Hostname: page-component-76fb5796d-45l2p Total loading time: 0 Render date: 2024-04-25T07:28:57.768Z Has data issue: false hasContentIssue false
  • English
  • Français

Boundedness of Pseudo-Differential Operators on Lp, Sobolev and Modulation Spaces

Published online by Cambridge University Press:  28 January 2013

S. Molahajloo  and
G.E. Pfander
S. Molahajloo*
Affiliation:
Department of Mathematics and Statistics, Queen’s University, Kingston, Ontario K7L3N6, Canada
G.E. Pfander
Affiliation:
School of Engineering and Science, Jacobs University, 28759 Bremen, Germany
*
∗Corresponding author. E-mail: molahajloo@mast.queensu.ca
Get access

Abstract

We introduce new classes of modulation spaces over phase space. By means of the Kohn-Nirenberg correspondence, these spaces induce norms on pseudo-differential operators that bound their operator norms on Lp–spaces, Sobolev spaces, and modulation spaces.

Keywords

pseudo-differential operatorsmodulation spacesSobolev spacesshort–time Fourier transforms
Type
Research Article
Information
Mathematical Modelling of Natural Phenomena , Volume 8 , Issue 1: Harmonic analysis , 2013 , pp. 175 - 192
Copyright
© EDP Sciences, 2013

