关键词: Hardy-Littlewood theorem, Fourier-Jacobi transform, Fourier-Bessel transform

Abstract: Let q > 2,f is measurable function on Rⁿ such that f(x)|x|^n(1-2/q) ∈L^q(Rⁿ), then its Fourier transform f can be defined and there exists a constant A_q such that the inequality||f||_q ≤ A_q||f|·|^n(1-2/q)||_q holds.This is the Hardy-Littlewood Theorem. This paper considers the corresponding result for the Fourier-Bessel transform and Fourier-Jacobi transform.It is interesting that we can deal with theses two cases in the same way,and the function corresponding to|x|ⁿ is tw(t), where w(t) is the weight,w(t)=t^2α+1 for Fourier-Bessel transform,and w(t)=(2 sinh t)^2α+1 (2 cosh t)^2β+1 for Fourier-Jacobi transform.

Key words: Hardy-Littlewood theorem, Fourier-Jacobi transform, Fourier-Bessel transform