notin [a-\varepsilon,b+\varepsilon]. \end{cases} \]

Let also

\[ \overline m=\frac{2\gamma}{1+\gamma}(l_2-l_1)-2. \]

Then, making the substitution \(t_1=t-x\) and applying Lemmas 1 and 2, we obtain

\[ J_3\leq c\sum_{s=0}^{l_2}\int_0^\infty\int_{a-\varepsilon}^{b+\varepsilon} y^{\alpha_1+\frac{p(1+\gamma)(2-\overline m)}{2}} \left|D_y^s f(x,y+\psi(x))\right|^p\,dxdy + \]

\[ + c\int_0^\infty\int_{a-\varepsilon}^{b+\varepsilon} y^{\alpha_1}\left|D^{l_1}f(x,y+\psi(x))\right|^p\,dxdy \leq \]

\[ \text{(finiteness of the function \(f\) for \(y\geq 1\) and Theorem 1 [10])} \]

\[ \leq c\int_0^\infty\int_{a-\varepsilon}^{b+\varepsilon} y^{\alpha_1+\gamma(l_2-l_1)} \left|D_y^{l_2}f(x,y+\psi(x))\right|^p\,dxdy + \]

\[ + c\|f\|_{W_{p,\alpha}^{\,l}(G_\varepsilon)} \leq c\|f\|_{W_{p,\alpha}^{\,l}(G_\varepsilon)} . \tag{33} \]

since \(\alpha_1+\gamma(l_2-l_1)\ge \alpha_1+l_2-\gamma l_1=\alpha_2\). Taking into account (26), and the estimates (27), (29), (30), (33), we obtain the assertion of the theorem.

Theorem 2. Let \(f\in W^l_{p,\alpha}(\Box^+)\), where

\[ \Box^+=(x_n\ge 0,\quad a_i\le x_i\le b_i,\quad i=1,2,\ldots,n-1). \]

We shall also assume that \(f\) admits an extension (we keep the same notation for it) to the half-strip

\[ \Box^+_\varepsilon=(x_n\ge 0,\quad a_i-\varepsilon\le x_i\le b_i+\varepsilon,\quad i=1,2,\ldots,n-1) \]

with preservation of the class. Denote

\[ \gamma_i=\frac{\lambda_n-\lambda_i}{l_i}+1,\qquad \beta=\sum_{i=1}^{n} k_i\gamma_i-\lambda_n, \]

where \(k_i\) are integers, \(\varkappa_i=l_i\gamma_i-\sum_{i=1}^{n}k_i\gamma_i\). Then if \(\lambda_n+\dfrac{\alpha_i}{p}>0\) \((i=1,2,\ldots,n)\), \(\varkappa_i>0\) for those \(i\) for which \(k_i\ne 0\), \(\varkappa_n\ge 0\), then

\[ x_n^{\tilde\beta}D^k f\in L_p(\Box^+), \]

\[ \bigl\|x_n^{\tilde\beta}D^k f\bigr\|_{L_p(\Box^+)} \le c\|f\|_{W^l_{p,\alpha}(\Box^+_\varepsilon)}, \tag{34} \]

where

\[ |k|=\sum_{i=1}^{n}k_i,\qquad \tilde\beta= \begin{cases} \beta, & \text{if } \beta\ge 0,\\ 0, & \text{if } \beta<0. \end{cases} \]

Proof. Let the function \(\eta(x_1,\ldots,x_{n-1})\) be equal to one on \(\Box^+\) and finite outside the strip \(\Box_\varepsilon=(a_i-\varepsilon\le x_i\le b_i+\varepsilon,\ i=1,2,\ldots,n-1)\). Without loss of generality we may assume that \(f=0\) for \(x_n\ge 1\).

