MATHEMATICS

N. Ya. KRUPNIK

ON MULTIDIMENSIONAL SINGULAR OPERATORS IN SPACES OF TEST AND GENERALIZED FUNCTIONS

(Presented by Academician N. I. Muskhelishvili on 13 III 1964)

In the monograph of S. G. Mikhlin (¹), along with a number of other problems, the problem is posed of studying multidimensional singular operators in spaces of generalized functions. In this direction a number of results are already available. In (⁷) the continuity of singular operators* (with symbol independent of the pole) in the space \(D'_{L_p}\) (⁹) is proved, and the composition formulas for singular integrals are extended to the case in which their densities are generalized functions. In (⁵) the boundedness of singular operators in the spaces of S. L. Sobolev \(W_p^{(l)}\) \((1<p<\infty)\), as well as in the spaces conjugate to them, is proved. In addition, it is shown that the operator \(A_1A_2-A_3\), where \(A_3\) is the singular operator with symbol equal to the product of the symbols of the operators \(A_1\) and \(A_2\), acts from \(W_p^{(l)}\) into \(W_p^{(l+1)}\). In (⁶) a number of estimates are derived for singular integral operators whose kernels are generalized functions. In the present paper the space \(\mathfrak M\) of test functions and \(\mathfrak M^*\) of generalized functions \((\mathfrak M^*\supset D'_{L_p}\) for \(p<\infty)\) are introduced, and the question of the normal solvability and the index of singular operators (as well as of matrix singular operators) in the spaces \(\mathfrak M\), \(D_{L_p}\), \(W_p^{(l)}\), \(W_p^{(l)*}\), \(D'_{L_p}\), and \(\mathfrak M^*\) is investigated.

Moreover, the impossibility is proved (in a certain sense) of extending singular operators to spaces broader than \(\mathfrak M^*\).

Let us agree on the following notation: \(E_m\) is \(m\)-dimensional Euclidean space; \(q=(q_1,\ldots,q_m)\), where the \(q_i\) are nonnegative integer coordinates, \(|q|=q_1+\cdots+q_m\); \(D^q\varphi=\partial^{|q|}\varphi/\partial x_1^{q_1}\cdots\partial x_m^{q_m}\); \(C\) is one of the singular operators (¹) \(A\), \(B\), where

\[ A\varphi=a(x)\varphi(x)+\int_{\dot E_m}\frac{f(x,\theta)}{r^m}\varphi(y)\,dy,\qquad B\varphi=\bar a(x)\varphi(x)+\int_{\dot E_m}\frac{\bar f(y,-\theta)}{r^m}\varphi(y)\,dy; \]

\(f^{(q)}(x,\theta)\) is the derivative \(D^q f(x,\theta)\), computed under the assumption that \(\theta\) does not depend on \(x\); \(W_p^{(l)}(E_m)\) is the S. L. Sobolev space with norm
\[ \|\varphi\|=\sum_{|q|\le l}\|D^q\varphi\|_{L_p(E_m)}, \]
if \(l\ge0\), and
\[ W_p^{(l)}(E_m)=[W_s^{(-l)}(E_m)]^* \]
\((s^{-1}+p^{-1}=1)\), if \(l<0\); \(M=\bigcup_\alpha A_\alpha\) \((0<\alpha<1)\) (see (¹), p. 64). We shall say that \(\varphi\in M^{(n)}\) if \(D^q\varphi\in M\) for all \(|q|\le n\). Everywhere, unless otherwise stated, we shall assume that the necessary condition for the existence of the singular integrals (¹) is fulfilled,

\[ \int_S f(x,\theta)\,dS=0. \]

* In the present paper, the notation and terminology of S. G. Mikhlin (¹) are used for the most part.

1. Let \(M_p=(1+|x|)^{m-\frac1{p+1}}\); denote by \(\mathfrak M\) the set of complex infinitely differentiable functions \(\varphi(x)\) on \(E_m\) for which the products \(M_p|D^q\varphi|\), for \(|q|\leq p\) \((p=0,1,\ldots)\), are bounded on all of \(E_m\). The topology introduced by means of the countable collection of norms
   \[ \|\varphi\|_p=\sup_{|q|\leq p} M_p|D^q\varphi|\quad (p=0,1,\ldots) \]
   turns \(\mathfrak M\) into a complete countably normed space of type \(K\{M_p\}\) \((^2)\).

