×

\(L^{p}\) boundedness and compactness of localization operators. (English) Zbl 1111.47041

The localization operator is a pseudodifferential operator of the form \[ D(f)(t) = \int_{{\mathbb R}^{2d}} F(x,\omega) \langle f, \phi_{x,\omega} \rangle \psi_{x, \omega}\; dx\, d\omega, \] where \(t \in {\mathbb R}^d\) and \(\phi_{x, \omega}(t) = e^{it\omega} \phi(t-x)\). Here, \(D = D_{F, \phi, \psi}\) depends on parameters which are functions, namely, \(F\), \(\phi\), and \(\psi\). The paper under review provides conditions on the functions \(F\), \(\phi\), and \(\psi\) which guarantee that the associated operator \(D\) is bounded (resp., compact) on \(L^q({\mathbb R}^d)\). The main result states that if \(\phi, \psi \in L^1({\mathbb R}^d) \cap L^\infty(\mathbb R)\) and \(F \in L^p({\mathbb R}^d)\), \(p \in [1,\infty]\), then \(D\) is bounded on \(L^q({\mathbb R}^d)\) for \(q \in [2p/(p+1), 2p/(p-1)]\). An analogous result for \(F\) in the Sobolev space \(H^{s, p}\) is also given.

MSC:

47G30 Pseudodifferential operators
42C40 Nontrigonometric harmonic analysis involving wavelets and other special systems
42B35 Function spaces arising in harmonic analysis
46E30 Spaces of measurable functions (\(L^p\)-spaces, Orlicz spaces, Köthe function spaces, Lorentz spaces, rearrangement invariant spaces, ideal spaces, etc.)
47B38 Linear operators on function spaces (general)
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Boggiatto, P., Localization operators with \(L^p\) symbols on modulation spaces, (Ashino, R.; Boggiatto, P.; Wong, M. W., Advances in Pseudo-Differential Operators (2004), Birkhäuser), 149-163 · Zbl 1078.46019
[2] Boggiatto, P.; Buzano, E.; Rodino, L., Global Hypoellipticity and Spectral Theory (1996), Akademie-Verlag · Zbl 0878.35001
[3] Boggiatto, P.; Rodino, L., Quantization and pseudo-differential operators, Cubo Mat. Educ., 5, 237-272 (2003) · Zbl 1369.47065
[4] Boggiatto, P.; Wong, M. W., Two-wavelet localization operators on \(L^p(R^n)\) for the Weyl-Heisenberg group, Integral Equations Operator Theory, 49, 1-10 (2004) · Zbl 1072.47046
[5] Cordero, E.; Gröchenig, K., Time-frequency analysis of localization operators, J. Funct. Anal., 205, 107-131 (2003) · Zbl 1047.47038
[6] Daubechies, I., Time-frequency localization operators: A geometric phase space approach, IEEE Trans. Inform. Theory, 34, 605-612 (1988) · Zbl 0672.42007
[7] Daubechies, I., Ten Lectures on Wavelets (1992), SIAM · Zbl 0776.42018
[8] Gröchenig, K., Foundations of Time-Frequency Analysis (2001), Birkhäuser · Zbl 0966.42020
[9] Kaiser, G., A Friendly Guide to Wavelets (1994), Birkhäuser · Zbl 0839.42011
[10] Lerner, N., The Wick calculus of pseudo-differential operators and some of its applications, Cubo Mat. Educ., 5, 213-236 (2003) · Zbl 1369.47066
[11] Lieb, E. H., Integral bounds for radar ambiguity functions and Wigner distributions, J. Math. Phys., 31, 594-599 (1990) · Zbl 0704.46050
[12] Shubin, M. A., Pseudodifferential Operators and Spectral Theory (2001), Springer-Verlag · Zbl 0980.35180
[13] Stein, E. M., Singular Integrals and Differentiability Properties of Functions (1970), Princeton Univ. Press · Zbl 0207.13501
[14] Wong, M. W., Weyl Transforms (1998), Springer-Verlag · Zbl 0908.44002
[15] Wong, M. W., An Introduction to Pseudo-Differential Operators (1999), World Scientific · Zbl 0940.35216
[16] Wong, M. W., Wavelet Transforms and Localization Operators (2002), Birkhäuser · Zbl 1016.42017
[17] Wong, M. W., Symmetry-breaking for Wigner transforms and \(L^p\)-boundedness of Weyl transforms, (Ashino, R.; Boggiatto, P.; Wong, M. W., Advances in Pseudo-Differential Operators (2004), Birkhäuser), 107-116 · Zbl 1081.47052
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.