Introduce the function

\[ f_1(x,x_n)= \begin{cases} \eta f, & x\in \Box^+_\varepsilon,\\ 0, & x\notin \Box^+_\varepsilon. \end{cases} \]

Since

\[ \gamma_i=\frac{\lambda_n-\lambda_i}{l_i}+1=\lambda_n+\frac{\alpha_i}{p}>0, \]

the function \(f\), for points \((x,x_n)\in\Box^+\), has the representation

\[ f=k\int_{0}^{\infty}\int_{R_{n-1}} f_1(t_1,\ldots,t_{n-1},x_n)\, \frac{\partial K}{\partial v}(x-t,v)\,dR_{n-1}, \]

where \(K(x,v)\) has the form (21). We may always suppose that all \(k_i>0\) \((i=1,2,\ldots,n-1)\), since if for some \(j,\ 1\le j\le n-1,\ k_i=0\), then we could regard \(f\) as a function only of those variables for which \(k_i\ne 0\). Then, using Lemma 2 [9], for \(D^k f\) we obtain

\[ \int_{\Box^+} |D^k f|^p x_n^{\tilde\beta p}\,dR_n^+ \le \]

\[ \le c\sum_{i=1}^{n-1}\sum_{j=0}^{l_i}\sum_{\nu=1}^{l_n} \int_{R_n^+} x_n^{\tilde\beta p} \int_{\Box^+_\varepsilon} p_i(t)D^{l_i-j}f\bigl(t,(2^\nu+ \]

\[ +1)v+x_n)v^{-k_n}D^{k-k_n}D_{t_i}^{\,l-l_i}\Phi_i\,dv\,dt\biggr|^p\,dR_n^+ + \]

\[ + c\sum_{\nu=1}^{l_n}\int_{R_n^+} x_n^{\tilde\beta p} \left|\int_{R_n^+} v^{-k_n}\int_v^{(2^\nu+1)v} du\,[(2^\nu+ \]

\[ +1)v-u]^{l_n-1}D^{l_n}f_1(t,u+x_n)D^{k-k_n}D_vK\,dv\,dt\right|^p dR_n^+ + \]

\[ +\beta_0\int_{R_n^+} x_n^{\tilde\beta p} \left|\int_0^\infty\int_{R_{n-1}} v^{-k_n}\int_0^v u^{l_n-1}D^{l_n}f(t,u+ \]

\[ +x_n)duD^{k-k_n}D_vK\,dv\,dt\right|^p dR_n^+. \tag{35} \]

Consider the kernel

\[ \Phi_{i,s}=v^{-k_n}D^{k-k_n}D_{t_i}^{\,l-l_i-s}\Phi_i. \]

From the construction of the function \(\Phi_i\) it follows that it is \(a=\left(\dfrac{n\gamma_1}{\sum_{i=1}^n\gamma_i},\ldots,\dfrac{n\gamma_{n-1}}{\sum_{i=1}^n\gamma_i},\right.\)

\[ \left.\dfrac{n}{\sum_{i=1}^n\gamma_i}\right) \]

homogeneous of degree \(m\ge \dfrac{\varkappa_i n}{\sum_{i=1}^n\gamma_i}-n\). Since \(\varkappa_i\ge0\), we have \(m\ge -n\).

The kernel also satisfies conditions 2)—4) of Lemma 2. Then we have

\[ J_1=\int_{R_n^+}x_n^{\tilde\beta p} \left|\int_{\Box_\varepsilon^+}P_j(t)D^{l_i-j}f(t,(2^\nu+ \]

\[ +1)v+x_n)v^{-k_n}D^{k-k_n}D_{t_i}^{\,l-l_i}\Phi\,dv\,dt\right|^p dR_n^+ \le \]

\[ \text{(transferring the differentiation from the kernel \(\Phi_i\) to \(f\))} \]

\[ \le c\sum_{s=0}^{l_i}\int_{R_n^+}x_n^{\tilde\beta p} \left|\int_{\Box_\varepsilon^+}\tilde p_s(t)D^{l_i}f(t,(2^\nu+1)v+x_n)\Phi_{i,s}\,dv\,dt\right|^p dR_n^+ + \]

