A. Kh. GUDIEV

THE PROBLEM OF S. L. SOBOLEV AND S. M. NIKOLSKII FOR THE LIMITING EXPONENT

(Presented by Academician S. L. Sobolev on 6 X 1962)

Denote by \(\mathbf{L}_{(p_1,p_2)}(R^n)\) the class of functions \(f(x)\), defined on \(R^n\), for which the norm is bounded

\[ \|f\|_{\mathbf{L}_{(p_1,p_2)}(R^n)} = \left[ \int_{R_x^{\,n-s}} \left( \int_{R_x^{\,s}} |f(\bar{x})|^{p_1}\, d\bar{x}_s \right)^{p_2/p_1} d\bar{x}_{n-s} \right]^{1/p_2}. \]

The aim of the present note is to prove the following theorem:

Theorem 1. If \(f(\bar{x}) \in W_p^{(l)}(\Omega)\),

\[ \frac{n}{p}-l=\frac{n-s}{p_2}+\frac{s}{p_1}, \qquad p_2 \geq p_1 > p > 1, \]

then \(f(\bar{x}) \in \mathbf{L}_{(p_1,p_2)}(\Omega)\) and, moreover, the inequality

\[ \|f\|_{\mathbf{L}_{(p_1,p_2)}(\Omega)} \leq c \|f\|_{W_p^{(l)}(\Omega)} \]

holds, where \(c\) is a constant independent of \(f\) (\(\Omega\) may also be unbounded).

This theorem gives a solution of the problem of S. L. Sobolev and S. M. Nikolskii for the limiting exponent, posed by them at the Fourth All-Union Mathematical Congress.

For the proof of Theorem 1 it suffices to establish the validity of the following theorem, which, in turn, generalizes some results obtained by S. L. Sobolev \((^1)\), V. I. Kondrashev \((^2)\), V. P. Il’in \((^3)\), and L. V. Kantorovich \((^4)\).

Theorem 2. If \(f(\bar{y}) \in \mathbf{L}_p(R^n)\),

\[ \lambda=\frac{n}{p'}+\frac{n-s}{p_2}+\frac{s}{p_1}, \qquad p_2 \geq p_1 > p > 1, \]

then

\[ U(\bar{x})=\int_{R^n} f(\bar{y}) r^{-\lambda}\, d\bar{y} \in \mathbf{L}_{(p_1,p_2)}(R^n) \tag{1} \]

and, moreover, the inequality

\[ \|U\|_{\mathbf{L}_{(p_1,p_2)}(R^n)} \leq c \|f\|_{\mathbf{L}_p(R^n)} \tag{2} \]

holds, where \(c\) is a constant independent of \(f\).

If one uses a known theorem of functional analysis, then the proof of inequality (2) is equivalent to the proof of the inequality

\[ I = \int_{R_x^{\,n}} \int_{R_y^{\,n}} \frac{ f(\bar{y})\, \varphi_2(\bar{x})\, \varphi_1^{1/p_1}(\bar{x}_{n-s}) }{ s^\lambda } \, d\bar{y}\, d\bar{x} \leq c\|\varphi_2\|_{\mathbf{L}_{p_1'}(R^n)} \|f\|_{\mathbf{L}_p(R^n)} \|\varphi_1\|_{\mathbf{L}_{(p_1/p_1)'}(R)^{n-s}}^{1/p_1}, \tag{3} \]

where

\[ (p_2/p_1)^{-1}+[(p_2/p_1)']^{-1}=1. \]

Introduce the notation:

\[ r_1=\left[\sum_1^s y_i^2\right]^{1/2}; \qquad r_2=\left[\sum_1^s x_i^2\right]^{1/2}; \qquad r_3=\left[\sum_1^s (x_i-y_i)^2\right]^{1/2}; \]

\[ r_4=\left[\sum_{s+1}^n (x_i-y_i)^2\right]^{1/2}; \qquad \bar{x}_s=(x_1,x_2,\ldots,x_s); \qquad \bar{x}_{n-s}=(x_{s+1},x_{s+2},\ldots,x_n). \]

