Abstract
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MATHEMATICS
A. KHAIMOVICH
ON SOME APPLICATIONS OF A THEOREM OF F. RIESZ
(Presented by Academician S. L. Sobolev, 26 X 1956)
F. Riesz’s theorem has been applied to many problems of functional analysis. Here we shall give some applications concerning functional equations.
§ 1. Let \(A\) be a linear operator whose domain of definition \(D_A\) is a subspace of some Hilbert space \(L_p(\Omega)\), where \(\Omega\) is the domain of definition of the functions forming this subspace. Let \(D\) be the range of the operator \(A\); consider the equation
\[ Au(P)=f(Q),\quad u(P)\in D_A,\quad f(Q)\in D,\quad P\in\Omega . \tag{1} \]
Assume that, when \(f(Q)\) is given, equation (1) has a unique solution in \(D_A\). Then one may assert that the value \(u(P)\) at a certain point \(P_0\) of \(\Omega\) is a functional \(\Phi\), additive and homogeneous with respect to \(f(Q)\). If, moreover,
\[ |u(P_0)|\leq M\|f(Q)\|_{L_p}, \tag{2} \]
then the functional \(\Phi\) is linear. It then follows from F. Riesz’s theorem that there exists a function of \(P\), say \(R_{P_0}(P)=R(P,P_0)\), belonging to \(L_p\) and such that
\[ u(P_0)=\int_{\Omega} R(P_0,P) f(P)\,dP, \tag{3} \]
and the general solution of equation (1) is written in this form.
Special cases. A. If the operator \(A\) is the Laplacian \(\Delta\), \(f\) is a continuous function, and \(u(P)\) belongs to the class of functions defined on \(\Omega\), twice differentiable and vanishing on the boundary of \(\Omega\) (the Dirichlet problem), then equation (1) is Poisson’s equation
\[ \Delta u=f(Q); \]
then inequality (2) can be proved. In this case \(R\) is the Green’s function, whose existence is thus proved directly.
B. If the operator \(A\) is the integral operator
\[ u(P)+\lambda\int_{\Omega} K(P,Q)u(Q)\,dQ, \]
where \(K\) is a Carleman kernel and \(u(P)\) is a function belonging to \(L_2\), then it is easy to prove that
\[ |u(P_0)-f(P_0)|\leq M\|u(P)\|_{L_2}. \]
In the case of a Hilbert–Schmidt kernel \((^5)\), this inequality is valid for every point \(P_0\) of \(\Omega\), except possibly for a set of measure zero.
Using one more known theorem on completely continuous operators (([1], p. 177), we obtain
\[ |u(P)-f(P)|\leq M_1\|f(P)\|_{L_2} \]
and, consequently, (3). In this case the function \(R\) is the resolvent of \(\Gamma(P_0,P;\lambda)\).
In both cases the characteristic properties of the Green function and of the resolvent are obtained directly.
§ 2. Let now \(A\) be a linear operator whose domain of definition \(D_A\) is a subspace of some Banach function space. As in § 1, \(u(P_0)\), satisfying (1), is an additive and homogeneous functional of \(f(P)\). If, in addition,
\[ |u(P_0)|\leq M\|f(Q)\|_B, \]
then, again by Riesz’s theorem, there exists a measure \(R_{P_0}(P)\) such that
\[ u(P)=\int_{\Omega} f(P)\,dR_{P_0}. \tag{3′} \]
Special cases. A. Let \(A\) be a second-order differential operator
\[ \sum_{h,k=1}^{n} a_{hk}(P)\frac{\partial^2 u}{\partial x_h\,\partial x_k} +\sum_{i=1}^{n} b_i(P)\frac{\partial u}{\partial x_i} +c(P)u\equiv Au, \tag{4} \]
which satisfies the conditions: 1) \(\Omega\) is bounded; 2) one can define a function \(\omega(P)\), regular with respect to \(A\), and such that \(A\omega<0\); 3) there exists a function \(w(P)\), nonnegative, regular in \(\Omega+\operatorname{fr}\Omega\), for which \(Aw\leq -1\). Then for every solution of (4), regular in \(\Omega+\operatorname{fr}\Omega\), equal to zero on the boundary of the domain \(\Omega\), we have
\[ |u(P)|\leq w(P)\sup_{\Omega}|f|=w(P)\|f\|_B \]
(see [3], p. 694). In this case the existence of the measure \(R\) holds.
B. If \(A\) is the linear operator defined above, and the class \(D_A\) consists of functions \(u\) satisfying the condition \(\partial u/\partial \nu+bu=0\) and continuous on the boundary \(\Omega\) (see [2]), where the coefficient \(c\) is negative, then we have
\[ |u(P)|\leq \frac{1}{\min |c|}\,\|f\|_B. \]
Then, as above, the measure \(R\) exists.
One can prove that the measure \(R\), depending on the parameter \(P_0\), has the following properties: it is equal to zero when \(P_0\) belongs to the boundary \(\Omega\), and it can be written in the form
\[ dR=\varphi(P_0,P)\,dP, \]
where \(\varphi\) is a function satisfying the condition \(A\varphi=0\) for \(P\ne P_0\), while for \(P=P_0\) it has a singularity of the same type as the elementary solution (in the sense of L. Schwartz (see [4], p. 126 and following)).
Mathematical Seminar named after A. Myller
University of Iași
Iași, Romania
Received
22 X 1956
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