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Lp-boundedness of Multilinear 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 operators on Zn and Tn = (Rn/ 2πZn) the n-torus. More precisely, we give sufficient conditions and sometimes necessary conditions for Lp-boundedness of these classes of operators. L2-boundedness results for multilinear pseudo-differential operators on Zn and Tn with L2-symbols are stated. The proofs of these results are based on elementary estimates on the multilinear Rihaczek transforms for functions in L2(Zn) respectively L2(Tn) which are also introduced.

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

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