Abstract
Let M p,q ω be the modulation space with parameters p,q ∈ [1,∞] and weight function ω. Also let M p,q = M p,q ω0 , ω0=1. We prove that for certain w, there is a canonical homeomorphism M p,q ω → M p,q → M p,q. and use this result to extend well-known embeddings for Mm-spaces to embeddings between certain M p,q ω -spaces and Sobolev-Besov spaces. We also give a convenient definition for convolutions between elements in M p,q ω -spaces, and prove certain Hölder-Young properties.
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
J. Bergh and J. LöfströmInterpolation Spaces: An IntroductionSpringer-Verlag, Berlin Heidelberg NewYork, 1976.
P. Boggiatto, Localization operators withLPsymbols on modulation spaces, preprint.
A. Boulkhemair, L2estimates for Weyl quantizationJ. Funct. Anal.165 (1999), 173–204.
E. Cordero and K. Gröchenig, Time-frequency analysis of localization operatorsJ. Funct. Anal.205 (2003), 107–131.
H.G. Feichtinger, Banach spaces of distributions of Wiener’s type and interpolation, in Proc. Conf. OberwolfachFunctional Analysis and ApproximationAugust 1980, Vol 69, Int. Ser. Num. Math., Editors: P. Butzer, B. Sz. Nagy and E. Görlich, Birkhäuser-Verlag, Basel, Boston, Stuttgart, 1981, pp. 153–165.
E. Cordero and K. Gröchenig, Banach convolution algebras of Wiener’s type, inProc. Functions Series Operatorsin Budapest, Colloquia Math. Soc. J. Bolyai, North Holland Publ. Co., Amsterdam Oxford NewYork, 1980.
E. Cordero and K. GröchenigModulation spaces on locally compact abelian groupsTechnical report, University of Vienna, Vienna, 1983.
H.G. Feichtinger and P. Gröbner, Banach spaces of distributions defined by decomposition methods, IMath. Nachr.123 (1985), 97–120.
P. GröbnerBanachräume Glatter Funktionen und ZerlegungsmethodenThesis, University of Vienna, 1992.
K. GröchenigFoundations of Time-Frequency AnalysisBirkhäuser, Boston, 2001.
L. HörmanderThe Analysis of Linear Partial Differential Operators I IIISpringer-Verlag, Berlin Heidelberg New York Tokyo, 1983, 1985.
K.A. Okoudjou, Embeddings of some classical Banach spaces into modulation spaces, preprint.
J. Sjöstrand, An algebra of pseudo-differential operatorsMath. Res. Lett. 1(1994), 185–192.
J. ToftContinuity and Positivity Problems in Pseudo-Differential CalculusThesis, University of Lund, 1996.
J. Toft, Subalgebras to a Wiener type algebra of pseudo-differential operatorsAnn. Inst. Fourier (5) 51(2001), 1347–1383.
J. ToftModulation spaces and pseudo-differentianl operatorsResearch Report 2002:05 (2002), Blekinge Institute of Technology, Karlskrona.
J. ToftWeighted modulation spaces and pseudo-differential operatorsResearch Report 2003:05 (2003), Blekinge Institute of Technology, Karlskrona.
J. Toft, Continuity properties for modulation spaces with applications in pseudo-differential calculus, IJ. Funct. Anal.207 (2004), 399–429.
M.W. WongWavelet Transforms and Localization OperatorsBirkhäuser, Basel, 2002.
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2004 Springer Basel AG
About this paper
Cite this paper
Toft, J. (2004). Convolutions and Embeddings for Weighted Modulation Spaces. In: Ashino, R., Boggiatto, P., Wong, M.W. (eds) Advances in Pseudo-Differential Operators. Operator Theory: Advances and Applications, vol 155. Birkhäuser, Basel. https://doi.org/10.1007/978-3-0348-7840-1_10
Download citation
DOI: https://doi.org/10.1007/978-3-0348-7840-1_10
Publisher Name: Birkhäuser, Basel
Print ISBN: 978-3-0348-9590-3
Online ISBN: 978-3-0348-7840-1
eBook Packages: Springer Book Archive