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A boundedness criterion for generalized Calderón-Zygmund operators. (English) Zbl 0567.47025
In 1965, A. P. Calderón showed the \(L^ 2\)-boundedness of the socalled first Calderón commutator. Many other nonconvolution operators in analysis, such as certain classes of pseudo-differential operators and the Cauchy integral operator on a curve, are also associated with kernels satisfying size and smoothness properties which imply together with the \(L^ 2\)-boundedness the \(L^ p-(p>1)\) boundedness of the operators in question. By these facts the aim of the paper is motivated: to give necessary and sufficient conditions for certain generalized Calderón- Zygmund operators to be bounded on \(L^ 2\). In the main theorem it’s about a continuous operator T from the class \({\mathcal S}(R^ n)\) of Schwartz functions to the dual class \({\mathcal S}'(R^ n)\), and the condition in order that T could be extended from \(L^ 2(R^ n)\) to itself is essentially that the images of the function 1 under the actions of the operator and its adjoint both lie in BMO, i.e. in the dual of \(H^ 1.\)
There are still some applications on the functional calculus connected with a second order partial differential operator, as well as on a conjecture of T. Kato, contained in a book of him [Perturbation theory for linear operators (1966; Zbl 0148.126)]. Cf. also a recent monography of the second author: Calderón-Zygmund operators, pseudo- differential operators and the Cauchy integral of Calderón, Lecture Notes Math. 994 (1983; Zbl 0508.42021).
Reviewer: M.Mikolás

MSC:
47B38 Linear operators on function spaces (general)
47A60 Functional calculus for linear operators
47F05 General theory of partial differential operators
42B20 Singular and oscillatory integrals (Calderón-Zygmund, etc.)
47B47 Commutators, derivations, elementary operators, etc.
47Gxx Integral, integro-differential, and pseudodifferential operators
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