UDC 517.948.32

I. B. SIMONENKO

CONVOLUTION-TYPE OPERATORS IN CONES

(Presented by Academician V. I. Smirnov on 30 XII 1966)

In the present paper the general method set forth in \((^{1,2})\) is applied to multidimensional convolution operators. It gives, for example, necessary and sufficient Noetherian conditions for multidimensional Wiener—Hopf equations in smooth cones. An essential role here is played by an expansion of Euclidean space specially adapted to the given problems \((^{20})\). In the paper new classes of operators are constructed (generalized convolutions, composite convolutions), for which necessary and sufficient Noetherian conditions are obtained. We note that the proposed method also provides the possibility of numerical solution by regularization followed by solution of Fredholm equations.

The study of convolution-type operators makes it possible to consider the associated boundary-value problems for analytic functions of many complex variables in tubular domains. In an analogous way one can also consider discrete convolutions and the boundary-value problems associated with them, for example, the problem of linear conjugation for bicylinders. We shall not touch upon this question here.

One-dimensional Wiener—Hopf equations and their discrete analogues have by now been considered with great completeness. Their solution was carried out by the method of factorization of the coefficients of the corresponding boundary-value problem \((^{3-8})\). Exceptions are the works \((^{4,5})\), whose analysis was one of the starting points of the theory expounded here. For many variables only the case of equations on a half-space has been completely investigated \((^{9,10})\). In those same works (for example \((^{11,12})\)), which were devoted to the study of convolutions in other domains (cones, angles) and of the boundary-value problems associated with them, the authors also used the Wiener—Hopf method, i.e. the method of factorization. This forced one to impose on the kernel of the equation requirements connected exclusively with the method and therefore very far from necessary.

In this paper we proceed in the opposite direction, namely: we investigate convolutions in themselves, simultaneously obtaining information about the boundary-value problems associated with them (without resorting to factorization).

\(1^\circ.\) The ring \(w_p^n\). The space of vector-functions (matrix-functions) of dimension \(n\), defined on the Euclidean space \(E_m\) and summable to the power \(p\) \((1 \le p < \infty)\), will be denoted by \(\mathscr L_p^n(E_m)\) \((\mathscr L_p^{n,n}(E_m))\). It is known that operators \(K\) of the form

\[ (Kf)(x)=cf(x)+\int_{E_m} k(x-y)f(y)\,dy, \tag{1} \]

where \(k \in \mathscr L_1^{\,n,n}(E_m)\), are linear operators \(\mathscr L_p^n \to \mathscr L_p^n\). We shall denote by \(w_p^n\) the closure, in the operator norm, of the set of operators of the form (1). It is clear that operators from \(w_p^n\) are invariant with respect to shifts. In \((^{13})\) it was established that every operator \(A\), invariant with respect to shifts, after the Fourier transform becomes an operator of multiplication by a measurable bounded matrix-function \(\hat A(\xi)\), which, following \((^{14})\), we shall call the symbol of the operator \(A\). The space consisting of functions of the form \(\hat A\), where \(A \in w_p^n\), will be denoted by \(m_p^n\). The norm in \(m_p^n\) is introduced by the equality \(\|\hat A\|_{m_p^n}=\|A\|_p\). In the space \(m_2^1\), obviously, there holds the equality

\[ \|A\|_2=\operatorname{ess\,max}|\hat A(\xi)|,\quad \xi\in \dot E_m . \]
It follows that the functions from \(m_p^n\) are continuous on \(\dot E_m\), and the space \(m_2^n\) coincides (also topologically) with the space of matrix-functions continuous on \(\dot E_m\). Here \(\dot E_m\) denotes the Euclidean space compactified by one point at infinity.

\(2^\circ\). We carry out the compactification of \(E_m\) as follows. To each ray issuing from the origin we assign a point at infinity. On the union of the set of points of \(E_m\) and the points at infinity we introduce a topology. As a fundamental system of neighborhoods of each point at infinity \(x_\infty\) we take sets of the form \((M\cap \Gamma)\cup \Gamma_\infty\). Here \(\Gamma\) is any open cone containing the ray corresponding to the point \(x_\infty\); \(\Gamma_\infty\) is the collection of points at infinity corresponding to rays lying inside \(\Gamma\); \(M\) is the exterior of an arbitrary closed disk. The space \(E_m\) compactified in this way is homeomorphic to the closed ball of dimension \(m\). We shall denote the indicated compactification by \(\tilde E_m\), and the collection of points at infinity by \(\mathfrak N\). We extend the measure to the compactified space \(\tilde E_m\) by declaring \(\operatorname{mes}\mathfrak N=0\).

In what follows, without further reservations we shall use the concepts and notation of \((^1,^2)\).

Lemma 1. An operator \(A(\in w_p^n)\) is an operator of local type.

Lemma 2. If two operators from \(w_p^n\) are locally equivalent at some point at infinity, then they are equal. If the point \(x\) is finite, then
\[ A\overset{x}{\sim} c_1 I \quad (I\text{ is the identity operator},\ c_1=\hat A(\infty)). \]

\(3^\circ\). An operator
\[ A\bigl(\mathcal L_p^n(E_m)\to \mathcal L_p^n(E_m)\bigr) \]
is called a generalized convolution if at each point \(x\in \tilde E_m\) it is locally equivalent to some operator from \(w_p^n\). We shall denote the space of such operators by \(S_p^n\). For a point at infinity \(\eta\), by Lemma 2, there exists a unique operator \(A_\eta\) locally equivalent to \(A\) at \(\eta\). Moreover, the dependence of \(A\) on \(\eta\) is continuous in norm. At finite points, however, the representative \(A_x(A\overset{x}{\sim}A_x)\) is not uniquely determined. But the value \(A_x(\infty)\), by Lemma 2, is determined by the point \(x\); moreover \(A_x(\infty)\) is continuous on \(E_m\), admits extension by continuity from \(E_m\) to \(\tilde E_m\), and there coincides with \(A_\eta(\infty)\).

