Abstract
We use sharp convolution estimates for weighted Lebesgue and modulation spaces to obtain an extension of the celebrated Cordero-Gröchenig theorems on boundedness and Schatten–von Neumann properties of localization operators on modulation spaces. We also give a new proof of the Weyl connection based on the kernel theorem for Gelfand–Shilov spaces.
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Abreu, L.D., Dörfler, M.: An inverse problem for localization operators, Inverse Problems 28(11), pp. 115001, 16 (2012)
Berezin F.A.: Wick and anti-wick symbols of operators. Mat. Sb. N. S. 86(128), 578–610 (1971)
Boggiatto P., Cordero E., Gröchenig K.: Generalized anti-wick operators with symbols in distributional Sobolev spaces. Integral Equ. Oper. Theory 48, 427–442 (2004)
Boggiatto P., Oliaro A., Wong M.W.: L p boundedness and compactness of localization operators. J. Math. Anal. Appl. 322(1), 193–206 (2006)
Calderón A.-P., Vaillancourt R.: On the boundedness of pseudo-differential operators. J. Math. Soc. Japan 23, 374–378 (1971)
Cordero E., Gröchenig K.: Time-frequency analysis of localization operators. J. Funct. Anal 205(1), 107–131 (2003)
Cordero E., Gröchenig K.: Symbolic calculus and Fredholm property for localization operators. J. Fourier Anal. Appl 12(4), 371–392 (2006)
Cordero E., Okoudjou K.A.: Multilinear localization operators J. Math. Anal. Appl. 325(2), 1103–1116 (2007)
Cordero E., Pilipović S., Rodino L., Teofanov N.: Localization operators and exponential weights for modulation spaces. Mediterr. J. Math 2(4), 381–394 (2005)
Cordero E., Pilipović S., Rodino L., Teofanov N.: Quasianalytic Gelfand-Shilov spaces with application to localization operators. Rocky Mt. J. Math 40(4), 1123–1147 (2010)
Cordero E., Rodino L.: Wick calculus: a time-frequency approach. Osaka J. Math 42(1), 43–63 (2005)
Cordero, E., Rodino, L., Gröchenig, K.: Localization operators and time-frequency analysis, in Harmonic, wavelet and p-adic analysis, World Sci. Publ., pp. 83–110 (2007)
Daubechies I.: Time-frequency localization operators: a geometric phase space approach. IEEE Trans. Inform. Theory 34(4), 605–612 (1988)
Feichtinger, H.G.: Modulation spaces on locally compact abelian groups, Technical Report, University Vienna, 1983. and also in Wavelets and Their Applications, Krishna, M., Radha, R., Thangavelu, S. (eds.) Allied Publishers, pp. 99–140 (2003)
Feichtinger, H.G., Nowak, K.: A first survey of Gabor multipliers. In: Advances in Gabor Analysis, Birkhäuser, 99–128 (2003)
Folland, G.B.: Harmonic analysis in phase space. Princeton Univ. Press, Princeton (1989)
Gelfand, I.M., Shilov, G.E.: Generalized functions II. Academic Press, New York (1968)
Gröchenig, K.: Foundations of time-frequency analysis. Birkhäuser, Boston (2001)
Gröchenig, K.H.: Weight functions in time-frequency analysis. In: Rodino, L., Wong M.W. (eds.) Pseudodifferential operators: partial differential equations and time-frequency analysis. Fields Inst. Comm. 52, 343–366 (2007)
Gröchenig K.H., Heil C.: Modulation spaces and pseudo-differential operators. Integr. Equ. Oper. Th. 34, 439–457 (1999)
Gröchenig, K.H., Heil, C.: Modulation spaces as symbol classes for pseudodifferential operators, In: Krishna, M., Radha, R., Thangavelu, S. (eds.) Wavelets and their applications, Allied Publishers Private Limited, New Dehli, pp. 151–170 (2003)
Gröchenig K., Zimmermann G.: Hardy’s theorem and the short-time Fourier transform of Schwartz functions. J. Lond. Math. Soc. 63, 205–214 (2001)
Hörmander, L.: The analysis of linear partial differential Operators, vol. I, Springer, Berlin (1983)
Langenbruch M.: Hermite functions and weighted spaces of generalized functions. Manuscr. Math 119, 269–285 (2006)
Lozanov–Crvenkovic, Z., Perisic, D., Taskovic, M.: Gelfand-Shilov spaces, Structural and Kernel theorems, Preprint. arXiv:0706.2268v2
Pilipović S., Teofanov N., Toft J.: Micro-local analysis in Fourier Lebesgue and modulation spaces, II. J. Pseudo Differ. Oper. Appl. 1, 341–376 (2010)
Ramanathan J., Topiwala P.: Time-frequency localization via the Weyl correspondence. SIAM J. Math. Anal. 24(5), 1378–1393 (1993)
Shubin, M.A.: Pseudodifferential Operators and Spectral Theory. Springer, Berlin, second edition (2001)
Sjöstrand J.: An algebra of pseudodifferential operators. Math. Res. Lett. 1, 185–192 (1994)
Slepian D.: Some comments on Fourier analysis, uncertainty and modeling. SIAM Rev. 25(3), 379–393 (1983)
Teofanov, N.: Ultradistributions and time-frequency analysis. In: Pseudo-differential operators and related topics, operator theory: advances and applications, Boggiatto, P., Rodino, L., Toft, J., Wong, M.W. (eds) Birkhäuser, 164,173–191 (2006)
Toft J.: Continuity properties for modulation spaces with applications to pseudo-differential calculus, I. J. Funct. Anal 207, 399–429 (2004)
Toft J.: Continuity properties for modulation spaces with applications to pseudo-differential calculus, II. Ann. Global Anal. Geom. 26, 73–106 (2004)
Toft, J.: Continuity and Schatten properties for pseudo-differential operators on modulation spaces. In: Oper. Theory Adv. Appl., 172, 173–206, Birkhäuser (2007)
Toft, J.: Continuity and Schatten properties for Toeplitz operators on modulation spaces. In: Oper. Theory Adv. Appl., 172, 313–328, Birkhäuser (2007)
Toft J.: The Bargmann transform on modulation and Gelfand-Shilov spaces, with applications to Toeplitz and pseudo-differential operators. J. Pseudo Differ. Oper. Appl. 3(2), 145–227 (2012)
Toft, J., Johansson, K., Pilipović, S., Teofanov, N.: Sharp convolution and multiplication estimates in weighted spaces, analysis and applications, accepted for publication (2014)
Toft J., Khrennikov A., Nilsson B., Nordebo S.: Decompositions of Gelfand-Shilov kernels into kernels of similar class. J. Math. Anal. Appl. 396(1), 315–322 (2012)
Wong, M.W.: Weyl transforms, Springer (1998)
Wong, M.W.: Wavelet transforms and localization operators, Birkhäuser Basel (2002)
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This research was supported by MPNTR of Serbia, project No. 174024.
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Teofanov, N. Continuity and Schatten–von Neumann Properties for Localization Operators on Modulation Spaces. Mediterr. J. Math. 13, 745–758 (2016). https://doi.org/10.1007/s00009-014-0509-8
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DOI: https://doi.org/10.1007/s00009-014-0509-8