\[ +c\sum_{s=1}^{l_i}\int_{R_n^+}x_n^{\tilde\beta p} \left|\int_{\Box_\varepsilon^+}\tilde p_s(t)\tilde f(t,(2^\nu+1)v+x_n)\Phi_{is}\,dv\,dt\right|^p dR_n^+ \le \]

\[ \text{(by Lemmas 1 and 2)} \]

\[ \le c\sum_{s=0}^{l_i}\int_{R_n^+} x_n^{\tilde\beta p+\varkappa_i p} \left(|\tilde p_s|^p\cdot |D^{l_i}f|^p+|\tilde p_s|^p\cdot |f|^p\right)dR_n^+ \le \]

\[ \le c\|f\|_{W_{p,\alpha}^l(\Box_\varepsilon^+)}, \tag{36} \]

since \(\tilde\beta p+\varkappa_i p\ge (l_i\gamma_i-\lambda_n)p=\alpha_i\).

Since the estimates are analogous, we consider one of the remaining terms in (35):

\[ J_2=\int_{R_n^+} x_n^{\tilde\beta p} \left|\int_{R_n^+} v^{-k_n}\int_0^v u^{l_n-1}D^{l_n}f_1(t,u+x_n)\times \right. \]

\[ \left. \times\,du\,D^{k-k_n}D_vK\,dv\,dt\right|^p dR_n^+\leq \]

\[ \text{(integrating by parts)} \]

\[ \leq c\int_{R_n^+} x_n^{\tilde\beta p} \left|\int_{R_n^+} v^{l_n-1}D^{l_n}f_1 \int_v^\infty s^{-k_n}D^{k-k_n}D_sK\,ds\right|^p dR_n^+ . \]

Consider the kernel

\[ \Phi=v^{l_n-1}\int_v^\infty s^{-k_n}D^{k-k_n}D_sK(t-x,s)\,ds . \]

From the construction of the function \(K\) it follows that it is \(a=\left(-\dfrac{n\gamma_1}{\sum_{i=1}^n\gamma_i},\ldots,-\dfrac{n\gamma_n}{\sum_{i=1}^n\gamma_i}\right)\)-homogeneous of degree

\[ m=-\frac{n}{\sum_{i=1}^n\gamma_i}\varkappa_n-n\geq -n . \]

It is also easily checked that all the remaining conditions of Lemma 2 are satisfied. Then

\[ J_2\leq c\int_{R_n^+} x_n^{\tilde\beta p+\varkappa_n p} \left|D^{l_n}f_1(x,x_n)\right|^p dR_n^+\leq \]

\[ \leq c\int_{\square_\varepsilon^+} x_n^{\tilde\beta p+\varkappa_n p} \left|D^{l_n}f(x,x_n)\right|^p d\square_\varepsilon^+ \leq c\|f\|_{W_{p,a}^{\,l}(\square_\varepsilon^+)} . \tag{37} \]

From (35)—(37) we obtain (34), as was required to prove.

Corollary 1. Let \(f\in W_{p,a}^{\,l}(\square_\varepsilon^+)\), \(\lambda_n>\lambda_i\), \(a_n>a_i\geq 0\), \(i=1,2,\ldots,n\), \(\psi(x)\in C^1\). Then the function \(F(y,y_n)=f(y,\psi(y)+y_n)\), defined in the domain

\[ G=(-\psi(x)\leq x_n\leq\infty,\quad a_i\leq x_i\leq b_i,\quad i=1,2,\ldots,n), \]

belongs there to the class \(W_{p,a}^{\,l}(G)\) and

\[ \|F\|_{W_{p,a}^{\,l}(G)}\leq c\|f\|_{W_{p,a}^{\,l}(\square_\varepsilon^+)} . \tag{38} \]

Proof. Let

\[ \gamma_i=\frac{\lambda_n-\lambda_i}{l_i}+1,\qquad \varkappa_i=l_i\gamma_i-k_n\gamma_n-k_i\gamma_i, \]

\[ \beta=k_n\gamma_n+k_i\gamma_i-\lambda_n . \]

