Rearrangement estimates for Fourier transforms in L-p and H-p in terms of moduli of continuity

One of the main purposes of this paper is to obtain estimates for Fourier t ransforms of functions in L-p(R-n) (1 less than or equal to p less than or equal to 2) in terms of their moduli of continuity. More precisely, we stud y the following problem: find sharp conditions on the modulus of continuity of a function f is an element of L-p(R-n), under which the non-increasing rearragement of (f) over cap, the Fourier transform of f, is integrable aga inst a given non-negative weight function rho. We shall also study similar problems for the Fourier transforms of functions or distributions in the Ha rdy spaces H-p(R-n) (0 < p <less than or equal to> 1, n is an element of N) .