Geodesic
Lp-boundedness of Multilinear Pseudo-differential Operators on Zn and Tn
V. Catană1
1 Department of Mathematics and Informatics, University Politehnica of Bucharest Splaiul Independenţei 313, 060041, Bucharest, Romania
Mathematical modelling of natural phenomena, Tome 9 (2014) no. 5, pp. 17-38.

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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.
DOI : 10.1051/mmnp/20149502

V. Catană 1

1 Department of Mathematics and Informatics, University Politehnica of Bucharest Splaiul Independenţei 313, 060041, Bucharest, Romania 
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V. Catană. Lp-boundedness of Multilinear Pseudo-differential Operators on Zn and Tn. Mathematical modelling of natural phenomena, Tome 9 (2014) no. 5, pp. 17-38. doi : 10.1051/mmnp/20149502. http://geodesic.mathdoc.fr/articles/10.1051/mmnp/20149502/

[1] R. Auscher, M.J. Carro Studia Math. 1992 165 182

[2] Á. Bényi, K. Gröchenig, C. Heil, K. Okoudjou J. Operator Theory 2005 387 399

[3] 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.

[4] R.T. Carlos Andres J. Pseudo-Differ. Oper. Appl. 2011 367 375

[5] V. Catană, S. Molahajloo, M.W. Wong. Lp-boundedness of multilinear pseudo-differential operators. In Operator Theory: Advances and Applications. vol. 205, 167-180, Birhäuser Verlag, Basel, 2009.

[6] M. Charalambides, M. Christ. Near-extremizers of Young’s inequality for discrete groups. arXiv: 1112.3716v1 [math.CA], 16 Dec. 2011.

[7] L. Grafakos, R.H. Torres Advances in Mathematics 2002 124 164

[8] L. Grafakos, P. Honzik J. Aust. Math. Soc. 2006 65 80

[9] L. Grafakos. Classical Fourier Analysis. Second Edition, Springer, 2008.

[10] R.V. Kadison, J.R. Ringrose. Fundamentals of the Theory of Operator Algebras: Elementary Theory. Academic Press, 1983.

[11] S. Molahajloo, M.W. Wong. Pseudo-differential operators on S1. In Operator Theory: Advances and Applications, vol. 189, 297-306, Birhäuser Verlag, Basel, 2008.

[12] S. Molahajloo. Pseudo-differential operators on Z. In Operator Theory: Advances and Applications, vol. 205, 213–221, Birhäuser Verlag, Basel, 2009.

[13] M. Pirhayati. Spectral Theory of Pseudo-Differential Operators on S1. In Pseudo-Differential Operators: Analysis, Applications and Computations, Operator Theory: Advanced and Applications 213, Springer Basel AG 2011.

[14] M. Ruzhansky, V. Turunen J. Fourier Anal. Appl. 2010 943 982

[15] M. Ruzhansky, V. Turunen. Pseudo-Differential Operators and Symmetries. Birhäuser, 2010.

[16] M. Ruzhansky, V. Turunen Numerical Functional Analysis and Optimization 2009 1098 1124

[17] M.W. Wong. Discrete Fourier Analysis. Birhäuser, 2011.

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