P. I. LIZORKIN

CHARACTERIZATION OF BOUNDARY VALUES OF FUNCTIONS FROM \(L_p^r(E_n)\) ON HYPERPLANES

(Presented by Academician M. A. Lavrent’ev on 12 I 1963)

1. The scales of differentiable functions of many variables \(B_p^r(E_n)\) and \(L_p^r(E_n)\) are constructed on the basis of taking into account the differential properties of functions in the metrics \(L_p\). It is known \((^{1,2})\) that the spaces \(B_p^r(E_n)\) possess a well-ordered system of embedding theorems*, remarkable, in particular, for its closedness with respect to boundary embeddings. In the author’s preceding paper \((^4)\), analogous theorems were obtained for the spaces \(L_p^r(E_n)\), with the sole exception of the boundary embedding theorems just mentioned. In the present paper the boundary values of functions from \(L_p^r(E_n)\) are characterized in terms of \(B_p^{r'}\)-spaces. This complication of the relations (a departure from the scale!) is compensated by the greater precision of the assertions not only for the \(L_p^r\)-spaces themselves, but also for the spaces \(B_p^r\). It turns out, for example, that for \(p>2\) one can ensure a better quality of the extended function than is guaranteed in the theory of \(B_p^r\)-spaces.

2. According to the initial definition \((^4)\), the space \(L_p^r(E_n)\), \(1<p<\infty\), \(r>0\), consists of functions \(f(x)\), \(x=(x_1,\ldots,x_n)\in E_n\), summable with power \(p\) in \(E_n\) together with their unmixed generalized Liouville derivatives of order \(r\). It was shown that the membership of \(f(x)\) in \(L_p^r(E_n)\) is equivalent to its representability by a potential of the form

\[ f(x)=\int_{E_n} G_r^n(x-y)\,\varphi(y)\,dy,\qquad \varphi(x)\in L_p(E_n), \tag{1} \]

where

\[ G_r^n(x)=\frac{|x|^{(r-n)/2}K_{(n-r)/2}(|x|)} {2^{(n+r-2)/2}\pi^{n/2}\Gamma(r/2)}; \]
\(K_\nu(t)\) is the Macdonald function. The norm of \(f\) in \(L_p^r(E_n)\) is taken to be \(\|\varphi\|_{L_p(E_n)}\).

The space \(B_p^r(E_n)\), \(1\le p<\infty\), \(r>0\) \((^{1,2})\), is the closure of the set of smooth finite functions with respect to the norm

\[ \|f\|_{B_p^r(E_n)} = \|f\|_{L_p(E_n)} + \sum_{i=1}^{n} \left\{ \int_{E_n}\int_{E_n} \frac{ \left| f_i^{(\bar r)}(x) - 2f_i^{(\bar r)}\!\left(\frac{x+y}{2}\right) + f_i^{(\bar r)}(y) \right|^p } {|x-y|^{\,n+p(r-\bar r)}}\,dx\,dy \right\}^{1/p}, \]

where \(\bar r\) is the greatest integer less than \(r\), and \(f_i^{(\bar r)}=\partial^{\bar r}f/\partial x_i^{\bar r}\), \(i=1,\ldots,n\).

The relation between the spaces \(B_p^r(E_n)\) and \(L_p^r(E_n)\) is given by the theorem:

Theorem 1**. The following continuous embeddings hold

a) \(B_p^r(E_n)\subset L_p^r(E_n)\), \(\quad 1<p\le 2\);

b) \(L_p^r(E_n)\subset B_p^r(E_n)\), \(\quad 2\le p<\infty\).

1. Another aspect of the connection between the spaces \(L_p^r(E_n)\) and \(B_p^r(E_n)\) is the somewhat paradoxical fact that the boundary values of functions from \(L_p^r(E_n)\) (as well as of functions from \(B_p^r(E_n)\)!) are conversely characterized in terms of \(B_p^{r'}(E_m)\)-spaces. By the trace of \(f(x)\) on the hyperplane \(E_m\) we shall mean the collection of limiting values \(\tilde f(x)\)

* Similar theorems in such a complete form were first given by S. M. Nikol’skii \((^3)\) for \(H_p^r\)-classes.

** This result was reported by the author in Baku at the Second All-Union Conference on the Constructive Theory of Functions in October 1962.

as \(x\) tends to \(E_m\). Under our conditions the trace of a function is at the same time also the boundary value in the sense of convergence in the \(p\)-mean.

