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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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References

Auscher, R., Carro, M.J.. On relations between operators on ℝN, TN and ℤn. Studia Math., 101 (1992), no. 2, 165–182. Google Scholar
Bényi, Á., Gröchenig, K., Heil, C., Okoudjou, K., Modulation spaces and a class of bounded multilinear pseudodifferential operators. J. Operator Theory, 54 (2005), 387–399. Google Scholar
D. Bose, S. Madan, P. Mohanty, S. Shrivastava. Relations between bilinear multipliers on ℝn, Tn and ℤn. arXiv: 0903.4052v1 [math.CA], 24 Mar 2009.
Carlos Andres, R.T.. Lp-estimates for pseudo-differential operators on ℤn. J. Pseudo-Differ. Oper. Appl., 2 (2011), 367–375. CrossRefGoogle Scholar
V. Catană, S. Molahajloo, M.W. Wong. L p-boundedness of multilinear pseudo-differential operators. In Operator Theory: Advances and Applications. vol. 205, 167-180, Birhäuser Verlag, Basel, 2009.
M. Charalambides, M. Christ. Near-extremizers of Young’s inequality for discrete groups. arXiv: 1112.3716v1 [math.CA], 16 Dec. 2011.
Grafakos, L., Torres, R.H.. Multilinear Calderon-Zygmund theory. Advances in Mathematics, 165 (2002), no. 1, 124–164. CrossRefGoogle Scholar
Grafakos, L., Honzik, P.. Maximal transferance and summability of multilinear Fourier series. J. Aust. Math. Soc., 80 (2006), no. 1, 65–80. CrossRefGoogle Scholar
L. Grafakos. Classical Fourier Analysis. Second Edition, Springer, 2008.
R.V. Kadison, J.R. Ringrose. Fundamentals of the Theory of Operator Algebras: Elementary Theory. Academic Press, 1983.
S. Molahajloo, M.W. Wong. Pseudo-differential operators on S 1. In Operator Theory: Advances and Applications, vol. 189, 297-306, Birhäuser Verlag, Basel, 2008.
S. Molahajloo. Pseudo-differential operators on Z. In Operator Theory: Advances and Applications, vol. 205, 213–221, Birhäuser Verlag, Basel, 2009.
M. Pirhayati. Spectral Theory of Pseudo-Differential Operators on S 1. In Pseudo-Differential Operators: Analysis, Applications and Computations, Operator Theory: Advanced and Applications 213, Springer Basel AG 2011.
Ruzhansky, M., Turunen, V.. Quantization of pseudo-differential operators on the torus. J. Fourier Anal. Appl., 16 (2010), 943-982. CrossRefGoogle Scholar
M. Ruzhansky, V. Turunen. Pseudo-Differential Operators and Symmetries. Birhäuser, 2010.
Ruzhansky, M., Turunen, V.. On the toriodal quantization of periodic pseudo-differential operators. Numerical Functional Analysis and Optimization, 30 (2009), 1098-1124. CrossRefGoogle Scholar
M.W. Wong. Discrete Fourier Analysis. Birhäuser, 2011.