The class Bp for weighted generalized Fourier transform inequalities
Abstract
In the present paper, we prove weighted inequalities for the Dunkl transform (which generalizes the Fourier transform) when the weights belong to the well-known class Bp. As application, we obtain the Pitt’s inequality for power weights.
References
C. Abdelkefi, J.Ph. Anker, F. Sassi, M. Sifi, Besov-type spaces on Rd and integrability for the Dunkl transform, SIGMA Symmetry Integrability Geom. Methods Appl. 5 (2009), Paper 019, 15pp.
##plugins.generic.googleScholarLinks.settings.viewInGS##
C. Abdelkefi, Weighted function spaces and Dunkl transform, Mediterr. J. Math. 9 (2012), no. 3, 499–513.
##plugins.generic.googleScholarLinks.settings.viewInGS##
W. Beckner, Pitt’s inequality and the uncertainty principle, Proc. Amer. Math. Soc. 123 (1995), no. 6, 1897–1905.
##plugins.generic.googleScholarLinks.settings.viewInGS##
J.J. Benedetto, H.P. Heinig, Weighted Fourier inequalities: new proofs and generalizations, J. Fourier Anal. Appl. 9 (2003), no. 1, 1–37.
##plugins.generic.googleScholarLinks.settings.viewInGS##
C. Bennett, R. Sharpley, Interpolation of operators, Pure and Applied Mathematics 129, Academic Press, Inc., Boston, MA, 1988.
##plugins.generic.googleScholarLinks.settings.viewInGS##
M.F.E. de Jeu, The Dunkl transform, Invent. Math. 113 (1993), no. 1, 147–162.
##plugins.generic.googleScholarLinks.settings.viewInGS##
C.F. Dunkl, Differential-difference operators associated to reflection groups, Trans. Amer. Math. Soc. 311 (1989), no. 1, 167–183.
##plugins.generic.googleScholarLinks.settings.viewInGS##
C.F. Dunkl, Integral kernels with reflection group invariance, Canad. J. Math. 43 (1991), no. 6, 1213–1227.
##plugins.generic.googleScholarLinks.settings.viewInGS##
G.H. Hardy, Note on a theorem of Hilbert, Math. Z. 6 (1920), no. 3-4, 314–317.
##plugins.generic.googleScholarLinks.settings.viewInGS##
G.H. Hardy, J.E. Littlewood, G. Pólya, Inequalities, 2d ed., Cambridge University Press, 1952.
##plugins.generic.googleScholarLinks.settings.viewInGS##
H.P. Heinig, Weighted norm inequalities for classes of operators, Indiana Univ. Math. J. 33 (1984), no. 4, 573–582. Cited on 127. The class Bp for weighted generalized Fourier transform inequalities [133]
##plugins.generic.googleScholarLinks.settings.viewInGS##
M. Jr. Jodeit, A. Torchinsky, Inequalities for Fourier transforms, Studia Math. 37 (1970/71), 245–276.
##plugins.generic.googleScholarLinks.settings.viewInGS##
V.G. Maz’ja, Sobolev spaces, Springer Series in Soviet Mathematics, SpringerVerlag, Berlin, 1985.
##plugins.generic.googleScholarLinks.settings.viewInGS##
M. Rösler, M. Voit, Markov processes related with Dunkl operators, Adv. in Appl. Math. 21 (1998), no. 4, 575–643.
##plugins.generic.googleScholarLinks.settings.viewInGS##
M. Rösler, Dunkl operators: theory and applications, Orthogonal polynomials and special functions (Leuven, 2002), 93–135, Lecture Notes in Math., 1817, Springer, Berlin, 2003.
##plugins.generic.googleScholarLinks.settings.viewInGS##
E. Sawyer, Boundedness of classical operators on classical Lorentz spaces, Studia Math. 96 (1990), no. 2, 145–158.
##plugins.generic.googleScholarLinks.settings.viewInGS##
E.M. Stein, Harmonic analysis: real-variable methods, orthogonality, and oscillatory integrals, Princeton Mathematical Series, 43, Monographs in Harmonic Analysis III, Princeton University Press, Princeton, NJ, 1993. Cited on 122. Institute of Engineer Studies of Tunis 1089 Monfleury Tunis University of Tunis Tunisia E-mail: chokri.abdelkefi@ipeit.rnu.tn E-mail: rachdi.mongi@ymail.com Received: January 22, 2015; final version: September 8, 2015; available online: October 6, 2015.
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