We transform the left-hand side of inequality (3)

\[ I=\int_{R_x^{n-s}}\varphi_1^{1/p_1}(\bar x_{n-s}) \left[ \int_{R_y^{n-s}} \left( \int_{R_x^s}\int_{R_y^s} \frac{f(\bar y_s,\bar y_{n-s})\varphi_2(\bar x_s,\bar x_{n-s})}{r^\lambda} \,d\bar y_s\,d\bar x_s \right)d\bar y_{n-s} \right]d\bar x_{n-s}. \]

In estimating this integral we may assume that \(f,\varphi_1,\varphi_2\) are nonnegative. On the basis of S. L. Sobolev’s lemma \((^1)\) and its generalization (see \((^3)\)) we have

\[ I\leq \int_{R_x^{n-s}}\varphi_1^{1/p_1}(\bar x_{n-s}) \left[ \int_{R_y^{n-s}} \left( \int_{R_x^s}\int_{R_y^s} \frac{f^*(r_1,\bar y_{n-s})\varphi_2^*(r_2,\bar x_{n-s})}{r^\lambda} \,d\bar y_s\,d\bar x_s \right)d\bar y_{n-s} \right]d\bar x_{n-s}, \tag{4} \]

where \(f^*(r_1,\bar y_{n-s})\) is a nonincreasing function of \(r_1\) for all \(\bar y_{n-s}=(y_{s+1},y_{s+2},\ldots,y_n)\); \(\varphi_2^*(r_2,\bar x_{n-s})\) is a nonincreasing function of \(r_2\) for all \(\bar x_{n-s}=(x_{s+1},x_{s+2},\ldots,x_n)\).

We split the integral in the round brackets on the right-hand side of inequality (4) into three summands \(B_1,B_2\), and \(B_3\), where

\[ B_i=\iint_{\substack{r_k\leq r_i\\ r_j\leq r_i}} \frac{f^*(r_1,\bar y_{n-s})\varphi_2^*(r_2,\bar x_{n-s})}{r^\lambda} \,d\bar x_s\,d\bar y_s \tag{5} \]

\[ (i\ne k;\ i\ne j;\ k\ne j;\ k,j,i=1,2,3); \]

then

\[ I\leq \sum_{i=1}^3 \int_{R_x^{n-s}}\varphi_1^{1/p_1}(\bar x_{n-s}) \left[ \int_{R_y^{n-s}}B_i\,d\bar y_{n-s} \right]d\bar x_{n-s} = \sum_{i=1}^3 I_i, \]

where

\[ I_i= \int_{R_x^{n-s}}\varphi_1^{1/p_1}(\bar x_{n-s}) \left[ \int_{R_y^{n-s}}B_i\,d\bar y_{n-s} \right]d\bar x_{n-s}. \]

In view of the fact that the quantities \(I_1,I_2\), and \(I_3\) are estimated analogously, it is sufficient to estimate one of them, for example \(I_2\).

To estimate \(B_2\) we pass to polar coordinates and, using the lemma of S. L. Sobolev \((^1)\), p. 474, obtain

\[ B_2= \iint_{\substack{r_1\leq r_2\\ r_3\leq r_2}} \frac{f^*(r_1,\bar y_{n-s})\varphi_2^*(r_2,\bar x_{n-s})}{r^\lambda} \,d\bar x_s\,d\bar y_s \leq \]

\[ \leq c_1\int_0^\infty \varphi_2^*(r_2,\bar x_{n-s})r_2^{(s-1)/p'} F(r_2,\bar y_{n-s}) \left( \int_0^{r_2} \frac{r_3^{s-1}\,dr_3}{(r_3^2+r_4^2)^{1/2\lambda}} \,dr_2 \right), \tag{6} \]

where

\[ F(r_2,\bar y_{n-s}) = r_2^{-s/p'-1/p} \int_0^{r_2} f^*(r_1,\bar y_{n-s})r_1^{s-1}\,dr_1. \]

Taking into account inequality (6) and carrying out the necessary transformations, we shall have

\[ \int_{R_y^{n-s}}B_i\,d\bar y_{n-s} \leq c_1\int_0^\infty \varphi_2^*(r_2,\bar x_{n-s})r_2^{(s-1)/p'} \times \]

\[ \times \left[ \int_{R_y^{n-s}} F(r_2,\bar y_{n-s}) \left( \int_0^{r_2} \frac{r_3^{s-1}\,dr_3}{(r_3^2+r_4^2)^{1/2\lambda}} \right) d\bar y_{n-s} \right]dr_2. \tag{7} \]

