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UDC 513.88+517.397+517.948
MATHEMATICS
M. Sh. BIRMAN, M. Z. SOLOMYAK
DOUBLE OPERATOR STIELTJES INTEGRALS AND MULTIPLIER PROBLEMS
(Presented by Academician V. I. Smirnov on 6 III 1966)
In a note by the authors \((^1)\) (see also \((^2)\)), properties were studied of double operator Stieltjes integrals of the form
\[ Q \equiv \Phi T = \iint_{\Lambda M} \varphi(\lambda,\mu)\,F(d\mu)\,T E(d\lambda), \tag{1} \]
where \(\varphi\) is a complex function, \(T\) is a bounded operator in a Hilbert space \(\mathfrak H\), and \(F(\cdot)\) and \(E(\cdot)\) are orthogonal spectral measures. Here we shall present further results in this direction, using the notation and terminology of \((^1)\).
In Sec. 1 the properties are studied of the transformation (transformer) \(Q=\Phi T\), acting in various symmetric normed ideals \((^3)\) of the ring \(\mathfrak R\) of bounded operators in \(\mathfrak H\). The corresponding results of \((^1)\) are generalized by the fact that, first, we no longer assume \(\Lambda\) and \(M\) to be one-dimensional manifolds and, second, the smoothness conditions on the function \(\varphi\) are imposed not in the uniform metric but in the metric of the space \(L_p\). Such generalizations are essential for applications.
In Sec. 2 the connection of integrals of the form (1) with a certain class of multiplier problems is clarified. In particular, our scheme includes the question of the boundedness of multidimensional singular integrals in \(L_2\) (Sec. 3) and the trigonometric multiplier problem in \(l_p\) (Sec. 4). From the general theorems on transformers there follow both certain known and new results for these problems. Let us also note that, by writing a singular integral in the form (1), we give for it (apparently for the first time) an invariant definition.
- Let \((\Lambda,\Sigma,E(\cdot))\), \((M,\Sigma_1,F(\cdot))\) be two spaces with orthogonal spectral measures. The definition of the integral (1), given in \((^1)\) for \(\Lambda=M=R^1\), carries over automatically to the case under consideration. Without dwelling on this question in detail, we point out only that, as in \((^1)\), one can introduce a certain special orthogonal measure \(\mathfrak G(\cdot)\) (essentially, the tensor product of the measures \(E(\cdot)\) and \(F(\cdot)\)). The transformer \(\Phi\) is then defined as the integral of the function \(\varphi(\lambda,\mu)\) with respect to the measure \(\mathfrak G(\cdot)\). Below we shall assume the function \(\varphi(\lambda,\mu)\) to be measurable and bounded with respect to the measure \(\mathfrak G\):
\[ (\mathfrak G)\;-\;\sup|\varphi(\lambda,\mu)|<\infty. \tag{2} \]
The analogue of Theorem 1 from \((^1)\) is the following proposition.
Theorem 1. Under condition (2), the integral (1) defines a transformer \(\Phi\) of class \((\mathfrak S_2,\mathfrak S_2)\). The set of such transformers forms a commutative normed ring with involution, isomorphic and isometric to the ring of \(\mathfrak G\)-measurable functions with norm (2).
Denote by \(Q^m\) the unit cube in \(R^m\), and by \(W_p^\alpha(Q^m)\) the corresponding Sobolev–Slobodetskii functional space \((^4)\).
For \(m=1\) we shall also consider the space \(V_\beta\) (see, for example, \((^5)\)) of functions of bounded \(\beta\)-variation.
