On Lebesgue Integrability of Fourier Transforms in Amalgam Spaces

Let f be an element of the subspace (Lq,lp)(Rd) (1qp2) of the Wiener amalgam space (Lq,lp)(Rd). It is well-known that the Fourier transform f<^> of f belongs to (Lp,lq)(Rd) where p, q and are the conjugate exponents of p, q and respectively. We give sufficient conditions, in terms of a modulus of continuity of f, for f<^> to be in a Lebesgue space L(Rd) or in an amalgam space (Lp,ls)(Rd) other than (Lp,lq)(Rd). As an application, we obtain a sufficient condition for the solvability in [L2(Rd)<mml:mo stretchy=]d of the equation F=f .