UDC 517.948.32

MATHEMATICS

N. K. KARAPETYANTS, S. G. SAMKO

ON THE INDEX OF CERTAIN CLASSES OF INTEGRAL OPERATORS

(Presented by Academician A. A. Dorodnitsyn, 16 III 1970)

Operators of the form

\[ H\varphi \equiv \varphi(t)+\sum_{j=1}^{n}\int_{-\infty}^{\infty} a_j(t,\tau)h_j(t-\tau)\varphi(\tau)\,d\tau=f(t),\qquad -\infty<t<\infty, \tag{1} \]

are considered, where \(h_j(t)\in {\mathcal L}_1(-\infty,\infty)\), \(\varphi(t),\ f(t)\in {\mathcal L}_p(-\infty,\infty)\), \(1\le p\le\infty\), and \(a_j(t,\tau)\) belong to a certain class of essentially bounded functions measurable in the plane (Definition 3). The main result is formulated in Theorem 1. The results obtained in no. 2 are then applied to the study of a Riemann boundary-value problem with integral terms and of a certain class of integral equations with kernel of the type of a homogeneous function. The present paper is directly adjacent to the preceding work of the authors \((^{12})\), in which the case of degeneration of the functions \(a_j(t,\tau)\) was studied:

\[ a_j(t,\tau)=\sum_{k=1}^{n} a_{kj}(t)b_{kj}(\tau). \]

A class of equations close to (1) was considered in \((^7)\). Some special cases of equation (1) were considered in the works \((^{3-6})\). We also note that, in the case of continuity of the functions \(a_j(t,\tau)\), our Theorem 1 can also be proved with the aid of the results of I. B. Simonenko on the theory of operators of local type \((^8)\).

No. 1. The class \(M^{\mathrm{sup}}(\widetilde R_2)\). We define the class of functions \(a_j(t,\tau)\) admissible in equation (1). Roughly speaking, this will be the class of functions having (in a certain sense) at least one of the repeated limits at each of the infinitely remote points \((-\infty,-\infty)\) and \((+\infty,+\infty)\). Let us pass to the precise definition. Denote by \(\widetilde R_1\) the line \(R_1\) with two adjoined infinitely remote points. Denote by \(\widetilde R_2\) the plane \(R_2\), completed by the infinitely remote points \((+\infty,+\infty)\), \((-\infty,-\infty)\). As usual, \(M=M(R_1)\), \(M(R_2)\) will denote the corresponding class of essentially bounded measurable functions.

Definition 1. \(\varphi(t)\in M^{\mathrm{sup}}(\widetilde R_1)\), if \(\varphi(t)\in M(R_1)\) and there exist constants \(c_+, c_-\) such that*

\[ \lim_{n\to\infty}\sup_{|t|>n}\theta(\pm t)|\varphi(t)-c_\pm|=0,\qquad \text{where }\theta(t)=\tfrac12(1+\operatorname{sign}t). \]

We shall denote \(c_\pm=\varphi(\pm\infty)\).

Definition 2. We shall say that a measurable essentially bounded function \(a(t,\tau)\) has the value \(a(+\infty,+\infty)\), if there exists a function \(b(x)\in M^{\mathrm{sup}}(\widetilde R_1)\) such that \(b(+\infty)=a(+\infty,+\infty)\) and either

\[ \lim_{n\to\infty}\sup_{0<\tau<\infty}\sup_{t>n}|a(t,\tau)-b(\tau)|=0, \tag{2} \]

* Everywhere in what follows, \(\sup\) means \(\operatorname{ess\,sup}\).

or

\[ \lim_{n\to\infty}\sup_{0<t<\infty}\sup_{\tau>n}|a(t,\tau)-b(t)|=0. \tag{3} \]

The value \(a(-\infty,-\infty)\) is defined analogously. We note that Definition 2 is correct in the sense that if \(a(t,\tau)\) has the value \(a(+\infty,+\infty)\) simultaneously both in the sense of (2) and in the sense of (3), then it is one and the same.

