UDC 517.5

M. D. RAMAZANOV

THEOREMS ON TRACES

AND EXTENSIONS OF FUNCTIONS FROM SURFACES

(Presented by Academician S. L. Sobolev on 28 VIII 1968)

1°. Consider functions \(f(x)\), defined in \(n\)-dimensional Euclidean space \(R^n\), with finite \(\nu\)-norm

\[ \|f\|_\nu=\left(\int\left|\int f(x)\nu(x,\xi)\exp(-ix\xi)\,dx\right|^2\,d\xi\right)^{1/2}, \tag{1} \]

where \(\nu(x,\xi)\) is a certain generalized function; \(x=x_1,\ldots,x_n\); \(\xi=\xi_1,\ldots,\xi_n\). The aim of the present paper is to study the values of functions \(f(x)\) on \((n-1)\)-dimensional smooth surfaces \(\Gamma\).

If \(\nu(x,\xi)=(|\xi|+1)^r\) and (1) defines an isotropic \(\nu\)-norm coinciding with the norm \(W_2^r\) of the Sobolev space \((^1)\), then the problem is easily reduced to the special case where \(\Gamma\) is a coordinate hyperplane, and is solved in terms of inverse embedding theorems \((^1,^2)\). If the function \(\nu\) does not depend on \(x\), then the problem is solved when \(\Gamma\) is a hyperplane \((^3)\), and results are known for some more general surfaces \(\Gamma\) \((^4)\).

We specify \(\Gamma\) by the conditions

\[ x=\kappa(y');\quad y'=y_1,\ldots,y_{n-1};\quad -\infty<y_j>\infty,\quad j=1,\ldots,n-1. \tag{2} \]

For functions \(h(y')\) defined on \(\Gamma\), define a \(\lambda\)-norm of the form (1):

\[ \|h\|_\lambda'=\left(\int\left|\int h(y')\lambda(y',\xi')\exp(-iy'\xi')\,dy'\right|^2\,d\xi'\right)^{1/2}. \tag{1'} \]

In the paper, an equality (4) is obtained which relates the functions \(\nu\) and \(\lambda\). The fulfillment of this equality (4) is a necessary and sufficient condition in order that every function on \(R^n\) with finite \(\nu\)-norm should have on \(\Gamma\) a finite \(\lambda\)-norm, with \(\lambda\) turning out to be such that every function defined on \(\Gamma\) with finite \(\lambda\)-norm can be extended from \(\Gamma\) to all of \(R^n\) with finite \(\nu\)-norm.

In what follows we use the following notation: \(Ff=\int f(x)\exp(-ix\xi)\,dx\) is the Fourier transform operator; \(F_n[f(x)]=(F_nf)(x',\xi_n)\) is the Fourier transform with respect to the variable \(x_n\); \(\delta_n=\delta(x_n)\) is the \(\delta\)-function in the variable \(x_n\);

\[ A_{\nu1}f=\int f(x)\nu(x,\xi)\exp(-ix\xi)\,dx;\qquad A_{\nu2}f=F^{-1}A_{\nu1}f; \]

\(C_0^\infty(R^n)\) is the set of infinitely differentiable finite functions of \(n\) variables; \(\mathscr L_2\) is the space of square-summable functions;

\[ (f,g)=\int f(x)\overline{g(x)}\,dx,\qquad \|f\|=(f,f)^{1/2}. \]

Primes will denote operations on functions of \((n-1)\) variables, for example

\[ \mathscr L_2'=\mathscr L_2(R^{n-1}),\qquad A_{\lambda1}'h=\int h(x')\lambda(x',\xi')\exp(-ix'\xi')\,dx'; \]

\[ (f,g)'=\int f(x')\overline{g(x')}\,dx'. \]

\(I'\) denotes proposition I in which \(n\) is replaced by \((n-1)\).

