Abstract
Full Text
UDC 517.43
MATHEMATICS
V. B. KOROTKOV
ON THE CARLEMANNESS OF THE RESOLVENTS OF CERTAIN LINEAR OPERATORS
(Presented by Academician S. L. Sobolev, July 1, 1968)
In the theory of expansions in generalized eigenfunctions of self-adjoint operators, the Carleman property of the resolvents* of these operators is used in an essential way \((^{1-3})\).
The present article is devoted to finding necessary and sufficient conditions under which the resolvent of a linear unbounded operator is an integral operator of Carleman type (for the definition of a Carleman operator see \((^{4})\), p. 748).
Theorem 1. Let \(\Omega\) be a certain domain in the Euclidean space \(R_n\). Let \(T\) be a linear operator defined on an everywhere dense linear manifold \(D_T\) in \(L_2(\Omega)\), and let \(\lambda\) be a regular point of the operator \(T\). In order that the operator \(R_\lambda=(T-\lambda E)^{-1}\) be an integral operator of Carleman type, it is necessary and sufficient that there exist a partition of the domain \(\Omega\) into a finite or countable set of nonintersecting bounded measurable sets \(\Omega_m\), \(m=1,2,\ldots\), such that for every function \(f\in D_T\), \(f\in L_\infty(\Omega_m)\), \(m=1,2,\ldots\).
Proof. Necessity. Let \(R_\lambda=(T-\lambda E)^{-1}\) be an integral operator of Carleman type, i.e.
\[ (R_\lambda f)(s)=\int_{\Omega} K(s,t)f(t)\,dt,\qquad f\in L_2(\Omega), \]
where \(K(s,t)\) is a measurable complex-valued function defined on \(\Omega\times\Omega\), satisfying the condition
\[ \int_{\Omega}|K(s,t)|^2\,dt<\infty \qquad \text{for almost all } s\in\Omega. \]
Consider the function
\[ K(s)=\left(\int_{\Omega}|K(s,t)|^2\,dt\right)^{1/2}. \]
\(K(s)\) is a measurable, finite almost everywhere function. Let
\[ \Omega_m=\left(s=(s_1,\ldots,s_n),\ m-1<K(s)\le m,\ \max_{j=1,2,\ldots,n}|s_j|\le m\right), \]
\(m=1,2,\ldots\), be measurable, nonintersecting bounded sets. Let \(f\in D_T\). Then there exists a function \(g\in L_2(\Omega)\) such that \(f=R_\lambda g\). Moreover, for almost all \(s\in\Omega_m\), \(m=1,2,\ldots\),
\[ |f(s)|=|(R_\lambda g)(s)|=\left|\int_{\Omega}K(s,t)g(t)\,dt\right|\le \]
\[ \le \left(\int_{\Omega}|K(s,t)|^2\,dt\right)^{1/2}\|g\|=K(s)\|g\|\le m\|g\|. \]
* Or of some power of the resolvent.
** That is, the operator \((T-\lambda E)^{-1}\) exists, is defined on the whole space \(L_2(\Omega)\), and is bounded.
Sufficiency. Let \(\Omega_m,\ m=1,2,\ldots,\) be the partition of the domain \(\Omega\) indicated in the condition of Theorem 1. Define the operator \(P_m\) by the equality
\[
P_m f=\chi_{\Omega_m}(s)f(s),\qquad f\in L_2(\Omega),
\]
where \(\chi_{\Omega_m}\) is the characteristic function of the set \(\Omega_m\). The operator \(P_mR_\lambda\) is a restricted operator acting in \(L_2(\Omega)\). At the same time, the range of the operator \(P_mR_\lambda\) belongs to \(L_\infty(\Omega)\). It is not difficult to verify that the operator \(P_mR_\lambda,\ m=1,2,\ldots,\) is a closed operator acting from \(L_2(\Omega)\) into \(L_\infty(\Omega)\). Hence it follows ((5), p. 70) that \(P_mR_\lambda\) is a bounded operator acting from \(L_2(\Omega)\) into \(L_\infty(\Omega)\).
Thus,
\[
\operatorname*{vrai\,sup}_{x\in\Omega_m}|(R_\lambda g)(s)|\leq C_m\|g\|,\qquad m=1,2,\ldots .