Theorem 2. Let \(\Lambda=Q^m\), \(\varphi(\lambda,\mu)\in W_p^\alpha(Q^m)\) \((\alpha p>m)\) as a function of \(\lambda\) for almost all \(\mu\in M\), and suppose that
\[ (F)\ — \sup \|\varphi(\cdot,\mu)\|_{W_p^\alpha}<\infty . \tag{3} \]
Then, for \(p\leqslant 2\), the integral (1) generates a transformer belonging to each of the classes \((\mathfrak S_q,\mathfrak S_q)\), \(1\leqslant q\leqslant \infty\), and \((\mathfrak R,\mathfrak R)\). For \(m=1\) this result is also true if, for almost all \(\mu\in M\), \(\varphi\in V_\beta\), \(\beta<2\), and \(\varphi\in \operatorname{Lip}\varepsilon\), \(\varepsilon>0\), with respect to the variable \(\lambda\), and
\[ (F)\ — \sup \{\|\varphi(\cdot,\mu)\|_{\operatorname{Lip}\varepsilon} +\|\varphi(\cdot,\mu)\|_{V_\beta}\}<\infty . \tag{4} \]
Analogous assertions are valid when the roles of the variables \(\lambda\) and \(\mu\) are interchanged.
We note that for \(m=1\) condition (4) is broader than condition (3).
Corollary 1. Under the hypotheses of Theorem 2, \(\Phi\in(\mathfrak S,\mathfrak S)\), where \(\mathfrak S\) is any separable \((^3)\) (or conjugate to a separable) symmetric normed ideal in \(\mathfrak R\).
Corollary 2. Under the hypotheses of Theorem 2, the transformer \(\Phi\) induces on the factor-ring \(\mathfrak R/\mathfrak S_\infty\) a bounded linear transformation whose norm does not exceed \(\|\Phi\|_{1,1}\).
Theorem 3. Suppose that the hypotheses of the first part of Theorem 2 are satisfied, \(p>2\), and \(\alpha\leqslant m/2\). Then the integral (1) generates a transformer \(\Phi\in(\mathfrak S_q,\mathfrak S_q)\), where \(|q^{-1}-1/2|<\alpha m^{-1}\). If \(\alpha=m/2\), then \(\Phi\in(\mathfrak S_1,\mathfrak S_\infty)\) and \(\Phi\in(\mathfrak S_\infty,\mathfrak S_\infty)^*\). If, in the hypotheses of the second part of Theorem 2, \(\beta\geqslant 2\), then \(\Phi\in(\mathfrak S_q,\mathfrak S_q)\) for \(|q^{-1}-1/2|<\beta^{-1}\).
Remark. The assertions of Theorems 2 and 3 remain valid if \(\Lambda\) is any smooth compact \(m\)-dimensional manifold, without boundary or with boundary.
We shall give brief indications concerning the method of proof of Theorems 2 and 3. Let \(f,g\in\mathfrak H\), \(\|f\|=\|g\|=1\), and let the measures \(\sigma\) and \(\tau\) be given by the formulas \(\sigma(\cdot)=(E(\cdot)f,f)\), \(\tau(\cdot)=(F(\cdot)g,g)\). According to the scheme set forth in \((^2)\), to prove Theorem 2 it suffices to estimate, uniformly with respect to \(f\) and \(g\), the nuclear norm of the integral operator with kernel \(\varphi(\lambda,\mu)\) acting from \(L_2(\Lambda;\sigma)\) into \(L_2(M;\tau)\). The required uniform estimates were obtained in the authors’ paper \((^5)\). The proof of Theorem 3 uses the general idea of “interpolation by smoothness,” applied earlier by Khirman \((^6)\) in a simpler situation. In carrying out the interpolation, it is necessary to make essential use of a number of results of \((^5)\), including Theorem 1 of \((^5)\), as well as the concrete method described there for approximating functions from \(W_p^\alpha\) by piecewise polynomial functions. The proof of Corollary 1 is based on the interpolation theorem of B. S. Mityagin \((^7)\).