\(4^\circ\). Symbol of a generalized convolution. Let \(A\in S_p^n\). In the direct product \(\tilde E_m\times \dot E_m\), we single out two subspaces: \(\mathfrak N\times \dot E_m\) and \(\tilde E_m\times \infty\). We denote their union by \(\Delta\). Define on \(\Delta\) the matrix-function \(\hat A(x,\xi)\) by the rule
\[ \hat A(x,\xi)= \begin{cases} \hat A_x(\xi), & (x,\xi)\in \mathfrak N\times \dot E_m,\\ \hat A_x(\infty), & (x,\xi)\in \tilde E_m\times \infty. \end{cases} \]

We shall call this matrix the symbol of the operator \(A\). From \(3^\circ\) it follows that the symbol of an operator \(A\in S_p^n\) satisfies two conditions:

1) \(\hat A_\eta(\xi)=\hat A(\eta,\xi)\) \((\eta\in\mathfrak N,\ \xi\in \dot E_m)\) depends continuously on \(\eta\) in the metric \(m_p^n\)
\[ \bigl(\|\hat A_\eta(\xi)-\hat A_{\eta_0}(\xi)\|_{m_p^n}\to 0,\quad \eta\to\eta_0\bigr). \]
In the case \(p=2\) this means continuity in the sense of uniform convergence.

2) \(\hat A(x,\infty)\) is continuous on \(\tilde E_m\).

From the theorem on the enveloping operator \((^1,^2)\) one can also conclude the converse: if a matrix-function \(\Phi(x,\xi)\), defined on \(\Delta\), satisfies conditions 1), 2), then there exists an operator \(A(\in S_p^n)\) such that
\[ \hat A(x,\xi)=\Phi(x,\xi). \]
Moreover, the operator \(A\) is determined up to a completely continuous summand, and the two-sided estimates hold
\[ \|\hat A\|_p\le \|\,|A|\,\|_p\le (m+1)\|\hat A\|_p . \]
Here
\[ \|\hat A\|_p=\max\left[ \max_{\eta\in\mathfrak N}\|\hat A_\eta(\xi)\|_{m_p^n},\ \max_{x\in\tilde E_m}\|\hat A(x,\infty)I\|_p \right]. \]
In the case \(p=2\) this

means that there is an isomorphism between the quotient ring \(S_p^n\) by the ideal of completely continuous operators and the ring of continuous matrix-functions on \(\Delta\) with the topology of uniform convergence.

Theorem 1. The operator \(A\) is a Noether operator if and only if its symbol has a determinant that nowhere on \(\Delta\) vanishes.

5°. Composite convolution. Let the space \(\dot E_m\) be divided into a finite number of closed conic sets \(\Gamma_i\)*, not intersecting at interior points, and let the trace of the boundary \(\bigl(\bigcup(\Gamma_i \setminus \operatorname{int}\Gamma_i)\bigr)\) on the unit sphere consist of closed, nonintersecting smooth surfaces.

An operator \(B\) of the form

\[ B=\sum_{i=1}^{N} A_i P_{\Gamma_i}, \tag{2} \]

where \(A_i \in S_p^n\), will be called a composite convolution. Here \(P_{\Gamma_i}\) are operators defined by the rule:

\[ (P_{\Gamma_i} f)(x)=0 \quad \text{for } x\notin \Gamma_i,\qquad (P_{\Gamma_i} f)(x)=f(x) \quad \text{for } x\in \Gamma_i . \]

Just as for generalized convolution, one can introduce a symbol, defining it on the parts \(\Delta_i=[(\mathfrak R\cap\Gamma_i)\times E_m]\cup[\Gamma_i\times\infty]\) and setting it there equal to the symbol of the operator \(A_i\). The symbol also satisfies conditions of the type 1), 2). If, conversely, a matrix-function satisfies these conditions, then there exists a composite convolution of the form (2), unique up to a completely continuous summand, with this symbol. In the case \(p=2\), the space of piecewise-continuous matrix-functions on \(\Delta\) with discontinuities of the first kind on the surfaces separating the \(\Gamma_i\) is isomorphic to the quotient space of composite convolutions of the form (2) by the subspace of completely continuous operators.

Theorem 2. Let \(n=1,\ m>1\). In order that the operator \(B\) be a Noether operator, it is necessary and sufficient that its symbol nowhere on \(\Delta_i\) vanish.

6°. The Hopf–Wiener equation in the cone \(\Gamma\) has the form

\[ (P_\Gamma K P_\Gamma f)(x)=g(x),\qquad x\in \Gamma . \tag{3} \]

Here the unknown function \(f\) and the given \(g\) belong to \(\mathcal L_p\), and the operator \(K\) has the form (1). Equation (3) is reduced to an equation with the composite operator \(B=P_{E\setminus\Gamma}+KP_\Gamma\).

Theorem 3. If \(\Gamma\) is a smooth cone, then the condition \(\hat K(\xi)\ne0\) \((\xi\in \dot E_m)\) is a necessary and sufficient condition for the Noetherian property of equation (3); moreover, its index is equal to zero.

Rostov State University

Received
15 XII 1966

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* By a conic set in \(E_m\) we shall mean a set which, together with each point, contains the whole ray connecting this point with the origin (including the corresponding infinitely distant point of this ray, but, possibly, excluding the origin).