\tag{1}
\]
Define the function \(\Lambda(s)\) by the equality \(\Lambda(s)=C_m\) if \(s\in\Omega_m,\ m=1,2,\ldots\). The function \(\Lambda(s)\) is measurable and finite almost everywhere. From (1) and the definition of the function \(\Lambda(s)\) it follows that, for almost all \(s\in\Omega\),
\[
|(R_\lambda g)(s)|\leq \Lambda(s)\|g\|,\qquad g\in L_2(\Omega).
\]
Hence, by virtue of (4), it follows that \(R_\lambda\) is an integral operator of Carleman type.
Corollary 1. Let \(\lambda\) be a regular point of a linear manifold \(D_T\), dense everywhere in \(L_2(\Omega)\), of the linear operator \(T\). Suppose that for every function \(f\in D_T\) and every compact subset \(\widetilde{\Omega}\subset\Omega\) one has \(f\chi_{\widetilde{\Omega}}\in L_\infty(\Omega)\). Then \((T-\lambda E)^{-1}\) is an integral operator of Carleman type.
Proof. Cover the domain \(\Omega\) by a system of pairwise disjoint half-open cubes \(d_k\) lying strictly inside \(\Omega\). Consider the closure \(\overline{d}_k\) of the cube \(d_k\). Since, by the assumption, \(f\chi_{\overline{d}_k}\in L_\infty(\Omega)\), \(f\in D_T\), it follows by Theorem 1 that \((T-\lambda E)^{-1}\) is an integral operator of Carleman type.
Theorem 2. Let \(\lambda\) be a regular point of the operator \(T\), \(\overline{D}_T=L_2(\Omega)\). In order that the operator \((T-\lambda E)^{-1}\) be an integral operator of Carleman type, it is necessary and sufficient that, for every \(\varepsilon>0\), there exist a measurable set \(E_\varepsilon,\ mE_\varepsilon<\varepsilon\), such that for every function \(f\in D_T\) one has \(f\chi_{\Omega\setminus E_\varepsilon}\in L_\infty(\Omega)\).
Proof. Necessity. Let \(f\in D_T\). Then there exists \(g\in L_2(\Omega)\) such that
\[
f=(T-\lambda E)^{-1}g=\int_\Omega K(s,t)g(t)\,dt.
\]
Consider the function
\[
K(s)=\left(\int_\Omega |K(s,t)|^2\,dt\right)^{1/2}.
\]
Since \(K(s)\) is a measurable finite function almost everywhere, for any \(\varepsilon>0\) there exists a set \(E_\varepsilon,\ mE_\varepsilon<\varepsilon\), such that for all \(s\in\Omega\setminus E_\varepsilon\) one has \(K(s)\leq K\). But then, for almost all \(s\in\Omega\setminus E_\varepsilon\),
\[
|f(s)|=|(T-\lambda E)^{-1}g(s)|=\left|\int_\Omega K(s,t)g(t)\,dt\right|\leq K(s)\|g\|\leq K\|g\|.
\]
Sufficiency. From the condition of Theorem 2 it follows that there exists a sequence of measurable sets \(E_m,\ E_1\subset E_2\subset\cdots\subset E_m\subset\cdots\subset\Omega\), such that \(f\chi_{E_m}\in L_\infty(\Omega)\), \(f\in D_T\). Let
\[
\Omega_{1,m}=E(s:s=(s_1,\ldots,s_n),\ s\in E_1,\ m-1<\max_{j=1,\ldots,n}|s_j|\leq m).
\]
Set \(G_l=E_l\setminus E_{l-1}\), \(l=2,3,\ldots\), and
\[
\Omega_{l,j}=E(s:s=(s_1,\ldots,s_n),\ s\in G_l,\ j-1<\max_{i=1,\ldots,n}|s_i|\leq j).
\]
The sets \(\Omega_{l,j};\ l=1,2,\ldots,\ j=1,2,\ldots,\) are pairwise disjoint, measurable, bounded, and
\[
\Omega=\bigcup_{j=1,\ l=1}^{\infty}\Omega_{l,j}.
\]
At the same time \(f\chi_{\Omega_{l,j}}\in L_\infty(\Omega)\), \(f\in D_T\). By virtue of Theorem 1, it follows from this that \((T-\lambda E)^{-1}\) is an integral operator of Carleman type.
