TY - JOUR

T1 - Pitt's inequalities and uncertainty principle for generalized fourier transform

AU - Gorbachev, Dmitry V.

AU - Ivanov, Valerii I.

AU - Tikhonov, Sergey Y.

PY - 2016/1/1

Y1 - 2016/1/1

N2 - © 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.

AB - © 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.

U2 - 10.1093/imrn/rnv398

DO - 10.1093/imrn/rnv398

M3 - Article

VL - 2016

SP - 7179

EP - 7200

JO - International Mathematics Research Notices

JF - International Mathematics Research Notices

SN - 1073-7928

IS - 23

ER -