Suppose that for each fixed \(z\ne0\) the functions \(a(x)\) and \(f(x,z/|z|)\) are infinitely differentiable with respect to \(x\) and
\[ |D^q a(x)|\leq N_q,\qquad |f^{(q)}(x,\theta)|\leq N'_q \quad (N_q,N'_q=\mathrm{const}); \tag{1} \]
then one can show that the operators \(A\) and \(B\) are continuous in \(\mathfrak M\), whence it follows that \(A^*\) and \(B^*\) are continuous in \(\mathfrak M^*\). But on the linear set \(L_p(E_m)\), dense in \(\mathfrak M^*\), \(A^*u=Bu\) and \(B^*u=Au\); this makes it possible to define in \(\mathfrak M^*\) the operators \(A\) and \(B\), respectively, by the equalities \((AF,\varphi)=(F,B\varphi)\), \((BF,\varphi)=(F,A\varphi)\), where \(\varphi\in\mathfrak M\), \(F\in\mathfrak M^*\).

1. We formulate three lemmas which we shall use in the proof of Theorems 1–4.

Lemma 1. Suppose \(C_3\) is the operator with symbol equal to the product \(\Phi_1(x,\theta)\Phi_2(x,\theta)\) of the symbols of the operators \(C_1\) and \(C_2\). If for every \(|q|\leq2\) the functions \(\Phi_1^{(q)}(x,\theta)\) and \(\Phi_2^{(q)}(x,\theta)\), and their derivatives up to order \(m+[m/2]+1\) with respect to the Cartesian coordinates of the points \(\theta\), are continuous on \(\Sigma\times S^*\), then for any \(1<p<\infty\)
\[ T(M)\equiv (C_1C_2-C_3)(M)\subset W_p^{(1)}(E_m)\,^{**}. \]

We introduce the following notation. We say that \(\Phi(x,\theta)\in G_n\) if:
1) for every \(|q|\leq2\) the functions \(\Phi^{(q)}(x,\theta)\) and their derivatives up to order \(m+[m/2]+1\) with respect to the Cartesian coordinates of the points \(\theta\) are continuous on \(\Sigma\times S\);
2) for all \(|q|\leq n+1\), \(\Phi^{(q)}(x,\theta)\) are continuous on \(\Sigma\times S\), and 3) \(\Phi^{(q)}(x,\theta)\hat{\ }\in \hat W_p^{(l)}(S)\), where \(l\geq m+[m/2]+2\), if \(|q|\leq n\), and \(l\geq m+[m/2]+1\), if \(|q|=n+1\).

Lemma 2. If the symbol of the operator \(C\), \(\Phi(x,\theta)\in G_n\), \(\inf|\Phi|>0\), \(\varphi\in L_2(E_m)\), \(\psi\in M^{(n)}\), and \(C\varphi=\psi\), then \(\varphi\in M^{(n)}\).

Lemma 3. Suppose \(\Phi(x,\theta)\in G_n\) for some \(n\) and \(\inf|\Phi|>0\). If \(C\varphi=0\) and \(\varphi\in L_{p_0}\) for some \(1<p_0<\infty\), then \(\varphi\in L_p\) for every \(p\) \((1<p<\infty)\).

Theorem 1. If \(\Phi(x,\theta)\in G_n\) and \(\inf|\Phi|>0\), then the operator \(C\) is normally solvable in \(W_p^{(l)}(E_m)\) \((1<p<\infty,\ |l|\leq n)\), and its index is equal to zero.

We carry out the proof according to the following scheme: we show that the operators \(C'C-I\) and \(CC'-I\), where \(C'\) is the singular operator with symbol \(\Phi^{-1}(x,\theta)\), and \(I\) is the identity operator, are completely continuous in \(W_p^{(l)}(E_m)\), whence the normal solvability and finiteness of the index of the operator \(C\) follow \((^3)\). Then, using Theorem 4 of the work \((^4)\), we prove that the index of the operator \(C\) is equal to zero.