The space \(H_\nu\) is the set of functions from \(\mathscr L_2\) with finite \(\nu\)-norm (1). We impose on the function \(\nu(x,\xi)\) two basic requirements:

I. For some natural number \(r\) and any function from the Sobolev space \(W_2^r\), the norm (1) is finite.

II. The equation \(A_{\nu1}f=g\) is uniquely solvable in \(\mathscr L_2\) for any function \(g\in\mathscr L_2\).

When conditions I, II are fulfilled, we shall call the function $\nu$ admissible (respectively $\lambda$-admissible, if I′, II′ are fulfilled). From conditions I, II there follows the embedding $\mathscr L_2 \supset H_\nu \supset W_2^r$ together with the topology and the density of $W_2^r$ and $C_0^\infty$ in $H_\nu$.

Let $x=\varphi(y)$ be a smooth change of variables taking $\Gamma$ into the hyperplane $y_n=0$; more precisely, it is assumed that $\varphi(y',0)=\chi(y')$, and if $D\varphi=|(\partial\varphi_k/\partial y_j)|$ is the Jacobian of the change, then $0<c_1\le D\varphi\le c_2<\infty$. As a result of this change the space $H_\nu$ passes into $H_\omega$, where
\[ \omega(y,\xi)=\nu(\varphi(y),\xi)\exp[i(y-\varphi(y))\xi]D\varphi(y). \]
Clearly, the function $\omega(y,\xi)$ is again admissible. Therefore in $H_\nu$-spaces it suffices for us to consider the special case when $\Gamma$ is the hyperplane $x_n=0$.

Let $\Pi[f(x)]=f(x',0)$. Then our problem can be formulated as follows: indicate conditions under which a pair of admissible functions $\nu(x,\xi)$ and $\lambda(x',\xi')$ defines spaces $H_\nu$ and $H_\lambda'$; the operator $\Pi:H_\nu\to H_\lambda'$ is bounded and acts onto all of $H_\lambda'$. A necessary condition for the solvability of this problem is

III. $H_\nu^*\supset a(x')\delta(x_n)$ for all $a(x')\in\mathscr L_2'$, where $H_\nu^*$ is the space adjoint to $H_\nu$.

We shall say that an admissible function $\mu(x',\xi')$ defines an equivalent norming of the space $H_\lambda'$, if $H_\lambda'=H_\mu'$. On the set $C_0^\infty(R^{n-1})$ define the operator
\[ B\equiv B_{\nu\mu}=A_{\nu2}^{-1}A_{\nu2}^{-1*}\delta_n A_{\mu2}^{\prime *}:C_0^\infty(R^{n-1})\to H_\nu. \tag{3} \]

The main result of the work is now formulated in the following two assertions.

Theorem 1. The operator $\Pi$ maps all of $H_\nu$ onto all of $H_\lambda'$ if and only if in the space $H_\lambda'$ one can prescribe an equivalent norming by a function $\mu(x',\xi')$ such that, for the pair of functions $(\nu,\mu)$, on the set $C_0^\infty(R^{n-1})$ the relation
\[ \Pi B_{\nu\mu}=I' \quad\text{— the identity operator} \tag{4} \]
holds.

Theorem 2. a) If $\mu$ is given with properties I′, I″, then equation (4) with respect to $\nu$ has a solution with properties I, II, III.

b) If $\nu$ is given with properties I, II, III, then equation (4) with respect to $\mu$ has a solution with properties I′, II′.

2°. We shall prove the sufficiency in Theorem 1, then Theorem 2, and then the necessity in Theorem 1.

Sufficiency of condition (4) in Theorem 1. Let $h(x')\in C_0^\infty(R^{n-1})$ and $f(x)=Bh$. Then
\[ \|f\|_\nu^2=\|A_{\nu2}Bh\|^2 =\|A_{\nu2}^{-1*}\delta_n A_{\mu2}^{\prime *}A_{\mu2}'h\|^2 \]
\[ =(\delta_n A_{\mu2}^{\prime *}A_{\mu2}'h,Bh) =(A_{\mu2}^{\prime *}A_{\mu2}'h,\Pi Bh)'=\|h\|_\mu^{\prime 2}. \]

Thus, the operator $B$, acting from $H_\mu'$ into $H_\nu$, is bounded and can be extended by continuity to all of $H_\mu'$. From condition III it follows that on all of $H_\nu$ the operator $\Pi$ is defined with range in $\mathscr L_2'$.

