© The Author(s) 2016. We study the two-parameter family of unitary operators (equation presented) which are called (κ, a)-generalized Fourier transforms and defined by the a-deformed Dunkl harmonic oscillator δk,a = |x|2-aδκ -|x|a, a > 0, where δk is the Dunkl Laplacian. Particular cases of such operators are the Fourier and Dunkl transforms. The restriction of Fk,a to radial functions is given by an a-deformed Hankel transform Hλ,a. We obtain necessary and sufficient conditions for the weighted (Lp, Lq) Pitt inequalities to hold for the a-deformed Hankel transform. Moreover, we prove two-sided Boas-Sagher type estimates for the general monotone functions. We also prove sharp Pitt's inequality for Fk,a transform in L2(ℝd) with the corresponding weights. Finally, we establish the logarithmic uncertainty principle for Fk,a.
Gorbachev, D. V., Ivanov, V. I., & Tikhonov, S. Y. (2016). Pitt's inequalities and uncertainty principle for generalized fourier transform. International Mathematics Research Notices, 2016(23), 7179-7200. https://doi.org/10.1093/imrn/rnv398