MATHEMATICS

L. S. RAKOVSHCHIK

INTEGRAL EQUATIONS WITH ALMOST DIFFERENCE KERNELS

(Presented by Academician V. I. Smirnov, March 24, 1960)

1. By integral equations with almost difference kernels we shall mean equations of the form

\[ a(t)\varphi(t)+\int_{-\infty}^{\infty} k(t,t-\tau)\varphi(\tau)\,d\tau=f(t). \tag{1} \]

If, for any \(c\), the convolution theorem is applicable to the pair \(k(c,t)\) and \(\varphi(t)\), where \(\varphi(t)\) is a finite infinitely differentiable function, then the operator (1) can be represented in the form

\[ k\varphi=\frac{1}{2\pi}\int_{-\infty}^{\infty} K(t,\lambda)\Phi(\lambda)\exp(-i\lambda t)\,d\lambda, \tag{2} \]

where \(\Phi(\lambda)\) is the Fourier transform of the function \(\varphi(t)\), and

\[ K(t,\lambda)=a(t)+\int_{-\infty}^{\infty} k(t,u)\exp(i\lambda u)\,du. \]

If, moreover, the operator (2) is bounded in the norm of the space \(L_p(-\infty,\infty)\) on the set \(D\), dense in it, of finite infinitely differentiable functions, then equation (1), considered in \(L_p\), can be written in the form

\[ \frac{1}{2\pi}\int_{-\infty}^{\infty} K(t,\lambda)\Phi(\lambda)\exp(-i\lambda t)\,d\lambda=f(t). \tag{3} \]

The operator on the left-hand side of (3) is understood here as the extension of the operator (2) to all of \(L_p(-\infty,\infty)\).

A special case of equation (3) was considered by I. M. Rapoport \((^{1,2})\) in the study of the Wiener–Hopf equation and paired equations. Multidimensional equations similar to equation (3) have been studied in a number of works by S. G. Mikhlin (for the bibliography, see \((^3)\)). Between the multidimensional equations considered in the works of S. G. Mikhlin and the one-dimensional equations of the form (3), despite a number of common features, there are essential differences, which will be apparent from what follows.

1. Theorem 1. If for almost all \(t\), \(-\infty<t<\infty\): 1) the function \(K(t,\lambda)\) is absolutely continuous in \(\lambda\) on any finite interval, there exists \(\lim_{\lambda\to+\infty} K(t,\lambda)=K(t,+\infty)\), and \(\sup_t |K(t,+\infty)|\le C\); 2) for some one-to-one and continuously differentiable mapping \(\mu=\mu(\lambda)\) of the whole axis onto a finite segment the inequality

\[ \sup_t\int_{-\infty}^{\infty} \left|\frac{\partial K(t,\lambda)}{\partial\lambda}\right|^q |\mu'(\lambda)|^{q-1}\,d\lambda\le C_1, \qquad \frac{1}{p}+\frac{1}{q}=1, \]

holds, then the operator (2) is bounded in the norm of \(L_p\) on the set \(D\).

Theorem 2. The assertion of the preceding theorem remains valid if condition 2) is replaced by the condition: \(2')\) there exists a function \(\Omega(\lambda)\in L\) such that for almost all \(t\)
\[ |\partial K(t,\lambda)/\partial\lambda|\leq \Omega(\lambda). \]

Theorem 3. Suppose that, for fixed \(t\), the function \(K(t,\lambda)\) has discontinuities of the first kind at the points of a fixed sequence \(\{c_k\}\). If
\[ \sum_k \sup_t |K(t,c_k+0)-K(t,c_k-0)|<\infty \]
and the function
\[ K(t,\lambda)-\sum_k \theta(\lambda-c_k)[K(t,c_k+0)-K(t,c_k-0)] \]
satisfies the conditions of one of the preceding theorems, then the operator (2) is bounded in \(L_p\) (\(\theta(\lambda)\) is the Heaviside function).