Since, by the hypothesis of the theorem, \(p_2 \geqslant p_1 > p > 1\) and

\[ \lambda=\frac{n-s}{p_2}+\frac{s}{p_1}+\frac{n}{p'} =(n-s)\left(\frac1{p_2}+\frac1{p'}\right) +s\left(\frac1{p_1}+\frac1{p'}\right), \]

after simple estimates we obtain

\[ \int_0^\infty \varphi_2^*(r_2,\bar x_{n-s})\,r_2^{(s-1)/p'} \left[ \int_{R_y^{\,n-s}} F(r_2,\bar y_{n-s}) \left( \int_0^{r_2}\frac{r_3^{s-1}\,dr_3}{(r_3^2+r_4^2)^{1/2\lambda}} \right) d\bar y_{n-s} \right]dr_2 \]

\[ \leqslant c_2\int_0^\infty \varphi_2^*(r_2,\bar x_{n-s})\, r^{\,s/p_1'-1/p'} \left[ \int_{R_y^{\,n-s}} F(r_2,\bar y_{n-s})\, r_4^{-(n-s)(1/p_2+1/p')}\,d\bar y_{n-s} \right]dr_2 . \tag{8} \]

Let us now estimate the expression

\[ I_2= \int_{R_x^{\,n-s}}\varphi_1^{1/p_1}(\bar x_{n-s}) \left[ \int_{R_y^{\,n-s}} B_2\,d\bar y_{n-s} \right]d\bar x_{n-s}. \tag{9} \]

Substituting in (9), in place of the expression in square brackets, its estimate from (7) and (8), and changing the order of integration in the resulting expression, we shall have

\[ I_2\leqslant c_3\int_0^\infty r^{\,s/p_1'-1/p'} \left[ \int_{R_x^{\,n-s}} \varphi_1^{1/p_1}(\bar x_{n-s}) \varphi_2^*(r_2,\bar x_{n-s}) \right. \]

\[ \left. {}\times \left( \int_{R_y^{\,n-s}} F(r_2,\bar y_{n-s})\, r_4^{-(n-s)(1/p_2+1/p')}\, d\bar y_{n-s} \right)d\bar x_{n-s} \right]dr_2 . \]

To the integral standing in square brackets we apply Hölder’s inequality. Taking into account the properties of integrals of potential type for the limiting exponent, we obtain

\[ I_2\leqslant c_3\int_0^\infty r_2^{\,s/p_1'-1/p'} \left[ \int_{R_x^{\,n-s}} \left|\varphi_1^{1/p_1}(\bar x_{n-s}) \varphi_2^*(r_2,\bar x_{n-s})\right|^{p_2'}d\bar x_{n-s} \right]^{1/p_2'} \]

\[ {}\times \left( \int_{R_y^{\,n-s}} |F(r_2,\bar y_{n-s})|^p\,d\bar y_{n-s} \right)^{1/p}dr_2 . \tag{10} \]

By hypothesis \(p_2>p_1>p\) (for \(p_2=p_1\) the theorem was proved by S. L. Sobolev \((^1)\)); therefore we consider the positive numbers

\[ \lambda=\frac{p_1(p_2-1)}{p_2-p_1}; \qquad \lambda'=\frac{p_1(p_2-1)}{p_2(p_1-1)} \qquad \left(\frac1\lambda+\frac1{\lambda'}=1\right) \]

and apply Hölder’s inequality to the integral standing in square brackets on the right-hand side of inequality (10), which, for convenience, we denote by \(\bar I_2^{\,p_2'}\). Then

\[ \bar I_2\leqslant \|\varphi_1\|_{L_{(p_2/p_1)'}(R^{\,n-s})}^{1/p_1} \left\{ \int_{R_x^{\,n-s}} [\varphi_2^*(r_2,\bar x_{n-s})]^{p_1'}\,d\bar x_{n-s} \right\}^{1/p_1}. \tag{11} \]

