The class Bp for weighted generalized Fourier transform inequalities

Chokri Abdelkefi, Mongi Rachdi

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

Keywords


Dunkl operators; Dunkl transform; class Bp; Pitt's inequality

Mathematics Subject Classification (2010)


42B10; 46E30; 44A35

References


Abdelkefi, C. et al. "Besov-type spaces on Rd and integrability for the Dunkl transform." SIGMA Symmetry Integrability Geom. MethodsAppl. 5 (2009): Paper 019.

Abdelkefi, C. "Weighted function spaces and Dunkl transform." Mediterr. J. Math. 9.3 (2012): 499–513.

Beckner, W. "Pitt’s inequality and the uncertainty principle." Proc. Amer. Math. Soc. 123.6 (1995):1897–1905.

Benedetto, J.J. and H.P. Heinig. "Weighted Fourier inequalities: new proofs and generalizations." J. Fourier Anal. Appl. 9.1 (2003): 1–37.

Bennett, C. and R. Sharpley. "Interpolation of operators." Pure and Applied Mathematics 129. Boston, MA: Academic Press, Inc., 1988.

de Jeu, M.F.E. "The Dunkl transform." Invent. Math. 113.1 (1993): 147–162.

Dunkl, C.F. "Differential-difference operators associated to reflection groups." Trans. Amer. Math. Soc. 311.1 (1989): 167–183.

Dunkl, C.F. "Integral kernels with reflection group invariance." Canad. J. Math. 43.6 (1991): 1213–1227.

Hardy, G.H. "Note on a theorem of Hilbert." Math. Z. 6.3-4 (1920): 314–317.

Hardy, G.H., J.E. Littlewood and G. Pólya. "Inequalities." 2d ed., Cambridge: Cambridge University Press, 1952.

Heinig, H.P. "Weighted norm inequalities for classes of operators.", Indiana Univ. Math. J. 33.4 (1984): 573–582.

Jodeit, M. Jr. and A. Torchinsky. "Inequalities for Fourier transforms.", Studia Math. 37 (1970/71): 245–276.

Maz’ja, V.G. "Sobolev spaces." Springer Series in Soviet Mathematics. Berlin: Springer-Verlag, 1985.

Rösler, M. and M. Voit. "Markov processes related with Dunkl operators." Adv. in Appl. Math. 21.4 (1998): 575–643.

Rösler, M. "Dunkl operators: theory and applications." Orthogonal polynomials and special functions (Leuven, 2002). Lecture Notes in Math. 1817. Berlin: Springer, 2003: 93–135.

Sawyer, E. "Boundedness of classical operators on classical Lorentz spaces." Studia Math. 96.2 (1990): 145–158.

Stein, E.M. "Harmonic analysis: real-variable methods, orthogonality, and oscillatory integrals." Princeton Mathematical Series 43. Monographs in Harmonic Analysis III. Princeton, NJ: Princeton University Press, 1993.


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