Abstract
Results on the L p-boundedness of multilinear pseudo-differential operators are given. The proofs are based on elementary estimates on the multilinear Rihaczek transforms, the multilinear Wigner transforms and the multilinear Weyl transforms.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
W. Beckner, Inequalities in Fourier analysis, Ann. Math. 102 (1975), 159–182.
Á. Bényi and K.A. Okundjou, Bilinear pseudodifferential operators on modulation spaces, J. Fourier Anal. Appl. 10 (2004), 301–313.
Á. Bényi and K.A. Okundjou, Modulation space estimates for multilinear pseudodifferential operators, Studia Math. 172 (2006), 169–180.
Á. Bényi, K. Gröchenig, C. Heil and K.A. Okundjou, Modulation spaces and a class of bounded multilinear pseudo-differential operators, J. Operator Theory 54 (2005), 389–401.
P. Boggiatto, E. Buzano and L. Rodino, Global Hypoellipticity and Spectral Theory, Akademie Verlag, 1996.
P. Boggiatto, G. De Donno and A. Oliaro, A class of quadratic time-frequency representations based on the short-time Fourier transform, in Modern Trends in Pseudo-Differential Operators, Editors: J. Toft, M. W. Wong and H. Zhu, Operator Theory: Advances and Applications 172, Birkhäuser, 2007, 235–249
P. Boggiatto, G. De Donno and A. Oliaro, A unified point of view on time-frequency representations and pseudo-differential operators, in Pseudo-Differential Operators: Partial Differential Equations and Time-Frequency Analysis, Editors: L. Rodino, B.-W. Schulze and M.W. Wong, Fields Institute Communications 52, American Mathematical Society, 2007, 383–399.
L. Cohen, Time-Frequency Analysis, Prentice-Hall, 1995.
E. Cordero and K.A. Okundjou, Multilinear localization operators, J. Math. Anal. Appl. 325 (2007), 1103–1116.
L. Hörmander, The Analysis of Linear Partial Differential Operators III, Springer-Verlag, 1985.
J.E. Moyal, Quantum mechanics as a statistical theory, Proc. Cambridge Philos. Soc. 45 1949, 99–124.
J.J. Kohn and L. Nirenberg, An algebra of pseudo-differential operators, Comm. Pure Appl. Math. 18 1965, 269–305.
A. Mohammed and M.W. Wong, Rihaczek transforms and pseudo-differential operators, in Pseudo-Differential Operators: Partial Differential Equations and Time-Frequency Analysis, Editors: L. Rodino, B.-W. Schulze and M. W. Wong, Fields Institute Communications 52, American Mathematical Society, 2007, 375–382.
A. W. Rihaczek, Signal energy distribution in time and frequency, IEEE Trans. Inform. Theory 14 (1968), 369–374.
E.M. Stein, Introduction to the Theory of Fourier Integrals, Third Edition, Chelsea, 1986.
M.W. Wong, Weyl Transforms, Springer-Verlag, 1998.
M.W. Wong, An Introduction to Pseudo-Differential Operators, Second Edition, World Scientific, 1999.
M.W. Wong, Symmetry-breaking for Wigner transforms and L p-boundedness of Weyl transforms, in Advances in Pseudo-Differential Operators, Editors: R. Ashino, P. Boggiatto and M.W. Wong, Operator Theory: Advances and Applications 155, Birkhäuser, 2004, 107–116.
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2009 Birkhäuser Verlag Basel/Switzerland
About this paper
Cite this paper
Catană, V., Molahajloo, S., Wong, M.W. (2009). L p-Boundedness of Multilinear Pseudo-Differential Operators. In: Schulze, BW., Wong, M.W. (eds) Pseudo-Differential Operators: Complex Analysis and Partial Differential Equations. Operator Theory: Advances and Applications, vol 205. Birkhäuser Basel. https://doi.org/10.1007/978-3-0346-0198-6_9
Download citation
DOI: https://doi.org/10.1007/978-3-0346-0198-6_9
Publisher Name: Birkhäuser Basel
Print ISBN: 978-3-0346-0197-9
Online ISBN: 978-3-0346-0198-6
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)