UDC 517.944

MATHEMATICS

R. DENCHEV

ON THE SPECTRUM OF SINGULAR INTEGRALS ON DOMAINS WITH BOUNDARY

(Presented by Academician S. L. Sobolev on 22 X 1968)

Let \(\Omega\) be a domain of \(m\)-dimensional Euclidean space, bounded by a finite number of simple closed surfaces of Lyapunov type, not intersecting one another. Introduce the operator of extension by zero outside \(\Omega\)

\[ P_{\Omega}: L_{2}(\Omega) \ni f(x) \to (P_{\Omega}f)(x)= \begin{cases} f(x) & \text{for } x \in \Omega,\\ 0 & \text{for } x \notin \Omega \end{cases} \in L_{2}(E_{m}) \]

and the operator of restriction to \(\Omega\)

\[ R_{\Omega}: L_{2}(E_{m}) \ni f(x) \to (R_{\Omega}f)(x)=\{f(x) \text{ for } x \in \Omega\}\in L_{2}(\Omega). \]

Let \(\mathcal A\) be a singular integral operator with symbol \(\sigma(x,\xi)\). The function \(\sigma(x,\xi)\) is defined for all \(x\) and \(\xi \in E_m\), \(\xi \ne 0\), is positively homogeneous in \(\xi\) of degree zero, and on the unit sphere \(\Sigma=\{\xi:|\xi|=1\}\) belongs to the space \(H_2(\Sigma)\). Suppose that, with respect to \(x\), the function \(\sigma(x,\xi)\) is sufficiently smooth. As is known, the operator \(\mathcal A\) maps \(L_2(E_m)\) into itself.

The present work is devoted to the study of the essential spectrum of the operator

\[ A=R_{\Omega}\mathcal A P_{\Omega}. \]

Obviously, \(A\) maps \(L_2(\Omega)\) into itself. A point \(\lambda\) is called a point of the essential spectrum of the operator \(A\) if the operator \(A-\lambda I\) is not Noetherian.

We shall use the necessary and sufficient condition for the Noetherian property of singular integral operators contained in \((^1)\).

Let \(m \ge 2\), and let \(x_0\) be a point of the boundary of \(\Omega\). Draw at \(x_0\) the unit normals: the inward \(n_{x_0}^{i}\) and the outward \(n_{x_0}^{e}\). Move \(n_{x_0}^{i}\) and \(n_{x_0}^{e}\) to the origin of coordinates. Their endpoints mark on the unit sphere \(\Sigma\) two points. Connect these points by all possible semicircles \(l_{x_0}\). Introduce the quantity

\[ d_{x_0}^{\,l_{x_0}}(\lambda)=\{\arg[\sigma(x,\xi)-\lambda]\}_{l_{x_0}}. \]

The braces denote the change of the quantity in the braces when \(\xi\) varies along the semicircle \(l_{x_0}\).

In the case \(m>2\) all paths \(l_{x_0}\) are homotopic, and \(d_{x_0}^{\,l_{x_0}}(\lambda)\) does not depend on \(l_{x_0}\). We denote the common value \(d_{x_0}^{\,l_{x_0}}(\lambda)\) by \(d_{x_0}(\lambda)\).

In the case \(m=2\) there are two classes of nonhomotopic paths, and we obtain two numbers \(d_{x_0}^{+}(\lambda)\) and \(d_{x_0}^{-}(\lambda)\). In the case \(m=1\) we denote \(d_{x_0}(\lambda)=\arg[\sigma(x_0,1)-\lambda]-\arg[\sigma(x_0,-1)-\lambda]\).

Using the results of \((^1)\), we obtain the following proposition:

Theorem. The essential spectrum of the operator \(A\) consists of the values of the function \(\sigma(x,\xi)\) for \(x \in \Omega\), \(\xi \in \Sigma\), and of those points \(\lambda\) for which \(|d_{x_0}(\lambda)| \ge \pi\) \((|d_{x_0}^{\pm}(\lambda)| \ge \pi)\) for some \(x_0\) on the boundary of \(\Omega\).