From Theorem 2 it follows that, if \(\varkappa_i\geq 0\), then for the partial derivative \(D^{k_n+k_i}f\) the estimate

\[ \left\|\rho^{\tilde\beta}D^{k_n+k_i}f\right\|_{L_p(\square^+)} \leq c\|f\|_{W_{p,a}^{\,l}(\square_\varepsilon^+)}, \]

where

\[ \tilde\beta= \begin{cases} \beta, & \text{for } \beta\geq 0,\\ 0, & \text{for } \beta<0. \end{cases} \]

Denote by

\[ \widetilde D_j^k f=\left.\frac{\partial^k}{\partial y_j^k}\,f(x_1,\ldots,x_{j-1},y_j,\ldots,x_{n-1},\psi(x)+x_n)\right|_{y_j=x_j}. \]

Then we have

\[ D^{l_j}F=\sum_{k=1}^{l_j}\sum_{s_1=0}^{k-1}\sum_{s_2=0}^{l_j-k-1} p_{k,s_1,s_2}\,\widetilde D_j^{\,l_j-k-s_2}D_n^{k-s_1}f(x,\psi(x)+x_n). \tag{39} \]

We shall show that for any \(k\le l_j\) and \(s\le k\) one has

\[ \left\|\rho^{\frac{\alpha_j}{p}}\,\widetilde D_j^{\,l_j-k}D_n^{k-s}f\right\|_{L_p(G)} \le c\|f\|_{W_{p,\alpha}^{l}(\square_\varepsilon^+)}. \tag{40} \]

For this it is enough to show that \(\chi_j=l_j\gamma_j-(l_j-k)\gamma_j-k\gamma_n\ge0\) and
\(\beta=k\gamma_n+(l_j-k)\gamma_j-\lambda_n\le \dfrac{\alpha_j}{p}\). We have
\(\chi_j=k(\gamma_j-\gamma_n)\ge0\) by virtue of the condition \(\lambda_n\ge\lambda_i\) \((i=1,2,\ldots,n)\).

We also have \(\beta=k(\gamma_n-\gamma_j)+\dfrac{\alpha_j}{p}\le\dfrac{\alpha_j}{p}\). From (39), (40) follows (38).

Corollary 2. Let \(f\in W_{p,\alpha}^{l}(G)\), where

\[ G=(\varphi(x)\le x_n<\infty,\quad x=(x_1,\ldots,x_{n-1})\in R_{n-1}). \]

Then, under the hypotheses of Theorem 2 and \(\lambda_n\ge\lambda_i>0,\ \alpha_n\ge\alpha_i\ge0,\ i=1,\ldots,n,\)

\[ \|\rho^{\widetilde\beta}D^k f\|_{L_p(G)} \le c\|f\|_{W_{p,\alpha}^{l}(G)}. \tag{41} \]

Proof. By Theorem 1 the function

\[ F(x,x_n)=f(x,x_n+\varphi(x))\in W_{p,\alpha}^{l}(R_n^+) \]

and

\[ \|F\|_{W_{p,\alpha}^{l}(R_n^+)} \le c\|f\|_{W_{p,\alpha}^{l}(G)}. \tag{42} \]

We shall prove that for the function \(f(x,x_n)=F(x,x_n-\varphi(x))\), under the hypotheses of Theorem 2, the estimate

\[ \|\rho^{\widetilde\beta}D^k f\|_{L_p(G)} \le c\|F\|_{W_{p,\alpha}^{l}(R_n^+)} \]

holds.

