Lp-boundedness of Multilinear Pseudo-differential Operators on Zn and Tn

V. Catană

doi:10.1051/mmnp/20149502

All issues

Volume 9 / No 5 (2014)

Math. Model. Nat. Phenom., 9 5 (2014) 17-38

Abstract

Free Access

Issue		Math. Model. Nat. Phenom. Volume 9, Number 5, 2014 Spectral problems


Page(s)		17 - 38
DOI		https://doi.org/10.1051/mmnp/20149502
Published online		17 July 2014

Math. Model. Nat. Phenom. Vol. 9, No. 5, 2014, pp. 17-38

L^p-boundedness of Multilinear Pseudo-differential Operators on Zⁿ and Tⁿ

V. Catană^⋆

Department of Mathematics and Informatics, University Politehnica of Bucharest Splaiul Independenţei 313, 060041, Bucharest, Romania

^⋆ Corresponding author. E-mail: catana_viorel@yahoo.co.uk

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 L^p-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 L^p_j(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 / L^p-boundedness / L^p-compactness / Young convolution inequality for Zⁿ / Hausdorff-Young inequality for the torus Tⁿ / Hilbert-Schmidt operators

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