Example 1. Consider the operator

\[ T:L_2(0,1)\ni u(x)\to m(x)u(x)+\int_0^1 \frac{K(x,t)}{x-t}\,u(t)\,dt, \]

where \(m(x)\) and \(K(x,t)\) have continuous first derivatives. This operator was studied in \((^2)\). Write \(T\) in the form

\[ T=A+C, \]

where

\[ A:L_2(0,1)\ni u(x)\to m(x)u(x)+k(x)\int_0^1 \frac{u(t)}{x-t}\,dt,\qquad k(x)=K(x,x); \]

\[ C:L_2(0,1)\ni u(x)\to \int_0^1 \frac{K(x,t)-K(x,x)}{x-t}\,u(t)\,dt. \]

It is easy to see that \(C\) is a completely continuous operator. Consequently, the operators \(T\) and \(A\) have the same essential spectrum. Let

\[ \mathcal A:L_2(-\infty,\infty)\ni u(x)\to m(x)u(x)+k(x)\int_{-\infty}^{\infty}\frac{u(t)}{x-t}\,dt. \]

Then

\[ A=R_\Omega \mathcal A P_\Omega, \]

where \(\Omega=(0,1)\). For the symbol of the operator \(\mathcal A\) we obtain

\[ \sigma(x,\xi)=m(x)+i\pi k(x)\operatorname{sgn}\xi. \]

The set of values of \(\sigma(x,\xi)\) for \(x\in(0,1)\) and \(|\xi|=1\) consists of two curves

\[ A_1B_1:\ m(x)+i\pi k(x),\qquad 0\le x\le 1, \]

\[ A_2B_2:\ m(x)-i\pi k(x),\qquad 0\le x\le 1. \]

It is easy to see that the points \(\lambda\) for which \(\left|d_{x_0}(\lambda)\right|\ge \pi\) fill two rectilinear segments \(A_1A_2\) and \(B_1B_2\), where

\[ A_1=m(0)+i\pi k(0),\qquad A_2=m(0)-i\pi k(0), \]

\[ B_1=m(1)+i\pi k(1),\qquad B_2=m(1)-i\pi k(1). \]

Thus, the essential spectrum of the operator \(T\) consists of the points of the curvilinear quadrilateral \(A_1B_1A_2B_2\). This agrees with the result \((^2)\), obtained with the aid of the theory of normed rings.

Example 2. Let \(m\ge 2\), and let \(\Omega\) be a bounded domain in \(m\)-dimensional Euclidean space with sufficiently smooth boundary \(\partial\Omega\). Denote by \(G(x;y)\), \(x=(x_1,\ldots,x_m)\), \(y=(y_1,\ldots,y_m)\), the Green’s function of the problem

\[ \Delta u=f,\qquad u|_{\partial\Omega}=0,\qquad \Delta=\partial^2/\partial x_1^2+\ldots+\partial^2/\partial x_m^2. \]

Consider the operator

\[ T:L_2(\Omega)\ni u(x)\to \int_\Omega \frac{\partial^2G(x;y)}{\partial y_1^2}\,u(y)\,dy. \]

Differentiation under the integral is performed in the space of generalized functions. This operator was studied in \((^{3-5})\).

It is not hard to see that \(T\) can be represented in the form

\[ T=R_\Omega \mathcal A P_\Omega+C, \]

where \(C\) is a completely continuous operator, and

\[ \mathcal{A}: L_2(E_m) \ni u(x) \to \begin{cases} \displaystyle \int_{E_m} \frac{\partial^2}{\partial y_1^2}\bigl(1-r^{m-2}\bigr)u(y)\,dy, & \text{if } m>2,\\[1.2em] \displaystyle \int_{E_m} \frac{\partial^2}{\partial y_1^2}(\ln r)u(y)\,dy, & \text{if } m=2. \end{cases} \]

The operator \(\mathcal{A}\) has the symbol

\[ \sigma(x,\xi)=\xi_1^2|\xi|^{-2}. \]

Applying the theorem, we obtain that the essential spectrum of the operator \(T\) fills the interval \([0,1]\).

Joint Institute for Nuclear Research

Received
2 IX 1969

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