Using the notation introduced above, we shall have

\[ D^k f= \sum_{|j|=1}^{|k|} \sum_{|s_1|\le |k-1|} \sum_{|s_2|\le |j-1|} p_{j,s_1,s_2}\,\widetilde D^{\,k-s_1-j}D^{\,j-s_2}F(x,-\varphi(x)+x_n), \tag{43} \]

where \(p_{j,s_1,s_2}(x)\) are certain polynomials in \(\varphi\) and its derivatives. Since \(\gamma_i\ge\gamma_n\) \((i=1,2,\ldots,n)\), it follows that

\[ \bar\chi_\nu=l_\nu\gamma_\nu-\sum_{i=1}^n(k_i-j_i)\gamma_i-|j|\gamma_n \ge \chi_\nu+\sum_{i=1}^n j_i(\gamma_i-\gamma_n)\ge \chi_\nu \tag{44} \]

and

\[ \begin{aligned} \bar\beta &=\sum_{i=1}^n(k_i-j_i)\gamma_i+|j|\gamma_n-\lambda_n \\ &=\beta-\sum_{i=1}^n j_i(\gamma_i-\gamma_n)\le \beta. \end{aligned} \tag{45} \]

Then for the mixed derivatives \(\widetilde D^{k-s_1-j}D_n^{j-s_2}F\) the estimate holds
\[ \int_G |x_n-\varphi(x)|^{\tilde\beta p} \left|\widetilde D^{k-s_1-j}D_n^{j-s_2}F(x,x_n-\varphi)\right|^p\,dG = \]
\[ =\int_{R_n^+} x_n^{\tilde\beta p} \left|D^{k-s_1-j}D_n^{j-s_2}F(x,x_n)\right|^p\,dR_n^+ \le \]
\[ \text{(taking into account (44), (45), and Theorem 2)} \]
\[ \le c\|F\|_{W_{p,\alpha}^{\,l}(R_n^+)} \le \]
\[ \text{(taking into account (42))} \]
\[ \le c\|f\|_{W_{p,\alpha}^{\,l}(G)}. \tag{46} \]

From (43) and (46) follows (41).

Theorem 3. Let \(G\) be a bounded domain with smooth boundary \(\partial G\in C^1\), and let \(f\in W_{p,\alpha}^{\,l}(G)\), \(l_i=l\), \(\alpha_i=\alpha\) \((i=1,\ldots,n)\). Denote
\[ \beta=\sum_{i=1}^n k_i-l+\frac{\alpha}{p},\qquad \varkappa=l-\sum_{i=1}^n k_i . \]
Then if \(\varkappa>0\), then
\[ \|\rho^{\tilde\beta}D^k f\|_{L_p(G)} \le c\|f\|_{W_{p,\alpha}^{\,l}(G)}, \tag{47} \]
where
\[ |k|=\sum_{i=1}^n k_i,\qquad \tilde\beta= \begin{cases} \beta, & \text{if } \beta>0,\\ 0, & \text{if } \beta<0. \end{cases} \]

Proof. Consider a sufficiently narrow \(n\)-dimensional strip \(G_\delta\) belonging to \(G\) and adjacent to \(\partial G\). By virtue of the smoothness of \(\partial G\), the strip \(G_\delta\) admits a partition into a finite number of intersecting domains \(G_j\) \((j=1,\ldots,N)\) of the following form:
\[ G_j=\bigl(\varphi_j(x_1,\ldots,x_{k-1},x_{k+1},\ldots,x_n)\le x_k\le \]
\[ \le \varphi_j(x_1,\ldots,x_{k-1},x_{k+1},\ldots,x_n)+\delta, \]
\[ a_i\le x_i\le b_i,\quad i=1,2,\ldots,n,\ i\ne k,\quad j=1,2,\ldots,N. \]
Extend the function \(f\) by zero from \(G_j\) to
\[ \overline G_j=(\varphi_j\le x_k<\infty,\quad a_i\le x_i\le b_i,\quad i=1,2,\ldots,n,\ i\ne k). \]
For the function \(f\) (defined on \(\overline G_j\)) it is possible to apply Theorems 1 and 2, from which the assertion of Theorem 3 follows.

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Received by the editors
June 17, 1966

Institute of Mathematics
SO AS USSR