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Lp-boundedness ofMultilinear Pseudo-differential Operators on Zn and Tn

Published online by Cambridge University Press:  17 July 2014

V. Catană
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Abstract

The aim of this paper is to introduce and study multilinear pseudo-differential operatorson Zn and Tn =(Rn/2πZn) then-torus.More precisely, we give sufficient conditions and sometimes necessary conditions forLp-boundedness of theseclasses of operators. L2-boundedness results for multilinearpseudo-differential operators on Zn and Tn withL2-symbols are stated. The proofs of theseresults are based on elementary estimates on the multilinear Rihaczek transforms forfunctions in L2(Zn)respectively L2(Tn)which are also introduced.

We study the weak continuity of multilinear operators on the m-fold product of Lebesguespaces Lpj(Zn),j =1,...,m and thelink with the continuity of multilinear pseudo-differential operators on Zn.

Necessary and sufficient conditions for multilinear pseudo-differential operators onZn or Tn to be aHilbert-Schmidt operators are also given. We give a necessary condition for a multilinearpseudo-differential operators on Zn to be compact. A sufficientcondition for compactness is also given.

Keywords

Fourier seriesmultilinear pseudo-differential operatorsmultilinear Rihaczek transformsLp-boundednessLp-compactnessYoung convolution inequality for ZnHausdorff-Young inequality for the torus TnHilbert-Schmidt operators
Type
Research Article
Information
Mathematical Modelling of Natural Phenomena , Volume 9 , Issue 5: Spectral problems , 2014 , pp. 17 - 38
Copyright
© EDP Sciences, 2014

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