\(H^ p\) continuity properties of Calderón-Zygmund-type operators
In a recent paper, G. David and J. L. Journé obtained a characterization of the Caldéron-Zygmund operators, in the sense of R. Coifman and Y. Meyer. These operators are an interesting generalization of the classical singular integral operators of Calderón-Zygmund. In particular, they include some classes of pseudodifferential operators, Cauchy integrals on Lipschitz curves, commutators of order \(n\), etc. The \(L^ p\) theory of these operators is now well understood. In this paper we consider the \(H^ p\) theory (in the sense of R. Coifman and G. Weiss) and obtain the following result: Let \(T\) be a \(\delta\) C-Z operator such that \(T^*(1)=0\). Then, \(T\) extends to a continuous operator from \(H^ p\) into itself for \(1\geq p>n/n+\delta\). The proof of this theorem consists in showing that the image under \(T\) of a \((p,\infty)\)-atom is what we call a \((p,q,\alpha)\)-molecule. In the second part of the paper we introduce the class of strongly singular Calderón-Zygmund operators. Our study of these operators is motivated by the multiplier operators whose symbols are given by \(\exp (i| \xi |^ a)/| \xi |^{\beta}\) away from the origin. These multiplier operators have been studied by several authors. We show that our strongly singular C-Z operators map \(L^{\infty}\) into BMO, extending a well-known result of C. Fefferman and E. M. Stein. Moreover, we show that a strongly singular C-Z operator \(T\) satisfying \(T^*(1)=0\) acts continuous on \(H^ p\) spaces for a certain range of values of \(p\), \(1\geq p>p_ 0\). For convolution operators on \({\mathbb{R}}\), R. Coifman has obtained the continuity for the critical index \(p_ 0\). We are not able to proof \(H^{p_ 0}\) continuity in the general case. However, we single out a class of convolution operators of strongly singular type, on \({\mathbb{R}}^ n\), for which a general version of Coifman's result holds. In the last part of the paper we consider a certain class of pseudodifferential operators. In fact, following a suggestion of E. M. Stein, we show that our class of strongly singular Calderón-Zygmund operators includes the pseudodifferential operators with symbols in \(S^{-b}_{a,\delta}\) where \(0<\delta \leq a<1\), \((1- a)n/2\leq b<n/2\). The case \(\delta <a\) was considered by C. Fefferman.
singular integral operators of Calderón-Zygmund
scientific article; zbMATH DE number 3960221
scientific article