Definition 3. \(a(t,\tau)\in M^{\mathrm{sup}}(\widetilde R_2)\) if \(a(t,\tau)\in M(R_2)\) and the values \(a(+\infty,+\infty)\), \(a(-\infty,-\infty)\) exist in the sense of Definition 2. It is clear that the class \(M^{\mathrm{sup}}(\widetilde R_2)\) contains the class \(C(\widetilde R_2)\) of continuous functions determined by the properties: 1) \(a(t,\tau)\) is bounded on \(R_2\) and continuous at every finite point; 2) one of the iterated limits \(\lim_{\tau\to\infty}\lim_{t\to\infty}a(t,\tau)\), \(\lim_{t\to\infty}\lim_{\tau\to\infty}a(t,\tau)\) exists, and the inner limit is uniform (with respect to \(\tau\), \(0<\tau<\infty\), and with respect to \(t\), \(0<t<\infty\), respectively); an analogous limit exists as \(t,\tau\to-\infty\). We note that Definition 3 imposes no requirements on the function \(a(t,\tau)\) in the second and fourth quadrants, apart from membership in \(M(R_2)\).

No. 2. The main theorem. Let \(h_j(t)\in\mathscr L_1(-\infty,\infty)\) and \(a_j(t,\tau)\in M^{\mathrm{sup}}(\widetilde R_2)\).

Theorem 1. In order that the operator \(H\) be a Noether operator in \(\mathscr L_p(-\infty,\infty)\), \(1\le p\le\infty\), it is necessary and sufficient that

\[ \sigma(\lambda)^{\pm}=1+\sum_{j=1}^{n}a_j(\pm\infty,\pm\infty)\mathscr H_j(\lambda)\ne0, \quad \text{where }\quad \mathscr H(\lambda)=\int_{-\infty}^{\infty}h(t)e^{i\lambda t}\,dt. \]

The index of the operator \(H\) is computed by the formula

\[ \varkappa_{\mathfrak B_p}(H)=-\frac{1}{2\pi}\Delta\left[\arg\frac{\sigma(\lambda)^{+}}{\sigma(\lambda)^{-}}\right]_{-\infty}^{\infty}. \tag{4} \]

The proof of the theorem is based on the following lemma.

Lemma 1. If \(h(t)\in\mathscr L_1(-\infty,\infty)\) and \(a(t,\tau)\in M^{\mathrm{sup}}(\widetilde R_2)\), then the operators

\[ \int_{-\infty}^{\infty}[a(t,\tau)-a(\infty,\infty)\theta(t)-a(-\infty,-\infty)\theta(-t)]h(t-\tau)\varphi(\tau)\,d\tau, \]

\[ \theta(t)\int_{-\infty}^{\infty}a(t,\tau)\theta(-\tau)h(t-\tau)\varphi(\tau)\,d\tau \]

are completely continuous in \(\mathscr L_p(-\infty,\infty)\), \(1\le p\le\infty\).

No. 3. Consider the following boundary-value problem

\[ \Phi^{+}(t)+\int_{-\infty}^{\infty}a(t,\tau)h_1(t-\tau)\Phi^{+}(\tau)\,d\tau = \]

\[ =G(t)\Phi^{-}(t)+\int_{-\infty}^{\infty}b(t,\tau)h_2(t-\tau)\Phi^{-}(\tau)\,d\tau+f(t), \qquad -\infty<t<\infty, \tag{5} \]

where \(\Phi^{\pm}(z)\) are analytic functions in the half-planes \(\operatorname{Im}z>0\), \(\operatorname{Im}z<0\), respectively, representable by a Cauchy-type integral with density from \(\mathscr L_p(-\infty,\infty)\); \(p>1\) (\(\Phi^{\pm}(t)\in\mathscr L_p^{\pm}\)). It is assumed that 1) \(h_1(t)\), \(h_2(t)\in\mathscr L_1(-\infty,\infty)\); 2) \(a(t,\tau)\), \(b(t,\tau)\in M^{\mathrm{sup}}(\widetilde R_2)\), with \(a(+\infty,+\infty)=a(-\infty,-\infty)\) and \(b(+\infty,+\infty)=b(-\infty,-\infty)\); 3) \(G(t)\in M^{\mathrm{sup}}\cap A_p\) and \(G(-\infty)=G(+\infty)\) (for the definition of the class \(A_p\), see (²)). In particular, one may take \(G(t)\) to be a function continuous on the closed axis, \(G(t)\ne0\).