Now take an arbitrary element $h\in H_\mu'$ and consider $f=Bh$ and $\Pi f=\Pi Bh$. The operator $\Pi B:H_\mu'\to\mathscr L_2'$ is bounded and on the set $C_0^\infty(R^{n-1})$, dense in $H_\mu'$, coincides with the identity $I'$. Therefore $\Pi B=I'$ on all $H_\mu'$, i.e. for every element $h\in H_\mu'$ there exists $f=Bh\in H_\nu$ for which $\Pi f=h$.

Conversely, let $f(x)\in H_\nu$. It is necessary to show that $\Pi f=h\in H_\mu'$. Note that $\Pi(f-Bh)=0$ and, as we have shown, $\|h\|_\mu'=\|Bh\|_\nu$. We have
\[ \|f\|_\nu^2=\|A_{\nu2}[(f-Bh)+Bh]\|^2= \]
\[ =\|A_{\nu2}(f-Bh)\|^2+\|A_{\nu2}Bh\|^2 +2\operatorname{Re}(A_{\nu2}(f-Bh),A_{\nu2}Bh)\ge \]
\[ \ge \|A_{\nu2}Bh\|^2 +2\operatorname{Re}((f-Bh),\delta_n A_{\mu2}^{\prime *}A_{\mu2}'h)=\|h\|_\mu^{\prime 2}, \]
which was required to be proved.

Proof of Theorem 2. a). Let \(\pi(\xi_n)\) be a smooth function satisfying
\(1\leq \pi(\xi_n)\leq (|\xi_n|+1)^r\) for some \(r>0\),
\[ \int \pi^{-2}(\xi_n)\,d\xi_n=1. \]
For a given admissible function \(\mu(x',\xi')\), construct
\(\nu(x,\xi)=\mu(x',\xi')\pi(\xi_n)\), which is again admissible. If
\(f\in H_\nu\), then
\[ A_{\nu_1}f=\int (F_nf)(x',\xi_n)\mu(x',\xi')\pi(\xi_n)\exp(-ix'\xi')\,dx' =g(\xi)\in \mathscr L_2, \]
\[ A'_{\mu_1}\Pi f=\int d\xi_n\int (F_nf)(x',\xi_n)\mu(x',\xi')\exp(-ix'\xi')\,dx' \]
\[ =\int g(\xi)\pi^{-1}(\xi_n)\,d\xi_n\in \mathscr L_2'. \]

Thus, \(\Pi f\in H_\mu'\). Therefore for all \(a(x')\in \mathscr L_2'\subset H_\mu'\),
\((\Pi f,a)'=(f,\delta_n a)\), i.e. \(a(x')\delta(x_n)\in H_\nu^*\), and III holds. Let us verify that relation (4) is satisfied:
\[ A_{\nu_2}^{-1}=F_n^{-1}\pi^{-1}(\xi_n)F_nA_{\mu_2}^{\prime -1},\qquad B=F_n^{-1}A_{\mu_2}^{\prime -1}A_{\mu_2}^{\prime -*}\pi^{-2}(\xi_n)F_n\delta_nA_{\mu_2}^*A_{\mu_2}'= \]
\[ =F_n^{-1}\pi^{-2}(\xi_n), \]
\[ \Pi Bh=\int d\xi_n\pi^{-2}(\xi_n)h=h. \]

b) Transform relation (4). For all \(f(x)\in C_0^\infty(R^n)\) we have
\[ \Pi A_{\nu_2}^{-1}A_{\nu_2}^{-1}\delta_nA_{\mu_2}^{\prime *}A_{\mu_2}'\Pi f=\Pi f, \]
or
\[ h(x')=A_{\mu_2}^{\prime *}A_{\mu_2}'\Pi f,\qquad g(x)=A_{\nu_2}^{-1}A_{\nu_2}^{-1*}\delta_nh,\qquad \Pi g=\Pi f. \tag{5} \]