Theorem 2. I. If a function \(f(x) \in L_p^r(E_n)\), then its trace on the hyperplane of \(m\) dimensions forms a function \(f'\) belonging to \(B_p^{r'}(E_m)\),
\[ r' = r - \frac{n-m}{p}, \]
and, moreover, the norm of \(f'\) in \(B_p^{r'}(E_m)\) is majorized by the norm
\[ \|f\|_{L_p^r(E_n)}. \]

II. If on the hyperplane \(E_m\) a function \(f' \in B_p^{r'}(E_m)\) is given, then there exists a function \(f(x) \in L_p^r(E_n)\) having \(f'\) as its trace on \(E_m\), and the norm of \(f\) in \(L_p^r(E_n)\) is majorized by the norm \(\|f'\|_{B_p^{r'}(E_m)}\).

The proof of both parts of the theorem may be carried out for the case \(m=n-1\); the further reduction is effected on the basis of embedding theorems for \(B_p^r\)-spaces \((^1,^2)\). Consider case I. Extend the function \(f \in L_p^r(E_n)\) metaharmonically* into the half-space \(E_{n+1}^+ \{(x,t);\, x \in E_n,\ t>0\}\) in the form
\[ u(x,t)=\frac{2t}{(2\pi)^{(n+1)/2}} \int_{E_n} \frac{K_{(n+1)/2}\bigl(\sqrt{|x-y|^2+t^2}\bigr)} {\bigl(|x-y|^2+t\bigr)^{(n+1)/4}} \,f(y)\,dy . \]

Using representation (1), we obtain
\[ u(x,t)=\int_{E_n} N_n(x-y,t)\varphi(y)\,dy, \]
\[ N_n(x,t)=\frac{2}{(2\pi)^{(n+1)/2}\Gamma(r)} \int_0^\infty \frac{(t+s)s^{r-1}K_{(n+1)/2}\bigl(\sqrt{|x|^2+(t+s)^2}\bigr)} {\{|x|^2+(t+s)^2\}^{(n+1)/4}}\,ds . \]

For the kernel \(N_n(x,t)\) the estimates
\[ |D^k N_n(x,t)| \leq \frac{c}{(|x|^2+t^2)^{(n-r+l)/2}}, \]
are valid, where \(D^k N_n(x,t)\) is any derivative of \(N_n\) of order \(k\), \(0 \leq k \leq l\). On the basis of these estimates, with the aid of the generalized Minkowski inequality and Theorem 319 from \((^5)\), one obtains the inequality
\[ \int_0^\infty t^\beta\,dt \int_{E_{n-1}} \left\{ \sum_{l_1+\cdots+l_{n-1}+m=l} \left| \frac{\partial^l u(x_1,\ldots,x_{n-1},0,t)} {\partial x_1^{l_1}\cdots \partial x_{n-1}^{l_{n-1}}\partial t^m} \right|^p + |u|^p \right\} \,dx' \leq c\|\varphi\|_{L_p}^p, \tag{*} \]
where \(E_{n-1}\) is the hyperplane of the variables \((x_1,\ldots,x_{n-1}) \equiv x'\), \(x_n=0\),
\[ \beta=(p-1)-(\gamma-[\gamma])p,\qquad l=[\gamma]+1,\qquad [\gamma]\text{ is the integer part},\qquad \gamma=r-\frac1p. \]
But the finiteness of the weighted integral \((*)\) ensures that the function
\[ \lim_{t\to0} u(x_1,\ldots,x_{n-1},0,t)=f'(x') \]
belongs to \(B_p^{r-1/p}(E_{n-1})\) and gives the corresponding estimate. It is also clear that
\[ f'(x')=\lim_{x_n\to0} f(x_1,\ldots,x_n), \]
since this last limiting passage can be replaced by a limiting passage in the function \(u(x,t)\) along a suitable \(\Pi\)-shaped contour for \(t>0\).

