Pitt's inequalities and uncertainty principle for generalized Fourier transform

2015/07/23 by Gorbachev, Dmitry, Ivanov, Valery, Tikhonov, Sergey · 1 citation
#33C45 #33C52 (Secondary) #42B10 (Primary) #Classical Analysis and ODEs (math.CA) #FOS: Mathematics

paper · doi:10.48550/arxiv.1507.06445

Abstract

We study the two-parameter family of unitary operators F_k,a=exp((iπ)/(2a) (2⟨ k⟩+d+a-2 )) exp((iπ)/(2a) Δ_k,a), which are called (k,a)-generalized Fourier transforms and defined by the a-deformed Dunkl harmonic oscillator Δ_k,a=|x|^2-aΔ_k-|x|^a, a>0, where Δ_k is the Dunkl Laplacian. Particular cases of such operators are the Fourier and Dunkl transforms. The restriction of F_k,a to radial functions is given by the a-deformed Hankel transform H_λ,a. We obtain necessary and sufficient conditions for the weighted (L^p,L^q) 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 F_k,a transform in L²(ℝ^d) with the corresponding weights. Finally, we establish the logarithmic uncertainty principle for F_k,a.

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