Theorem 3. Let \(\lambda\) be a regular point of the operator \(T\), \(\overline{D_T}=L_2(\Omega)\). In order that the operator \((T-\lambda E)^{-1}\) be an integral operator of Carleman type, it is necessary and sufficient that for every \(\varepsilon>0\) there exist a set \(E_\varepsilon\), \(mE_\varepsilon<\varepsilon\), such that, whatever function \(f\in D_T\) is taken, the restriction of this function to the set \(\Omega\setminus E_\varepsilon\) is a continuous bounded function after alteration on a set of measure zero.
Proof. Necessity. Let \((T-\lambda E)^{-1}\) be an integral operator of Carleman type, and let \(K(s,t)\) be its kernel. As was shown in \((^6)\), the kernel \(K(s,t)\) determines, by the equality \(\overline{K(s,\cdot)}=\varphi(s)\), a measurable abstract function \(\varphi(s)\) taking values in \(L_2(\Omega)\). By virtue of the known generalization of N. N. Luzin’s theorem to the case of abstract functions, for every \(\varepsilon>0\) there is a set \(H_\varepsilon\), \(mH_\varepsilon<\varepsilon/2\), such that the restriction of the function \(\varphi(s)\) to the set \(\Omega\setminus H_\varepsilon\) will be a continuous abstract function. Since the function
\[ K(s)=\|\varphi(s)\|=\left(\int_\Omega |K(s,t)|^2\,dt\right)^{1/2} \]
is measurable and finite almost everywhere, for every \(\varepsilon>0\) there is a set \(G_\varepsilon\subset\Omega\setminus H_\varepsilon\), \(mG_\varepsilon<\varepsilon/2\), such that the restriction of the function \(K(s)=\|\varphi(s)\|\) to the set \(\Omega\setminus H_\varepsilon\cup G_\varepsilon\) will be a bounded function. Let \(f\in D_T\). Then there is a \(g\in L_2(\Omega)\) such that, after alteration on a set of measure zero, for all \(s\in\Omega\setminus H_\varepsilon\cup G_\varepsilon\),
\[ f(s)=((T-\lambda E)^{-1}g)(s)=(g,\varphi(s)). \]
This implies the validity of the assertion being proved.
Sufficiency follows in an obvious way from Theorem 2.
Example 1. Let \(P\) be a maximal complete hypoelliptic operator with constant coefficients. Let \(\lambda\) be a regular point of \(P\). Then there exists a natural number \(k\) such that \((P-\lambda E)^{-k}\) is an integral operator of Carleman type.
Proof. As is known \(((^7), p. 105)\), there is a natural number \(k\) such that a function \(f\in D_{P^k}\), after alteration on a set of measure zero, will be a continuous function. We note that
\[ D_{P^k}=D_{(P-\lambda E)^k}, \]
and use Corollary 1 for the operator \((P-\lambda E)^{-k}\).
Example 2. Let \(P(x,D)\) be a linear differential operator of constant strength with coefficients from \(C^\infty(\Omega)\). Let the operator \(P_0(D)=P(x_0,D)\) be hypoelliptic at some point \(x_0\in\Omega\). Let zero be a regular point of the operator \(P(x,D)\). Then there is a natural number \(k\) such that \([P(x,D)]^{-k}\) is an integral operator of Carleman type.
This assertion follows from \((^8)\), 7.4.1, 2.3.7, and Corollary 1.
Example 3. Let \(L\) be a strongly elliptic operator of order \(2m\) with coefficients from \(C^\infty(\Omega)\), and let \(0\) be a regular point of \(L\). Then \(L^{-k}\) is an integral operator of Carleman type if \(4mk>n\).
By K. Friedrichs’ theorem \(((^9), p. 248)\), \(L^{-k}\) acts from \(L_2(\Omega)\) into \(W_2^{2mk}(\Omega)\). If \(4mk>n\), then by the imbedding theorem of S. L. Sobolev \(((^{10}), p. 64)\), \(W_2^{2mk}(\Omega)\) is imbedded in \(C(\Omega)\). By virtue of Corollary 1 it follows from this that \(L^{-k}\) is an integral operator of Carleman type.
Institute of Mathematics
Siberian Branch of the Academy of Sciences of the USSR
Received
10 V 1968
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