It is known that in the case of a finite contour and Fredholm kernels the index of a problem of the form (5) (as well as of more general integro-differential

problems does not depend on the integral (Fredholm) terms (see (1), p. 362). This, it turns out, is also true for problem (5), which contains integral terms with a difference kernel. However, the condition of normal solvability will depend on the integral terms. We shall also indicate a case when problem (5) is solvable in closed form.

Theorem 2. If \(1+a(\infty,\infty)\mathscr H_1(x)\ne0\) for \(0\le x\le\infty\) and \(G(\infty)+b(\infty,\infty)\mathscr H_2(x)\ne0\), \(-\infty\le x\le0\), where
\[ \mathscr H_j(x)=\int_{-\infty}^{\infty} h_j(t)e^{itx}\,dt,\qquad j=1,2, \]
then problem (5) is Noetherian in \(\mathscr L_p^{\pm}\), and its index is equal to the index of the coefficient \(G(t)\).

By virtue of the lemma, problem (5) differs only by completely continuous terms from the problem
\[ A\Phi\equiv \Phi^+(t)+a(\infty,\infty)\int_{-\infty}^{\infty}h_1(t-\tau)\Phi^+(\tau)\,d\tau- \]
\[ -G(t)\left[\Phi^-(t)+\frac{b(\infty,\infty)}{G(\infty)} \int_{-\infty}^{\infty}h_2(t-\tau)\Phi^-(\tau)\,d\tau\right]=f(t). \tag{6} \]

Denoting
\[ H_1\Phi^+\equiv a(\infty,\infty)\int_{-\infty}^{\infty}h_1(t-\tau)\Phi^+(\tau)\,d\tau=\Phi_1^+(t), \]
\[ H_2\Phi^-\equiv \frac{b(\infty,\infty)}{G(\infty)} \int_{-\infty}^{\infty}h_2(t-\tau)\Phi^-(\tau)\,d\tau=\Phi_1^-(t), \]
we see that (6) reduces to the successive solution of the Riemann problem
\[ \Phi^+(t)+\Phi_1^+(t)=G(t)\,[\Phi^-(t)+\Phi_1^-(t)] \]
and of convolution-type integral equations (on the whole axis) in the class of analytic functions. In other words, there is the representation \(A=B\cdot C\), where
\[ B=\tfrac12(I+S)+\tfrac12G(I-S),\qquad C=\tfrac12(I+H_1)(I+S)+\tfrac12(I+H_2)(I-S), \]
\(G\) is the operator of multiplication by the function \(G(t)\), and
\[ S\varphi=\frac1{\pi i}\int_{-\infty}^{\infty}\frac{\varphi(\tau)\,d\tau}{\tau-t}. \]
By the assumptions of the theorem, the operator \(C\) is invertible.

Naturally, problem (6) is solved in closed form. Consequently, problem (5) is solvable in closed form if \(a(t,\tau)=\mathrm{const}\), \(b(t,\tau)=\gamma G(t)\), \(\gamma=\mathrm{const}\). Note that equation (6) generalizes an equation of the form \(\varphi+\lambda S\varphi+H_1\varphi=f\), considered in \(\mathscr L_2(-\infty,\infty)\) by T. I. Savel’eva (9).

No. 4. Consider the equation
\[ K\psi\equiv \psi(x)+\int_0^a\gamma(x,y)k(x,y)\psi(y)\,dy=g(x),\qquad 0<x<a, \tag{7} \]
where \(k(x,y)\) is a homogeneous function of arbitrary order \(\alpha\): \(k(\lambda x,\lambda y)=\lambda^\alpha k(x,y)\), and it is assumed that there exists a number \(\beta\) such that one of the summability conditions is satisfied:
\[ \int_0^\infty |k(1,y)|\frac{dy}{y^\beta}<\infty,\qquad \int_0^\infty |k(x,1)|\frac{dx}{x^{1-\beta}}<\infty. \tag{8} \]

The function \(\gamma(x,y)\) will belong to a certain subclass of measurable functions in the fundamental square, bounded everywhere except, possibly, at the origin of the coordinates.