Let us pass to the second part of Theorem 2. The definition of \(L_p^r(E_n)\) may be expressed by the requirements
\[ f \in L_p(E_n); \]
\[ \frac{\partial^{[r]+1}}{\partial x_i^{[r]+1}} (-\Delta_{x'}+1)^{-\alpha/2} f \in L_p(E_n), \qquad i=1,\ldots,n-1; \]
\[ \frac{\partial^{[r]+1}}{\partial x_n^{[r]+1}} \left(-\frac{\partial^2}{\partial x_n^2}+1\right)^{-\alpha/2} f \in L_p(E_n), \qquad \alpha=[r]+1-r, \]

\[ \text{* That is, by the solution of the problem } \Delta u-u=0,\quad u(x,0)=f(x). \]

where

\[ (-\Delta_{x'}+1)^{-\alpha/2}f=\int_{E_{n-1}}G_\alpha^{\,n-1}(x'-y')f(y',x_n)\,dy', \]

\[ \left(-\frac{\partial^2}{\partial x_n^2}+1\right)^{-\alpha/2}f =\int_{-\infty}^{\infty}G_\alpha^{\,1}(x_n-y_n)f(x',y_n)\,dy_n. \]

Now let \(f'(x')\in B_p^{r'}(E_{n-1})\), \(r'=r-1/p\). Take the metaharmonic extension \(u(x',x_n)\) of the function \(f'(x')\) into the half-space \(E_n^+\) \((x_n>0)\) and extend it by the method of Whitney and Hestenes, preserving smoothness, to the whole space \(E_n\). The function \(U(x)\) thus obtained gives the desired extension. The proof is reduced to estimating the integral

\[ \int_{E_n}\left\{|U|^p+ \left|\frac{\partial^{[r]+1}}{\partial x_n^{[r]+1}} \left(-\frac{\partial^2}{\partial x_n^2}+1\right)^{-\alpha/2}U\right|^p +\sum_{i=1}^{n}\left| \frac{\partial^{[r]+1}}{\partial x_i^{[r]+1}} (-\Delta_{x'}+1)^{-\alpha/2}U \right|^p\right\}dx, \]

which is obtained on the basis of Hilbert’s two-parameter inequality and Theorem 329 from \((^5)\). Various properties of metaharmonic extension are also used, in particular the equality

\[ \int_{E_{n-1}}G_\alpha^{\,n-1}(x'-y')u(y',x_n)\,dy' =\frac{1}{\Gamma(\alpha)}\int_{x_n}^{\infty} \frac{u(x',\tau)\,d\tau}{(\tau-x_n)^{1-\alpha}}, \qquad x_n>0. \]

In the intermediate computations there occur weighted integrals of \(u(x',x_n)\).

Finally, let us formulate a general theorem on the extension of a system of functions. Let \(E_m\) be the hyperplane obtained by fixing the last \(n-m\) coordinates; for definiteness we shall assume \(x_{m+1}=0,\ldots,x_n=0\).

Theorem 3. Let on the hyperplane \(E_m\) there be given a system of functions

\[ f_{\lambda_{m+1},\ldots,\lambda_n}^{\lambda}(x_1,\ldots,x_m) \in B_p^{\,r-\lambda-(n-m)/p}(E_m), \]

\[ \lambda=0,\ldots,l;\qquad r-\lambda-\frac{n-m}{p}>0;\qquad \lambda_i\ge 0,\quad \lambda_{m+1}+\cdots+\lambda_n=\lambda. \]

Then there exists a function of \(n\) variables \(f(x)\) having the properties:

\[ \text{a) }\ f(x)\in L_p^r(E_n);\qquad \text{b) }\left. \frac{\partial^\lambda f} {\partial x_{m+1}^{\lambda_{m+1}}\cdots\partial x_n^{\lambda_n}} \right|_{E_m} =f_{\lambda_{m+1},\ldots,\lambda_n}^{\lambda}(x_1,\ldots,x_m); \]

\[ \text{c) }\ \|f\|_{L_p^r(E_n)} \le c\sum_{\lambda=0}^{l} \sum_{\lambda_{m+1}+\cdots+\lambda_n=\lambda} \left\|f_{\lambda_{m+1},\ldots,\lambda_n}^{\lambda}\right\|_ {B_p^{\,r-\lambda-\frac{n-m}{p}}(E_m)}. \]

The proof is carried out according to the schemes of work \((^6)\).

Let us note in conclusion that, on the whole, the methods of the present paper rely essentially on the use of weighted spaces, following the example of L. D. Kudryavtsev \((^8)\). Let us also recall (see \((^4)\)) that the spaces \(L_p^r(E_n)\) give a natural extension of the \(W_p^{(l)}\)-classification of function spaces according to S. L. Sobolev \((^9)\) to noninteger indices of differentiation; in particular (for integer \(r=l\)),

\[ L_p^l(E_n)\equiv W_p^{(l)}(E_n). \]

Steklov Mathematical Institute
Academy of Sciences of the USSR

Received
25 III 1962

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