1. Let \(T\) be a completely continuous operator in \(L_p\). Following S. G. Mikhlin, we shall call the function \(K(t,\lambda)\) the symbol of the operator
   \[ A\varphi=\frac{1}{2\pi}\int_{-\infty}^{\infty} K(t,\lambda)\Phi(\lambda)\exp(-i\lambda t)\,dt+T\varphi. \tag{4} \]

The definition of the symbol of an operator is not unique, since there exist completely continuous operators representable in the form (2), for example the operator with symbol \((1+\lambda^2)^{-1}e^{-|t|}\). From a given symbol the operator is recovered up to a completely continuous summand.

We now consider the totality of all operators whose symbols satisfy the conditions of Theorem 1 and the following conditions: the functions \(K[t,\lambda(\mu)]\) and \(\partial K[t,\lambda(\mu)]/\partial\mu\), where \(\lambda(\mu)=-\operatorname{ctg}\frac{1}{2}\mu\), are continuous on the interval \([0,2\pi]\), and each of them assumes equal values at the endpoints of this interval; the modulus of continuity \(\omega(t,\delta)\) of the function \(\partial K[t,\lambda(\mu)]/\partial\mu\) (with respect to the variable \(\mu\)) tends uniformly with respect to \(t\) to zero as \(\delta\to0\); if the difference \(t-\tau\) is sufficiently small in absolute value, or if \(|t|\) and \(|\tau|\) are sufficiently large and \(t\tau>0\), then
\[ |K(t,\lambda)-K(\tau,\lambda)|\,e^{\mp i(t-\tau)}|t-\tau|^k\,\theta[\pm(t-\tau)]\leq \]
\[ \leq \sum_{j=0}^{n_k} c_k^{(j)} |t-\tau|^{-\alpha_j}(1+t^2)^{\alpha_j/2-1/p}(1+\tau^2)^{\alpha_j/2-1/q}, \]
where \(0\leq \alpha_j<1\); \(c_k^{(j)}\) are constants depending on the function \(K(t,\lambda)\), and \(k=0,1,\ldots\).

Theorem 4. The product of two operators of the form (4), whose symbols satisfy the conditions listed above, is an operator of the same form; moreover, the symbol of the product is equal to the product of the symbols of the factors.

Remark 1. If the function \(K(t,\lambda)\) satisfies the conditions of Theorem 4, then the functions \(\overline{K(t,\lambda)}\) and, if \(\inf_{t,\lambda}|K(t,\lambda)|>0\), \([K(t,\lambda)]^{-1}\) also satisfy these same conditions.

Remark 2. If the symbol \(K(t,\lambda)\) of an operator \(A\) satisfies the conditions of Theorem 4 for the exponents \(p\) and \(q\) simultaneously, then the adjoint operator is representable in the form (4), and its symbol is equal to \(\overline{K(t,\lambda)}\).

We introduce the following definitions:

We shall call a ring \(\sigma_1\) the ring of bounded operators of the form (4) if: a) the assertion of Theorem 4 holds for this ring; b) if the ring contains an operator with symbol \(K(t,\lambda)\), then it also contains the operators with symbols \(\overline{K(t,\lambda)}\) and, if \(\inf_{t,\lambda}|K(t,\lambda)|>0\), \([K(t,\lambda)]^{-1}\); c) the ring contains operators with symbols \(1\), \(\theta(t)\), \(\Omega(t)\), \(\Omega(\lambda)\), where \(\Omega\) is an arbitrary finite infinitely differentiable function, and \(\theta\) is the Heaviside function (instead of requiring that operators with symbols \(\Omega(t)\) and \(\Omega(\lambda)\) be present in the ring, it is sufficient to assume that the ring contains at least ...

as well as one completely continuous operator with a nonnegative symbol, whose modulus is bounded below by a positive constant in any bounded domain).

By the ring \(\sigma_2\) we shall mean the ring of bounded operators of the form (4) satisfying all the conditions of the preceding definition, but with the function \(\theta(t)\) in condition c) replaced by the function \(\theta(\lambda)\).

An example of the ring \(\sigma_1\) is furnished by the totality of operators satisfying the conditions of Theorem 4. Under certain additional conditions, the totality of operators satisfying the conditions of Theorem 3 forms the ring \(\sigma_2\). A simpler example of the ring \(\sigma_1\) (\(\sigma_2\)) is the totality of operators with symbols of the form
\[ a(t)+\sum_{i=1}^n b_i(t)K_i(\lambda)\left(c(t)+\sum_{i=1}^n d_i(t)\theta(\lambda-c_i)\right), \]
where \(a(t)\) and \(b_i(t)\) are piecewise constant bounded functions (\(c(t)\) and \(d_i(t)\) satisfy a Lipschitz condition on the whole axis, and for each of these functions there exists an interval outside of which it is constant), and \(K_i(\lambda)\) is the Fourier transform of a summable function.

