I. Ts. GOKHBERG

ON THE THEORY OF MULTIDIMENSIONAL SINGULAR INTEGRAL EQUATIONS

(Presented by Academician V. I. Smirnov on 2 IV 1960)

The present note contains some additions to the general theory of multidimensional singular equations in the space \(L_2(E_m)\), constructed by S. G. Mikhlin \((^{1-3})\). These additions are obtained with the aid of results and methods from the theory of normed commutative rings of I. M. Gelfand \((^4)\).

1. Under certain restrictions \((^{1-3})\), the singular integral equation over the whole \(m\)-dimensional space \(E_m\) \((m \geq 2)\) has the form

\[ (A\varphi)(x)= \sum_{n_1,n_2,\ldots,n_{m-1}=-\infty}^{\infty} a_{n_1 n_2 \ldots n_{m-1}}(x) (S_1^{n_1}S_2^{n_2}\ldots S_{m-1}^{n_{m-1}}\varphi)(x) +(T\varphi)(x)= \]
\[ = f(x)\quad (x\in E_m), \tag{1} \]

where \(\varphi(x)\) \((\in L_2(E_m))\) is the unknown function, \(f(x)\) and \(a_{n_1 n_2\ldots n_{m-1}}(x)\) are prescribed functions; the first of them belongs to the space \(L_2(E_m)\), while the others are continuous functions on the whole space \(E_m\) (including the point at infinity); \(T\) is a linear completely continuous operator acting in \(L_2(E_m)\); \(S_j\) \((j=1,2,\ldots,m-1)\) are the “simplest” \(m\)-dimensional singular integral operators. They admit the representation

\[ S_j\varphi = F^{-1}\exp[iv_j(x)](F\varphi)(x) \quad (j=1,2,\ldots,m-1), \]

in which \(v_1(x), \tfrac12 v_2(x),\ldots,\tfrac12 v_{m-1}(x)\) \((0\leq v_j\leq 2\pi,\ j=1,2,\ldots,m-1)\) denote the angular spherical coordinates of the point \(x\) \((\in E_m)\); \(F\) is the unitary Fourier transform operator in \(L_2(E_m)\); finally, the series in (1) converges in the operator norm.

According to the general theory \((^{1-3})\) of multidimensional singular integral equations, to each operator \(A\) of the form (1) (generally speaking, under certain additional restrictions ensuring, for example, convergence of the series \(\sum \max |a_{n_1n_2\ldots n_{m-1}}(x)|\)) there is associated a continuous function

\[ \mathcal{A}(x,\theta_1,\theta_2,\ldots,\theta_{m-1}) = \sum_{n_1,n_2,\ldots,n_{m-1}=-\infty}^{\infty} [[\unclear: coefficient]](x)\exp\left[i\sum_j n_j\theta_j\right] \quad (0\leq \theta_j\leq 2\pi), \tag{2} \]

called the symbolic function (symbol) of the operator \(A\). In \((^{1-3})\) it is proved that if the symbol (2) nowhere vanishes, then the Fredholm theorem holds for equation (1), i.e. the operator \(A\) is normally solvable and the equations \(A\varphi=0\), \(A^*\psi=0\) \((\varphi,\psi\in L_2(E_m))\) have equal finite numbers of linearly independent solutions. Analogous results were obtained in \((^{1-3})\) also for systems of equations of the form (1).

2. For simplicity we shall henceforth put \(m=2\) (\(E_2=E\)). Then the operator (1) takes the form

\[ A=\sum_{j=-\infty}^{\infty} a_j(x)S^j+T, \]

where*

\[ (S\varphi)(x)=\frac{1}{2\pi}\int_E \frac{e^{i\nu(y)}\varphi(y)}{|x-y|^2}\,dy, \]

and \(|x|\) and \(\nu(x)\) are the polar coordinates of the point \(x\in E\). Formula (2), defining the symbol of the operator \(A\), takes the form

\[ \mathcal A(x,\theta)=\sum_{j=-\infty}^{\infty} a_j(x)e^{ij\theta}\qquad (0\leqslant \theta\leqslant 2\pi). \]

Denote by \(\mathfrak T\) the set of all linear completely continuous operators and by \(\mathfrak S_k\) the set of all operators of the form

\[ A=\sum_{|j|\leqslant k_A} a_j(x)S^j+T\qquad (T\in\mathfrak T), \]

where \(k_A\) is a finite number depending on the operator \(A\), and \(a_j(x)\) \((j=0,\pm1,\ldots)\) are continuous functions on the whole space \(E\), including the point at infinity.

