Math. Model. Nat. Phenom.
Volume 9, Number 5, 2014Spectral problems
|Page(s)||17 - 38|
|Published online||17 July 2014|
Lp-boundedness of Multilinear Pseudo-differential Operators on Zn and Tn
Department of Mathematics and Informatics, University
Politehnica of Bucharest Splaiul Independenţei 313, 060041, Bucharest, Romania
Corresponding author. E-mail: firstname.lastname@example.org
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.
Mathematics Subject Classification: 47G30 / 47B38 / 47H60
Key words: Fourier series / multilinear pseudo-differential operators / multilinear Rihaczek transforms / Lp-boundedness / Lp-compactness / Young convolution inequality for Zn / Hausdorff-Young inequality for the torus Tn / Hilbert-Schmidt operators
© EDP Sciences, 2014
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