Corollary 1. Suppose the conditions of Theorem 2 are fulfilled and \(|l|\leq n\). If \(\psi\in M^{(l)}\) \(\bigl(\psi\in W_p^{(l)}\bigr)\), \(\varphi\in W_s^{(-n)}\) \((1<s,p<\infty)\), and \(C\varphi=\psi\), then \(\varphi\in M^{(l)}\) \(\bigl(\varphi\in W_p^{(l)}\bigr)\).

* Here \(S\) is the unit sphere in \(E_m\), and \(\Sigma\) is the unit sphere in \(E_{m+1}\), into which \(E_m\) is mapped under the stereographic transformation \((^1)\).

** Under the condition that the symbols are infinitely differentiable with respect to the Cartesian coordinates of the points \(\theta\) on \(S\), in \((^5)\) it is proved that \(T(L_p)\subset W_p^{(1)}\).

Corollary 2. If \(\Phi(x,\theta)\in G_n\) for all \(n\) and \(\inf|\Phi|>0\), then the operator \(C\) is normally solvable in the L. Schwartz spaces \(D'_{L_p}\) and \(D'_{L_p}\) \((^9)\), and its index is equal to zero.

Theorem 2. If \(\Phi(x,\theta)\in G_n\) for all \(n\) and \(\inf|\Phi|>0\), then the operator \(C\) is normally solvable in the spaces \(\mathfrak M\) and \(\mathfrak M^*\), and its index in these spaces is equal to zero.

We carry out the proof according to the following scheme: we represent the operator \(C\) in the form of a sum \(C=C_1+K\), where \(C_1\) is invertible in \(L_2(E_m)\), and \(K\) is a finite-dimensional operator. Then we show that the operator \(K\) acts in \(\mathfrak M\) and that \(C_1\) is invertible in \(\mathfrak M\); consequently \((^8)\), the operator \(C\) is normally solvable in \(\mathfrak M\) and its index is equal to zero. The corresponding assertions in the space \(\mathfrak M^*\) are proved by passing to adjoint operators.

1. Denote by \(\mathfrak M^k\) the set of vector-functions \(\varphi=(\varphi_1,\ldots,\varphi_k)\), where \(\varphi_i\in\mathfrak M\). If in \(\mathfrak M^k\) we introduce the countable system of norms
   \[ \|\varphi\|_{kp}=\left(\sum_{i=1}^{k}\|\varphi_i\|_p^2\right)^{1/2}, \]
   where \(\|\varphi_i\|_p\) is the \(p\)-th norm of the function \(\varphi_i\in\mathfrak M\), then \(\mathfrak M^k\) becomes a complete countably normed space. Consider the system
   \[ \sum_{j=1}^{k} C_{ij}\varphi_j=\psi_i\qquad (i=1,2,\ldots,k), \tag{2} \]
   where \(C_{ij}\) are operators of type \(A\) and \(B\). Let \(C\) be the matrix singular operator \((^1)\) corresponding to the system (2).

Theorem 3. If the symbols of the operators \(C_{ij}\), \(\Phi_{ij}\in G_n\), and \(\inf|\det\Phi|>0\), then for \(1<p<\infty\) and \(|l|\le n\) the operator \(C\) is normally solvable in \(W_p^{(l)k}\) and has a finite index, moreover
\[ \operatorname{ind}_{W_p^{(l)k}}(C)=\operatorname{Ind}_{L_2^k}(C). \]

Theorem 4. If \(\Phi_{ij}\in G_n\) for all \(n\) and \(\inf|\det\Phi|>0\), then the operator \(C\) is normally solvable in \(\mathfrak M^k\) and \((\mathfrak M^k)^*\), and
\[ \operatorname{Ind}_{\mathfrak M^k}(C)=\operatorname{Ind}_{(\mathfrak M^k)^*}(C)=\operatorname{Ind}_{L_2^k}(C). \tag{3} \]