- Let the space \(\mathfrak H\) be decomposed into a direct integral
\[ \mathfrak H=\int_{\Lambda}\oplus\,\mathfrak H(\lambda)\rho(d\lambda) \qquad \left(\mathfrak H=\int_{\widetilde M}\oplus\,\mathfrak H_1(\mu)\rho_1(d\mu)\right) \tag{5} \]
so that the action of the operator \(E(\delta)\), \(\delta\subset\Lambda\) \((F(\partial),\partial\subset M)\), on an element \(f\in\mathfrak H\) reduces to multiplication of the “representative” \(f(\lambda)\) \((f_1(\mu))\) by the characteristic function of the set \(\delta(\partial)\):
\[ (E(\delta)f)(\lambda)=\chi_\delta(\lambda)f(\lambda) \qquad ((F(\partial)f)_1(\mu)=\chi_\partial(\mu)f_1(\mu)). \]
To every operator \(A\in\mathfrak S_2\), in the decompositions (5) there corresponds an operator kernel \(A(\lambda,\mu)\), determined almost everywhere on \(\Lambda\times M\), of class \(\mathfrak S_2\), mapping
\[ \text{* For the definition of the ideals } \mathfrak S_\omega,\mathfrak S_\Omega \text{ see } ({}^3), \S 15. \]
\(\mathfrak H(\lambda)\) into \(\mathfrak H_1(\mu)\) and such that
\[ (Af)_1(\mu)=\int_\Lambda A(\lambda,\mu)f(\lambda)\rho(d\lambda), \]
\[ \|A\|_{\mathfrak S_2}^{2}=\iint_{\Lambda M}\|A(\lambda,\mu)\|_{\mathfrak S_2}^{2}\rho(d\lambda)\rho_1(d\mu). \]
It is easy to verify that, under the hypotheses of Theorem 1, the kernels corresponding to the operators \(T\) and \(Q\) are related by
\[ Q(\lambda,\mu)=\varphi(\lambda,\mu)T(\lambda,\mu). \]
Denote by \(\widetilde{\mathfrak S}_q\), \(q\le 2\), the class of kernels corresponding, in the decompositions (5), to operators of the class \(\mathfrak S_q\). Starting from the relation \(\mathfrak S_1^*=\mathfrak R\), one can associate a generalized kernel with every operator \(A\in\mathfrak R\). Thus we arrive at the classes of generalized kernels \(\widetilde{\mathfrak R}\), \(\widetilde{\mathfrak S}_q\), \(2<q\le\infty\), associated with the decompositions (5). The assertions of Theorems 2 and 3 can now be regarded as sufficient conditions for a scalar kernel \(\varphi(\lambda,\mu)\) to be a multiplier in some of the classes \(\widetilde{\mathfrak S}_q,\widetilde{\mathfrak R}\).
- In this section \(\mathfrak H=L_2(R^m)\), \(\Lambda=R^m\), \(M=S_{m-1}\) is the unit sphere in the space \(\Xi_m\) dual to \(R^m\). If \(\delta\subset S_{m-1}\), then \(\bar\delta(\subset \Xi_m)\) is the complete inverse image of \(\delta\) under central projection. Let \(F(\partial)\) be the operator of multiplication by the characteristic function of a set \(\partial\subset R^m\), let \(\mathcal F\) be the Fourier operator, and let \(E(\delta)=\mathcal F^*F(\bar\delta)\mathcal F\), \(\delta\subset S_{m-1}\). Suppose further that a function \(\sigma(\theta,x)\) \((\theta\in S_{m-1}, x\in R^m)\) defines the transformer
\[ \Phi=\iint \sigma(\theta,x)F(dx)(\cdot)E(d\theta) \]
of the class \((\mathfrak R,\mathfrak R)\). We introduce the following definition.
Definition. A singular integral (s.i.) operator in \(L_2(R^m)\) with symbol \(\sigma(\theta,x)\) is the operator \(J=\Phi I\) \((I\) is the identity operator).
The considerations of § 2 make it possible easily to establish the equivalence of this definition and the customary definitions of an s.i. operator in terms of its symbol by means of the Fourier transform \((^8{}^{-10})\) or by means of a certain repeated integral \((^8)\). The advantage of the definition proposed here is its invariance and symmetry with respect to \(\theta\) and \(x\).
Theorem 2 now leads to various criteria for boundedness of s.i. operators in \(L_2(R^m)\). We shall agree to write \(\sigma\in \widetilde W_p^\alpha(S_{m-1})\) if \(\sigma(\theta,x)\) belongs to \(W_p^\alpha\) with respect to \(\theta\), and the norm of \(\sigma\) as a function of \(x\) is essentially (with respect to Lebesgue measure) bounded. Analogously for other functional classes, and also when the roles of \(\theta\) and \(x\) are interchanged.