From (10), taking (11) into account, after elementary transformations we obtain

\[ I_2\leqslant c_4\|\varphi_1\|_{L_{(p_2/p_1)'}(R^{\,n-s})}^{1/p_1} \int_0^\infty [\chi(r_2)]^{p/p_1}\Phi(r_2) \left[I_2^{1/p}\chi(r_2)\right]^{1-1/p_1}\,dr_2, \tag{12} \]

where

\[ \chi(r_2)=\left(\int_{R_y^{\,n-s}} |F(r_2,\bar y_{n-s})|^p\,d\bar y_{n-s}\right)^{1/p}, \]

\[ \Phi(r_2)=r_2^{(s-1)/p'_1}\left\{\int_{R_x^{\,n-s}}[\varphi_2^*(r_2,\bar x_{n-s})]^{p'_1}\,d\bar x_{n-s}\right\}^{1/p'_1}. \]

Let us estimate the expression

\[ r_2^{1/p}\chi(r_2)=r_2^{-s/p'}\left[\int_{R_y^{\,n-s}}\left(\int_0^{r_2} f^*(r_1,\bar y_{n-s})\,r_1^{s-1}\,dr_1\right)^p d\bar y_{n-s}\right]^{1/p} \]
\[ = c_5 r_2^{-s/p'}\left[\int_{R_y^{\,n-s}}\left(\int_{C_{r_2}} f^*(r_1,\bar y_{n-s})\,d\bar y_s\right)^p d\bar y_{n-s}\right]^{1/p}, \]

where \(C_{r_2}\) is the ball of radius \(r_2\) in the \(s\)-dimensional space \((y_1,y_2,\ldots,y_s)\) with center at the origin. Applying Hölder’s inequality to the inner integral and taking into account that \(\|f^*\|_{L_p}=\|f\|_{L_p}\) \((^5)\), we obtain the estimate

\[ r_2^{1/p}\chi(r_2)\le c_6\|f\|_{L_p(R^n)}; \tag{13} \]

therefore

\[ I_2\le c_7\|\varphi_1\|_{L_{(p_2/p_1)'} }^{1/p_1}\|f\|_{L_p}^{\,1-p/p_1} \left\{\int_0^\infty |\chi(r_2)|^p\,dr_2\right\}^{1/p_1} \left\{\int_0^\infty |\Phi(r_2)|^{p'_1}\,dr_2\right\}^{1/p'_1}. \tag{14} \]

Let us estimate the integral

\[ \int_0^\infty |\chi(r_2)|^p\,dr_2 = \int_{R_y^{\,n-s}} \left[\int_0^\infty r_2^{-(s/p'+1/p)} \left(\int_0^{r_2} f^*(r_1,\bar y_{n-s})\, r_1^{(s-1)/p}r_1^{(s-1)/p'}\,dr_1\right)^p dr_2\right]d\bar y_{n-s} \tag{15} \]

Using Hardy’s inequality \((^6)\), for estimating the integral standing in the square brackets on the right-hand side of equality (15), we obtain

\[ \int_0^\infty |\chi(r_2)|^p\,dr_2 \le c_8\|f\|_{L_p(R^n)}^p. \tag{16} \]

It is also not difficult to show that

\[ \int_0^\infty |\Phi(r_2)|^{p'_1}\,dr_2 \le c_9\|\varphi_2\|_{L_{p'_1}(R^n)}^{p'_1}. \tag{17} \]

From (14), taking (16) and (17) into account, we shall have

\[ I_2\le c_{10}\|f\|_{L_p(R^n)}\|\varphi_2\|_{L_{p'_1}(R^n)}\|\varphi_1\|_{L_{(p_2/p_1)'}(R^{n-s})}^{1/p_1}. \]

Theorem 2 is completely proved.

I take this opportunity to express my gratitude to S. V. Uspenskii for the valuable comments he made in carrying out this work.

Institute of Mathematics with Computing Center
of the Siberian Branch of the Academy of Sciences of the USSR

Received
6 VII 1962

REFERENCES

\(^1\) S. L. Sobolev, Matem. sborn., 4 (46), No. 3 (1938).
\(^2\) V. I. Kondrashev, DAN, 48, 563 (1945).
\(^3\) V. P. Il’in, DAN, 96, 905 (1954).
\(^4\) L. V. Kantorovich, UMN, 11, 2 (68), 3 (1956).
\(^5\) G. H. Hardy, J. E. Littlewood, G. Pólya, Inequalities, 1948.
\(^6\) G. Pólya, G. Szegő, Isoperimetric Inequalities in Mathematical Physics, 1962.