Rearrangement estimates for fourier transforms in Lp and Hp in terms of moduli of continuity

One of the main purposes of this paper is to obtain estimates for Fourier transforms of functions in Lp(n) (1 p 2) in terms of their moduli of continuity. More precisely, we study the following problem: find sharp conditions on the modulus of continuity of a function f Lp(n), under which the non-increasing rearragement of f, the Fourier transform of f, is integrable against a given non-negative weight function . We shall also study similar problems for the Fourier transforms of functions or distributions in the Hardy spaces Hp(n) (0 < p 1, n )