As is known \({}^{(1)}\), the set \(\mathfrak S_k\) forms a ring, and \(AB-BA\in\mathfrak T\) for any pair of operators \(A,B\in\mathfrak S_k\). Hence, in particular, it follows that the symbol of the product of any pair of operators from \(\mathfrak S_k\) is equal to the product of their symbols.

Theorem 1. If the operator \(A\in\mathfrak S_k\), then**

\[ \max_{x\in E,\;0\leqslant\theta\leqslant 2\pi} |A(x,\theta)| \leqslant \inf_{T\in\mathfrak T}\|A+T\|. \tag{3} \]

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

Lemma. Let \(x_0(\in E)\) and \(\theta_0(0\leqslant\theta_0\leqslant 2\pi)\) be an arbitrary fixed pair of points. Then for any positive numbers \(\varepsilon_1,\varepsilon_2\) and \(\rho\) there exists an ort \(\varphi\in L_2(E)\) possessing the following properties:

a) \(\|S\varphi-e^{i\theta_0}\varphi\|<\varepsilon_1\);

b) \(\displaystyle \int_{|x-x_0|>\rho} |\varphi(x)|^2\,dx<\varepsilon_2.\)

3. Denote by \(\mathfrak S\) the closure, in the operator norm, of the ring \(\mathfrak S_k\). The set \(\mathfrak S\) forms a complete normed ring, and for any pair of operators \(A\) and \(B\in\mathfrak S\) the difference \(AB-BA\in\mathfrak T\).

Let \(A\in\mathfrak S\) and \(\|A_n-A\|\to 0\), where \(A_n\in\mathfrak S_k\). Then, by Theorem 1, the sequence of symbols \(\mathcal A_n(x,\theta)\) converges uniformly on \(E\times[0,2\pi]\) to some continuous function \(\mathcal A(x,\theta)\) \((x\in E;\ 0\leqslant\theta\leqslant 2\pi)\), independent of the choice of the sequence \(A_n\) tending to the operator \(A\). From Theorem 1 it also follows that

\[ \max_{x,\theta} |\mathcal A(x,\theta)| \leqslant \inf_{T\in\mathfrak T}\|A+T\|. \tag{4} \]

\[ \text{* The integral is understood in the sense of the principal value (see \({}^{(1)}\)).} \]

\[ \text{** We note that from this theorem, in particular, it follows that the operator } \sum_{|n|\leqslant k} a_n(x)S^n\in\mathfrak T \text{ if and only if } a_n(x)=0\ (x\in E;\ n=1,2,\ldots). \]
Thus, the symbol \(A(x,\theta)\) is uniquely determined by the operator \(A(\in\mathfrak S)\).

The function \(\mathcal A(x,\theta)\) is naturally called the symbol of the operator \(A\). It is easy to see that the symbol of a product of operators from \(\mathfrak S\) is equal to the product of the corresponding symbols.

Denote by \(\mathfrak R\) the quotient ring \(\mathfrak S/\mathfrak T\), with the norm defined by

\[ \|\hat A\|=\inf_{T\in\mathfrak T}\|A+T\|\qquad (\hat A\in\mathfrak R), \tag{5} \]

where \(\hat A\) denotes the residue class containing the operator \(A\).

From the properties of the ring \(\mathfrak S\) indicated above it follows that \(\mathfrak R\) forms a commutative normed ring. All operators belonging to one and the same residue class \(\hat A\) have one and the same symbol; we shall denote it by \(\hat{\mathcal A}(x,\theta)\) \((x\in E;\ 0\leq \theta\leq 2\pi)\).

Let \(x_0\in \widetilde E\,^*\) and \(0\leq \theta_0\leq 2\pi\); then the formula

\[ F(\hat A)=\hat{\mathcal A}(x_0,\theta_0)\qquad (\hat A\in\mathfrak R) \tag{6} \]

defines a linear multiplicative functional in \(\mathfrak R\). In view of (4) and (5), this functional is continuous. It is not difficult to prove that all continuous linear multiplicative functionals of the ring \(\mathfrak R\) are exhausted by functionals of the form (6). Hence it follows:

Theorem 2. The set \(M_{x_0,\theta_0}\) of all \(\hat A\in\mathfrak R\) for which \(\mathcal A(x_0,\theta_0)=0\), where \(x_0(\in E)\) and \(\theta_0\) \((0\leq \theta_0\leq 2\pi)\) is an arbitrary fixed point of \(E\times[0,2\pi]\), forms a maximal ideal in \(\mathfrak R\). These ideals exhaust all maximal ideals of the ring \(\mathfrak R\).

