## Conditionally convergent series of linear operators on $$L^ p$$-spaces and $$L^ p$$-estimates for pseudodifferential operators.(English)Zbl 0681.35091

The first aim of this article is to develop an $$L^ p$$-variant of the Cotlar-Stein lemma. The substitute for this lemma is an extension of the Littlewood-Paley theory to $$L^ p$$-spaces for non-translation-invariant operators. It is assumed that there exist suitable bounded linear operators $$Q_ j$$ in $$L^ p$$ such that $$\| f\|_ p\sim \| (\sum_{j}| Q_ jf|^ 2)^{1/2}\|_ p,$$ $$f\in L^ p$$. The $$Q_ j's$$ are Fourier multiplier operators on $$R^ n$$ with appropriate hypotheses. Let T be an operator with a decomposition $$T=\sum T_ j$$. The boundedness of T is deduced from suitable boundedness hypotheses for the operators $$Q_ iT_ kQ^*_ j$$. Applications to the theory of pseudodifferential operators are given.
Reviewer: P.Jeanquartier

### MSC:

 35S05 Pseudodifferential operators as generalizations of partial differential operators 42B15 Multipliers for harmonic analysis in several variables 47Gxx Integral, integro-differential, and pseudodifferential operators 42B20 Singular and oscillatory integrals (Calderón-Zygmund, etc.) 42B25 Maximal functions, Littlewood-Paley theory
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