We must find an admissible \(\mu\) such that, for all \(f\in C_0^\infty(R^n)\), the system of equations (5) is compatible with respect to
\(h\in H_\mu^{\prime *}\) and \(g\in H_\nu\). If \(\mu\) is found, then by conditions I′, II′ it is necessary that \(h\in \mathscr L_2'\). Therefore, considering (5) as a system with respect to \(f\) and \(g\) from \(H_\nu\) for a fixed function \(h\in\mathscr L_2'\), we impose on \(\mu\) more stringent restrictions than (4). We shall seek \(\mu\) under these new conditions, assuming \(h\in\mathscr L_2'\). Then \(g\) lies in the range \(R\) of the operator \(A_{\nu_2}^{-1}A_{\nu_2}^{-1*}\delta_n:\mathscr L_2'\to H_\nu\), restricted by virtue of III. The operator \(\Pi\), acting from \(R\subset H_\nu\) to \(\mathscr L_2'\), is bounded:
\[ \|\Pi\|=\sup_g(\|\Pi g\|'/\|g\|_\nu)=\sup_{g,h}(|(g,\delta_nh)|/\|g\|_\nu\|h\|') \leq \sup_h(\|\delta_nh\|_\nu^*/\|h\|')<\infty \]
(by condition III).

The range \(D\) of the operator \(\Pi:R\to\mathscr L_2'\) is dense in \(\mathscr L_2'\): if
\(0=(\psi,\Pi g)'\) for all \(g\in R\), then
\(0=[\psi,\Pi A_{\nu_2}^{-1}A_{\nu_2}^{-1*}\delta_nh)'\) for all \(h\in\mathscr L_2'\), and, taking \(h=\psi\), we obtain
\(0=A_2^{-1*}\delta_n\psi\), i.e. \(\psi=0\). On \(R\), \(\Pi\) has an inverse
\(\Pi^{-1}:\mathscr L_2'\supset D\to R\).

Indeed, if \(g\in R\) and \(\Pi g=0\), then for all \(\varphi\in\mathscr L_2'\)
\[ 0=(\varphi,\Pi g)'=(\varphi\delta_n,g)=(\varphi\delta_n,A_{\nu_2}^{-1}A_{\nu_2}^{-1*}h\delta_n) =(A_{\nu_2}^{-1*}\varphi\delta_n,A_{\nu_2}^{-1*}h\delta_n). \]
Taking \(\varphi=h\), we obtain \(A_{\nu_2}^{-1*}h\delta_n=0\), i.e. \(h=0\) and \(g=0\). Thus, if \(g\in R\), then \(\Pi g=u\in D\) and \(g=\Pi^{-1}u\). The second equation of system (5) is written in the form
\[ h=F_nA_{\nu_2}^*A_{\nu_2}g=F_nA_{\nu_2}^*A_{\nu_2}\Pi^{-1}u. \]

Thus, we seek the function \(\mu\) from the compatibility conditions, for all
\(h\in\mathscr L_2'\), of the system of equations with respect to \(u\in D\):
\[ h=A_{\mu_2}^{\prime *}A_{\mu_2}'u,\qquad h=F_nA_{\nu_2}^*A_{\nu_2}\Pi^{-1}u. \tag{6} \]