Thus the bicompact \(\mathfrak M\) of all maximal ideals of the ring \(\mathfrak R\) is topologically equivalent to the product \(\widetilde E\times[0,2\pi]\), and the symbol \(\hat{\mathcal A}(x,\theta)\) is the function of the element \(\hat A(\in\mathfrak R)\) on the bicompact of maximal ideals \(\mathfrak M\):

\[ \hat A(M_{x,\theta})=\hat{\mathcal A}(x,\theta)\qquad (x\in\widetilde E,\ 0\leq \theta<2\pi). \]

From the fact that the ring \(\mathfrak R\) is symmetric and has no radical it follows ((4), supplement IV, theorem 3) that every noninvertible element of \(\mathfrak R\) is a generalized divisor of zero. With the aid of this property of the ring \(\mathfrak R\), Theorem 2, and the results of the note \((^5)\), one proves:

Theorem 3. In order that an operator \(A\in\mathfrak S\) be normally solvable and that at least one of the equations

\[ A\varphi=0,\qquad A^*\psi=0\qquad (\varphi,\psi\in L_2(E)) \]

have no more than a finite number of linearly independent solutions, it is necessary and sufficient that the symbol of the operator \(A\) vanish nowhere:

\[ \mathcal A(x,\theta)\ne 0\qquad (x\in\widetilde E;\ 0\leq \theta\leq 2\pi). \tag{7} \]

If condition (7) is satisfied, then the equations \(A\varphi=0\), \(A^*\varphi=0\) \((\varphi,\psi\in L_2(E_m))\) have the same number of linearly independent solutions.

4. Denote by \(L_2^{(r)}(E)\) the Hilbert space of \(r\)-dimensional vector-functions \(f=\{f_j\}_1^r\), with components \(f_j\in L_2(E)\), and with scalar product defined by

\[ (f,g)=\sum_{j=1}^{r}(f_j,g_j). \]

Then the system of multidimensional singular integral equations will have the form \(A\varphi=f\) \((\varphi,f\in L_2^{(r)}(E))\), where the operator \(A\) is defined in \(L_2^{(r)}(E)\)

\(^*\) By \(\widetilde E\) is denoted the compact space obtained by adjoining to \(E\) a point at infinity.

by the matrix \(\|A_{jk}\|_1^r\) with elements from \(\mathfrak S\). The matrix \(\|\mathcal A_{jk}(x,\theta)\|_1^r\) plays the role of the symbol of the operator \(\mathbf A\).

Theorem 4. Let \(\mathbf A\) be a linear operator defined in the space \(L_2^{(r)}(E)\) by the matrix \(\|A_{jk}\|_1^r\) with elements from \(\mathfrak S\). In order that the operator \(\mathbf A\) be normally solvable and that at least one of the equations \(\mathbf A\varphi=0,\ \mathbf A^*\psi=0\) \((\varphi,\psi\in L_2^{(r)}(E))\) have no more than a finite number of linearly independent solutions, it is necessary and sufficient that

\[ \det \|\mathcal A_{jk}(x,\theta)\|_1^r \ne 0 \qquad (x\in \tilde E;\ 0\leq \theta \leq 2\pi). \tag{8} \]

If condition (8) is satisfied, then the equations \(\mathbf A\varphi=0,\ \mathbf A^*\psi=0\) \((\varphi,\psi\in L_2^{(r)}(E))\) have the same finite number of linearly independent solutions.

The theorems given above generalize to the case of any natural \(m>2\).

The author expresses his gratitude to I. A. Iitskovich, M. G. Krein, and G. E. Shilov for useful advice.

Moldavian Branch
of the Academy of Sciences of the USSR

Received
31 III 1960

CITED LITERATURE

¹ S. G. Mikhlin, UMN, 3, no. 3 (1948). ² S. G. Mikhlin, UMN, 8, no. 2 (1953). ³ S. G. Mikhlin, Vestn. Leningrad. Univ., no. 1, issue 1 (1956). ⁴ I. M. Gelfand, D. A. Raikov, G. E. Shilov, UMN, 1, no. 2 (1946). ⁵ I. Ts. Gokhberg, DAN, 76, no. 4 (1951).