Theorem 4. Let \(J\) be an s.i. operator with symbol \(\sigma(\theta,x)\), where \(\sigma\in \widetilde W_2^\alpha(S_{m-1})\), \(2\alpha>m-1\), or, for \(m=2\), \(\sigma\in \widetilde V_\beta(S_1)\cap \operatorname{Lip}\varepsilon(S_1)\), \(\beta<2\), \(\varepsilon>0\). Then \(J\in\mathfrak R\).
The first assertion of Theorem 4 was recently obtained, with the aid of expansions in spherical functions, by M. S. Agranovich \((^9)\), who refined the considerations of S. G. Mikhlin \((^8)\). The second assertion is apparently new.
Interchanging the roles of the variables \(x\) and \(\theta\) leads to another type of boundedness conditions. Let \(x=x(y)\) be the inverse stereographic mapping of the sphere \(S^m\) onto the extended space \(R^m\).
Theorem 5. Let \(J\) be an s.i. operator with symbol \(\sigma(\theta,x)\) and \(\hat\sigma(\theta,y)=\sigma(\theta,x(y))\). If \(\hat\sigma\in \widetilde W_2^\alpha(S^m)\), \(2\alpha>m\), then \(J\in\mathfrak R\).
If the symbol \(\sigma(\theta,x)\) is finite with respect to \(x\), then the assertion of Theorem 5 is valid under the condition \(\sigma\in \widetilde W_2^\alpha(R^m)\), \(2\alpha>m\). Imposing the latter condition locally, we arrive at a boundedness criterion for an s.i. operator in \(L_2\) on a compact \(m\)-dimensional manifold.
The approach described makes results such as Theorem 5 quite natural. At the same time, until very recently the possibility of imposing requirements on the symbol with respect to the variable \(x\), and not to \(\theta\), had not been noticed at all. Only recently, in a paper of Kohn and Nirenberg \({}^{10}\), was a similar condition proposed in terms of the Fourier transform of the symbol with respect to the variable \(x\). This condition and Theorem 5 do not cover one another, but, when formulated in terms of smoothness of the symbol, our conditions are less restrictive.
- Let \(\{c_k\}\), \(k=(k_1,\ldots,k_m)\), \(-\infty<k_j<\infty\), be an \(m\)-fold sequence of class \(l_q\), \(q\geqslant 1\). Multiply the function
\[ f(\lambda)=\sum_k c_k e^{i(k,\lambda)},\qquad \lambda=(\lambda_1,\ldots,\lambda_m) \]
(for \(q>2\), possibly generalized) by a fixed periodic function \(\psi(\lambda)\). Let \(\Psi\) be the linear transformation which sends the sequence \(\{c_k\}\) to the sequence of Fourier coefficients of the function \(\psi(\lambda)f(\lambda)\). It is required to indicate conditions under which \(\Psi\) is a bounded operator from \(l_q\) to \(l_q\). The formulated trigonometric problem of multipliers \({}^{11,6}\) is included in the scheme of § 2 if one sets \(\Delta=M=\mathscr T_m\), where \(\mathscr T_m\) is the \(m\)-dimensional torus; \(\mathfrak H=L_2(\mathscr T_m)\); \(E(\delta)=F(\delta)\) is the operator of multiplication by the characteristic function of the set \(\delta\subset\mathscr T_m\); \(T\) is the integral operator in \(L_2(\mathscr T_m)\) with kernel \(f(\mu-\lambda)\); \(\varphi(\lambda,\mu)=\psi(\mu-\lambda)\). Application of Theorems 2 and 3 leads to the following result.
Theorem 6. If \(\psi\in W_p^\alpha(\mathscr T_m)\), \(p\alpha>m\), then \(\Psi\in(l_q,l_q)\), where \(q\geqslant 1\) for \(p\leqslant 2\) and \(|q^{-1}-1/2|<\alpha m^{-1}\) for \(p>2\), \(2\alpha\leqslant m\).
As for the second assertion of Theorem 3, in the present case it leads to a result obtained earlier by Hirschman \({}^{6}\) on another technical basis.
Leningrad State University
named after A. A. Zhdanov
Received
1 III 1966
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