We seek solutions in the weighted space
\[ \mathscr L_p^\beta=\{\psi:x^{\beta-1/p}\psi(x)\in\mathscr L_p(0,a)\}. \]
If in (8) the admissible values are \(0\le\beta\le1\), then equation (7) may be considered in all \(\mathscr L_p(0,a)\), \(1\le p\le\infty\).

Denote
\[ Q=\{x,y:0<x<a,\ 0<y<a\}. \]

Definition 4. We shall say that \(\omega(x,y)\in M^{\mathrm{sup}}(Q)\) if:

1) \(\omega(x,y)\) is a measurable function, essentially bounded on \(Q\);

2) \(\omega(x,y)\) has the value \(\omega(0,0)\), defined analogously to the value \(a(\infty,\infty)\) in Definition 2.

Theorem 3. Let \(\gamma_1(x,y)=x^{1+\alpha}\gamma(x,y)\in M^{\mathrm{sup}}(Q)\). In order that the operator \(K\) be a Noether operator in the space \(\mathscr L_p^\beta\), \(1\le p\le \infty\), it is necessary and sufficient that
\[ \sigma(\lambda)=1+\gamma_1(0,0)\mathfrak M(i\lambda-\beta+1)\ne 0, \]
\[ -\infty\le \lambda\le \infty,\qquad \mathfrak M(s)=\int_0^\infty k(1,y)y^{s-1}\,dy. \]
The index of the operator \(K\) is computed by the formula
\[ \varkappa_{\mathscr L_p^\beta}(K)=-\frac{1}{2\pi}\Delta\,[\arg\sigma(\lambda)]_{-\infty}^{\infty}. \]

Theorem 3 is established by reducing equation (7) to equation (1). For simplicity, it is formulated for a kernel \(k(x,y)\) satisfying the first of the summability conditions (8), and is easily carried over to the case when \(k(x,y)\) satisfies the second of conditions (8). We note that an analogous result for \(\alpha=-1\) and \(\gamma(x,y)\in C(Q)\) was previously obtained by another method in the works of L. G. Mikhailov \((^{10,11})\). Finally, a theorem analogous to Theorem 3 can be obtained for the case \(a=\infty\), and also for an equation more general than (7),
\[ K\psi\equiv \psi(x)+\sum_{j=1}^n\int_0^a \gamma_j(x,y)k_j(x,y)\psi(y)\,dy=g(x),\qquad 0<x<a. \]

In conclusion, let us consider the example
\[ A\psi\equiv \psi(x)+\int_0^\infty \frac{\gamma(x,y)}{x+y}\psi(y)\,dy=f(x),\qquad x>0, \]
where \(\gamma(x,y)\in M^{\mathrm{sup}}(Q)\), \(f(x),\psi(x)\in \mathscr L_p(0,\infty)\), \(1<p<\infty\). Let \(\gamma_0=\gamma(0,0)\), \(\gamma_\infty=\gamma(\infty,\infty)\)—values in the sense of Definition (2). The Noether condition has the form: \(\gamma_0,\gamma_\infty>-1/\pi\) for \(p=2\), and \(\gamma_0,\gamma_\infty\ne -\dfrac{1}{\pi}\sin\dfrac{\pi}{p}\) for \(p\ne 2\), while the index \(\varkappa\) of the operator \(A\), when this condition is fulfilled, is equal to
\[ \varkappa= \begin{cases} 0, & \text{if } \gamma_0,\gamma_\infty>-\sin(\pi/p)/\pi \text{ or } \gamma_0,\gamma_\infty<-\sin(\pi/p)/\pi,\\ \operatorname{sign}(p-2), & \text{if } \gamma_0>-\sin(\pi/p)/\pi,\ \gamma_\infty<-\sin(\pi/p)/\pi,\\ \operatorname{sign}(2-p), & \text{if } \gamma_0<-\sin(\pi/p)/\pi,\ \gamma_\infty>-\sin(\pi/p)/\pi. \end{cases} \]

Rostov State University

Received
11 III 1970

CITED LITERATURE

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