1. Theorem 5. If the symbol \(K(t,\lambda)\) of the operator \(A\in\sigma\) \((\sigma=\sigma_{1,2})\), outside some circle \(t^2+\lambda^2\le R^2\), satisfies the inequality \(|K(t,\lambda)|\ge c>0\), then the operator \(A\) admits a regularization \((^3)\) by an operator from the same ring.

Corollary. If there exist uniform limits
\[ \lim_{t\to\pm\infty}K(t,\lambda)=K(\pm\infty,\lambda), \qquad \lim_{\lambda\to\pm\infty}K(t,\lambda)=K(t,\pm\infty) \]
and
\[ \min\left[\inf_t |K(t,\pm\infty)|,\ \inf_\lambda |K(\pm\infty,\lambda)|\right]>0, \]
then the assertion of Theorem 5 remains valid.

Theorem 6. Under the conditions of the preceding theorem or of its corollary, the known theorems of F. Noether \((^{4,5})\) hold for the operator \(A\).

1. Lemma. If the symbol \(K(t,\lambda)\) of the operator \(A\in\sigma\) satisfies the conditions of the corollary to Theorem 5 and if the closure of the set of values of the functions \(K(t,\pm\infty)\) and \(K(\pm\infty,\lambda)\) does not intersect a certain ray issuing from the origin, then the index of the operator \(A\) is equal to zero.

The proof of the lemma is similar to the proof of the analogous proposition in \((^6)\).

Theorem 7. Suppose that for the symbol of the operator \(A\in\sigma_1\) the following conditions are fulfilled: 1) there exist uniform limits \(K(t,\pm\infty)\) and \(K(\pm\infty,\lambda)\); the operators with symbols \(K(\pm\infty,\lambda)\) belong to the ring \(\sigma_1\); 2)
\[ \min\left[\inf_\lambda |K(\pm\infty,\lambda)|,\ \inf_t |K(t,\pm\infty)|\right]>0; \]
3) the closure of the set of values of the functions \(K(t,\lambda)/K(-\infty,\lambda)\) for \(t<0\) and \(K(t,\lambda)/K(+\infty,\lambda)\) for \(t>0\) does not intersect a certain ray issuing from the origin.

Under these conditions the index of the operator \(A\) is equal to the index of the operator
\[ A_\infty\varphi= \begin{cases} \displaystyle \frac{1}{2\pi}\int_{-\infty}^{\infty}K(-\infty,\lambda)\Phi(\lambda)\exp(-i\lambda t)\,d\lambda, & \text{for } t<0,\\[1.2em] \displaystyle \frac{1}{2\pi}\int_{-\infty}^{\infty}K(+\infty,\lambda)\Phi(\lambda)\exp(-i\lambda t)\,d\lambda, & \text{for } t>0. \end{cases} \]

Theorem 8. Suppose that for the symbol of the operator \(A\in\sigma_2\) the following conditions are fulfilled: 1) there exist uniform limits \(K(t,+\infty)\) and \(K(\pm\infty,\lambda)\); the operators with symbols \(K(t,\pm\infty)\) belong to the ring \(\sigma_2\); 2) the same as in Theorem 7; 3) the same as in Theorem 7, but with the roles of the variables \(t\) and \(\lambda\) interchanged.