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Benedek, A., Panzone, R.. The Space L p, with Mixed Norm. Duke Math. J. 28 (1961), 301-324. CrossRefGoogle Scholar
Bishop, S.. Mixed modulation spaces and their application to pseudodifferential operators. J. Math. Anal. Appl. 363 (2010) 1, 255–264. CrossRefGoogle Scholar
V. Catană, S. Molahajloo, M. W. Wong. L p-Boundedness of Multilinear Pseudo-Differential Operators, in Pseudo-Differential Operators : Complex Analysis and Partial Differential Equations . Operator Theory : Advances and Applications. 205, Birkhäuser, 2010, 167–180.
Cordero, E., Nicola, F.. Metaplectic Representation on Wiener Amalgam Spaces and Applications to the Schrödinger Equation. J. Funct. Anal. 254 (2008), 506-534. CrossRefGoogle Scholar
Cordero, E., Nicola, F.. Pseudodifferential Operators on L p, Wiener Amalgam and Modulation Spaces. Int. Math. Res. Notices 10 (2010), 1860-1893. Google Scholar
Concetti, F., Toft, J.. Trace Ideals for Fourier Integral Operators with Non-Smooth Symbols, in Pseudo-Differential Operators : Partial Differential Equations and Time Frequency Analysis. Fields Institute Communications, 52 (2007), 255–264. Google Scholar
Czaja, W.. Boundedness of Pseudodifferential Operators on Modulation Spaces. J. Math. Anal. Appl. 284(1) (2003), 389-396. CrossRefGoogle Scholar
Feichtinger, H. G.. Atomic Characterization of Modulation Spaces through the Gabor-Type Representations. Rocky Mountain J. Math. 19 (1989), 113-126. CrossRefGoogle Scholar
Feichtinger, H. G.. On a New Segal Algebra. Monatsh. Math. 92 (1981), 269-289. CrossRefGoogle Scholar
Feichtinger, H. G., Gröchenig, K.. Banach Spaces Related to Integrable Group Representations and Their Atomic Decompositions I. J. Funct. Anal. 86 (1989), 307-340. CrossRefGoogle Scholar
Feichtinger, H. G., Gröchenig, K.. Banach Spaces Related to Integrable Group Representations and Their Atomic Decompositions II. Monatsh. Math. 108 (1989), 129-148. CrossRefGoogle Scholar
H. G. Feichtinger, K. Gröchenig. Gabor Wavelets and the Heisenberg Group : Gabor Expansions and Short Time Fourier Transform from the Group Theoretical Point of View, in Wavelets : a tutorial in theory and applications. Academic Press, Boston, 1992.
Feichtinger, H. G., Gröchenig, K.. Gabor Frames and Time-Frequency Analysis of Distributions. J. Funct. Anal. 146 (1997), 464-495. CrossRefGoogle Scholar
K. Gröchenig. Foundation of Time-Frequency Analysis. Brikhäuser, Boston, 2001.
Gröchenig, K., Heil, C.. Counterexamples for Boundedness of Pseudodifferential Operators. Osaka J. Math. 41 (3) (2004), 681-691. Google Scholar
Gröchenig, K., Heil, C.. Modulation Spaces and Pseudodifferential Operators. Integr. Equat. Oper. th. 34 (4) (1999), 439-457. CrossRefGoogle Scholar
Hong, Y. M., Pfander, G. E.. Irregular and multi-channel sampling of operators. Appl. Comput. Harmon. Anal. 29 (2) (2010), 214-231. CrossRefGoogle Scholar
L. Hörmander. The Analysis of Linear Partial Differential Operators I. Second Edition, Springer-Verlag, Berlin, 1990.
Hörmander, L.. The Weyl Calculus of Pseudodifferential Operators. Comm. Pure Appl. Math. 32 (1979), 360-444. CrossRefGoogle Scholar
Hwang, I. L., Lee, R. B.. L p-Boundedness of Pseudo-Differential Operators of Class S0,0. Trans. Amer. Math. Soc. 346 (2) (1994), 489-510. Google Scholar
H. Kumano-Go. Pseudo-Differential Operators. Translated by Hitoshi Kumano-Go, Rémi Vaillancourt and Michihiro Nagase, MIT Press, 1982.
Okoudjou, K. A.. A Beurling-Helson Type Theorem for Modulation Spaces. J. Func. Spaces Appl. 7 (1) (2009), 33-41. CrossRefGoogle Scholar
Pfander, G. E., Walnut, D.. Operator Identification and Feichtinger’s Algebra. Sampl. Theory Signal Image Process. 5 (2) (2006), 151-168. Google Scholar
G. E. Pfander. Sampling of Operators. arxiv : 1010.6165.
Sjöstrand, J.. An Algebra of Pseudodifferential Operators. Math. Res. Lett. 1 (2) (1994), 185-192. CrossRefGoogle Scholar
J. Sjöstrand. Wiender Type Algebras of Pseudodifferential Operators, in Séminaire Équations aux dérivées Partielles. 1994-1995, exp. 4, 1–19.
Toft, J.. Continuity Properties for Modulation Spaces, with Applications to Pseudo-Differential Calculus I. J. Funct. Anal. 207 (2004), 399–429 CrossRefGoogle Scholar
Toft, J.. Continuity Properties for Modulation Spaces, with Applications to Pseudo-Differential Calculus II. Ann. Glob. Anal. Geom. 26 (2004), 73–106. CrossRefGoogle Scholar
Toft, J.. Fourier Modulation Spaces and Positivity in Twisted Convolution Algebra. Integral Transforms and Special Functions, 17 nos. 2-3 (2006), 193–198. CrossRefGoogle Scholar
Toft, J.. Pseudo-Differential Operators with Smooth Symbols on Modulation Spaces. CUBO. 11 (2009), 87-107. Google Scholar
Toft, J., Pilipovic, S., Teofanov, N.. Micro-Local Analysis in Fourier Lebesgue and Modulation Spaces. Part II, J. Pseudo-Differ. Oper. Appl. 1 (2010), 341-376. Google Scholar
M. W. Wong. An Introduction to Pseudo-Differential Operators. Second Edition, World Scientific, 1999.
Wong, M. W.. Fredholm Pseudo-Differential Operators on Weighted Sobolev Spaces. Ark. Mat. 21 (2) (1983), 271–282. CrossRefGoogle Scholar
M. W. Wong. Weyl Transforms. Springer-Verlag, 1998.