We carry out the proof according to the following scheme. Let \(S\) be a matrix singular operator with symbolic matrix \(\Phi^{-1}(x,\theta)\). Then the operators \(SC\) and \(CS\) are normally solvable in \(L_2^k\), and their indices are equal to zero. By the same scheme as in theorem 2, we prove that \(SC\) and \(CS\) are normally solvable in \(\mathfrak M^k\) and \((\mathfrak M^k)^*\), and their indices are equal to zero. Then \((^8)\) the operators \(S\) and \(C\) are normally solvable in \(\mathfrak M^k\) and \((\mathfrak M^k)^*\), and have finite indices. The validity of equality (3) follows from the fact that
\[ \operatorname{Ind}_{\mathfrak M^k}(C)\le \operatorname{Ind}_{L_2^k}(C),\qquad \operatorname{Ind}_{\mathfrak M^k}(S)=\operatorname{Ind}_{L_2^k}(S), \]
\[ \operatorname{Ind}_{\mathfrak M^k}(C)+\operatorname{Ind}_{\mathfrak M^k}(S) = \operatorname{Ind}_{L_2^k}(C)+\operatorname{Ind}_{L_2^k}(S). \]

Remark. In the monograph of S. G. Mikhlin \((^1)\) several sufficient conditions are given for the index of the system (2) in \(L_p^k(E_m)\) to be equal to zero (see Theorems 2.40—5.40). From theorem 3 \((^4)\) it follows that each of the conditions of S. G. Mikhlin gives rise to a corresponding sufficient condition for the index of the system (2) in \(W_p^{(l)k}\) \((\mathfrak M^k,(\mathfrak M^k)^*)\) to be equal to zero.

1. Since the space \(\mathfrak M\) contains all finite infinitely differentiable functions, the values of a regular functional \((F,\varphi)\) on basic functions uniquely (up to values on a set of measure zero) determine the corresponding locally integrable function \(F(x)\) \((^2)\). The space \(\mathfrak M^*\) contains all locally integrable func-

tions for which the products \(|F(x)|(1+|x|)^{-m+\varepsilon}\) (for some \(\varepsilon>0\)) are summable on \(E_m\).

Let \(Q\) be a linear topological space such that \(Q^*\) contains all locally integrable functions for which the products \(|F(x)|(1+|x|)^k\), where \(k<m\) is fixed, are summable on \(E_m\).

Consider the following problem. Is it possible, by the same method as above, to extend singular operators to the space \(Q^*\) (which is broader than \(\mathfrak{M}^*\))? In order to extend the operator \(A\) to \(Q^*\), one must find a space \(P\) such that \(B(P)\subset Q\) and define \((AF,\varphi)=(F,B\varphi)\), where \(F\in Q^*\), \(\varphi\in P\). Suppose that the symbol of the operator \(C\) does not depend on the pole. A negative answer to the problem posed above is given by

Theorem 5. If there exists a linear topological space \(P\subset Q\), sufficiently rich in functions, such that \(C(\dot P)\subset Q\), then the characteristic of the operator \(C\) satisfies \(f(\theta)\equiv 0\).

Kishinev State
University

Received
16 VIII 1963

CITED LITERATURE

¹ S. G. Mikhlin, Multidimensional Singular Integrals and Integral Equations, Moscow, 1962.
² I. M. Gel'fand, G. E. Shilov, Spaces of Basic and Generalized Functions, Moscow, 1958.
³ F. V. Atkinson, Mat. sborn., 28 (70), 1 (1951).
⁴ S. A. Freidkin, Uch. zap. Kishinevsk. gos. univ., 11, 12 (1954).
⁵ B. Malgrange, Sborn. per. Matematika, 6, 5, 87 (1962).
⁶ L. Hörmander, Estimates for Operators Invariant under Translation, IL, 1962.
⁷ J. Horváth, C. R., 237, 23, 1480 (1953); Trans. Am. Math. Soc., 82, 52 (1956).
⁸ H. Schaefer, Math. Zs., 66, 2, 147 (1956).
⁹ L. Schwartz, Théorie des distributions, 1, Paris, 1950; 2, Paris, 1951.

* The space \(P\) is said to be sufficiently rich in functions if from the equality \((F,\varphi)=0\), where \(F\) is locally integrable and \(\varphi\) ranges over all \(P\), it follows that \(F(x)=0\) almost everywhere.