Then the index of the operator \(A\) is equal to the index of the operator

\[ A^{\infty}\varphi = \frac{1}{2}[K(t,+\infty)+K(t,-\infty)] + \frac{1}{2\pi j}[K(t,+\infty)-K(t,-\infty)] \int_{-\infty}^{\infty}\frac{\varphi(\tau)\,d\tau}{t-\tau}. \]

It can be shown that if, in the conditions of Theorem 7 (8), the functions \(K(\pm\infty,\lambda)\) \((K(t,\pm\infty))\) are continuous on the closed axis and the quantity

\[ \operatorname{Arg}K(+\infty,\lambda)\big|_{-\infty}^{\infty} \quad (\operatorname{Arg}K(t,+\infty)) \]

is finite, then

\[ \operatorname{ind}A = \frac{1}{2\pi}\operatorname{Arg} \frac{K(-\infty,\lambda)}{K(+\infty,\lambda)} \bigg|_{-\infty}^{\infty} \]

\[ \left( \operatorname{ind}A = \frac{1}{2\pi}\operatorname{Arg} \frac{K(t,+\infty)}{K(t,-\infty)} \bigg|_{-\infty}^{\infty} \right). \]

Theorem 9. Let the operator \(A\in\sigma_1\cap\sigma_2\); let: 1) conditions 1) and 2) of Theorems 7 and 8 be satisfied; 2) \(A_\infty A\in\sigma_2\) (or \(A^\infty A\in\sigma_1\)); 3) the limits

\[ \lim_{t\to\pm\infty}K(t,\pm\infty) \quad\text{and}\quad \lim_{\lambda\to\pm\infty}K(\pm\infty,\lambda) \]

exist. Under these conditions the index of the operator \(A\) is equal to the sum of the indices of the operators \(A_\infty\) and \(A^\infty\).

If, moreover, the functions \(K(t,\pm\infty)\) and \(K(\pm\infty,\lambda)\) are continuous on the closed axis and the numbers

\[ \operatorname{Arg}K(t,\pm\infty)\big|_{-\infty}^{\infty} \quad\text{and}\quad \operatorname{Arg}K(\pm\infty,\lambda)\big|_{-\infty}^{\infty} \]

are finite, then

\[ \operatorname{ind}A = \frac{1}{2\pi} \operatorname{Arg} K(-\infty,\lambda)K(\lambda,+\infty) \,[K(+\infty,\lambda)K(\lambda,-\infty)]^{-1} \bigg|_{-\infty}^{\infty}. \]

In deriving the formulas for the index, results of the papers \((^{7,8})\) were used.

1. Suppose that for the symbols of the operators from the ring \(\sigma\) under consideration there exist uniform limits \(K(t,\pm\infty)\) and \(K(\pm\infty,\lambda)\), and that the sets of values of the functions \(K(t,\pm\infty)\) and \(K(\pm\infty,\lambda)\), \(-\infty<t,\lambda<\infty\), are closed. In addition, suppose that for complete continuity of an operator from the ring under consideration it is necessary and sufficient that

\[ K(t,\pm\infty)=K(\pm\infty,\lambda)=0. \]

(This condition is satisfied for the rings \(\sigma\) given as examples in item 3. Under fairly broad assumptions about the ring, one can prove the necessity of the condition

\[ K(t,\pm\infty)=K(\pm\infty,\lambda)=0 \]

for complete continuity of the corresponding operator.) Under the assumptions made, it can be shown, using methods of the theory of normed rings, that in the case \(p=2\), in order that there exist a bounded regularizer for an operator \(A\in\sigma\), it is necessary and sufficient that the conditions

\[ K(\pm\infty,\lambda)\ne0,\qquad K(t,\pm\infty)\ne0 \]

be satisfied. In the case \(p\ne2\), \(1<p<\infty\), this assertion remains valid, at any rate, for operators whose symbols are real or are even (odd) functions of at least one of the variables \(t\) or \(\lambda\).

1. Results analogous to those established hold for systems

\[ \sum_{j=1}^{n}\frac{1}{2\pi} \int_{-\infty}^{\infty} K_{lj}(t,\lambda)\Phi_j(\lambda)\exp(-i\lambda t)\,d\lambda = f_l(t), \qquad l=1,2,\ldots,n. \]

In particular, under certain conditions, Noether theorems hold for such systems, and their index is equal to the index of an operator of the form (4) with symbol

\[ \det\|K_{lj}(t,\lambda)\|. \]

In conclusion, the author considers it his duty to express gratitude to Prof. S. G. Mikhlin for suggesting the topic and for guidance.

Leningrad State University
named after A. A. Zhdanov

Received
24 III 1960

REFERENCES

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3. S. G. Mikhlin, Vestn. LGU, Ser. Mat., Mekh. i Astr., No. 1, 3 (1956).
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