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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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