## 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

### Citations:

Zbl 0148.126; Zbl 0508.42021
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