Consider
\[ P=F_nA_{\nu_2}^*A_{\nu_2}\Pi^{-1}:\mathscr L_2'\supset D\to\mathscr L_2'. \]
There exists
\[ P^{-1}=\Pi A_{\nu_2}^{-1}A_{\nu_2}^{-1*}\delta_n:\mathscr L_2'\to D \]
and
\[ (P^{-1}h,h)=\|A_{\nu_2}^{-1*}\delta_nh\|^2\geq 0. \]
Thus, \(P\)—

positive operator, and from it one can extract the square root

\[ \sqrt{\overline P}:D\to {\mathscr L}_2'. \tag{5} \]

The domain of definition of \(\sqrt{\overline P}\) can be extended to the set \(D_{\sqrt{\overline P}}\) by adding all those elements of \({\mathscr L}_2'\) which \(\sqrt{\overline P}\) maps into \({\mathscr L}_2'\). \(D_{\sqrt{\overline P}}\), with norm \(\|h\|_{\sqrt{\overline P}}'=\|\sqrt{\overline P}h\|'\), becomes a Hilbert space \(H_{\sqrt{\overline P}}'\), and for all \(\Pi f\in C_0^\infty(R^{n-1})\) condition (4) is fulfilled:

\[ \Pi A_{\nu_2}^{-1} A_{\nu_2}^{-1*}\delta_n \sqrt{\overline P}\sqrt{\overline P}\,\Pi f=\Pi f . \]

Therefore the operator \(\Pi\) maps all of \(H_\nu\) onto all of \(H_{\sqrt{\overline P}}'\), and, according to condition I and the embedding theorems for \(W_2^r(R^n)\), we have the embedding

\[ H_{\sqrt{\overline P}}'\supset W_2^{r-1/2}(R^{n-1}) \]

together with the topology. This means that the operator

\[ K=\sqrt{\overline P}\,F'^{-1}(|\xi'|+1)^{-r+1/2}F':{\mathscr L}_2'\to{\mathscr L}_2' \tag{5} \]

is bounded and is a Hilbert–Schmidt operator.

Therefore there is a function \(k(x',y')\in{\mathscr L}_2(R^{2n-2})\) such that

\[ (Kh)(x')=\int k(x',y')h(y')\,dy' . \]

Thus, also for the operator \(\sqrt{\overline P}\) we can write integral representations

\[ \begin{aligned} (\sqrt{\overline P}\,h)(x') &=KF'^{-1}(|\xi'|+1)^{r-1/2}F'h \\ &=\int (F_{y'}k)(x',\xi')(|\xi'|+1)^{r-1/2}(F'h)(\xi')\,d\xi', \end{aligned} \]

first defined on \(W_2^{r-1/2}(R^{n-1})\), and then extended by continuity to all of \(H_{\sqrt{\overline P}}'\).

Let

\[ \mu(z',\eta')=(2\pi)^{-n}\int dx'\int d\xi'\int dy'\, k(x',y')(|\xi'|+1)^{r-1/2} \exp\{i\xi'(z'-y')+i\eta'(z'-x')\}. \]

Then the operator \(F'^{-1}\sqrt{\overline P}\,h\) can be written as

\[ (F'^{-1}\sqrt{\overline P}\,h)(\xi') =\int h(x')\,\mu(x',\xi')\exp(-ix'\xi')\,dx' . \]

Let us now note that \(H_{\sqrt{\overline P}}'=H_\mu'\), and that the function \(\mu\) satisfies conditions I′, II′. This proves Theorem 2.

Necessity of condition (4) in Theorem 1. It is known that \(\Pi\) maps all of \(H_\nu\) onto all of \(H_\lambda'\). For the given \(\nu\), according to Theorem 2, we find some \(\mu\). Then for every \(h\in H_\lambda'\) there exists an extension \(f\) in \(H_\nu\), and the trace \(\Pi f=h\) lies in \(H_\mu'\). Hence \(H_\mu'\subset H_\lambda'\). But in exactly the same way every \(h\) from \(H_\mu'\) lies in \(H_\lambda'\). Therefore \(H_\mu'=H_\lambda'\).

Institute of Mathematics
Siberian Branch of the Academy of Sciences of the USSR

Received
12 VI 1968

REFERENCES

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