A Hardy-Littlewood theorem for spherical Fourier transforms on symmetric spaces

AbstractLet q ⩾ 2. If f is a measurable function on Rn such that f(x) ¦x¦n(1 − 2q) ϵ Lq(Rn), then its Fourier transform fɞ can be defined and there exists a constant Aq such that the inequality ∥fɞ∥q ∥ f ¦ · ¦n(1 − 2q ∥q holds. This result is called the Hardy-Littlewood theorem. This paper studies what the corresponding function to ¦x¦n is for the spherical Fourier transform on Riemannian symmetric spaces and gives an analogue of this theorem. Also it is shown that the spherical Fourier transform of function f having the corresponding property on GK extends holomorphically to a tube domain and that an analogue of Riemann-Lebesgue lemma holds there. The following three cases are treated: the Riemannian symmetric spaces of the noncompact type, those of the compact type and the symmetric spaces on which the Cartan motion groups are acting