On Lebesgue Integrability of Fourier Transforms in Amalgam Spaces

Abstract

Let f be an element of the subspace \((L^{q},l^{p})^{\alpha }({\mathbb {R}}^d)\) (\(1\le q \le \alpha \le p \le 2 \)) of the Wiener amalgam space \((L^{q},l^{p})({\mathbb {R}}^{d})\). It is well-known that the Fourier transform \(\widehat{f}\) of f belongs to \((L^{p'},l^{q'})^{\alpha '}({\mathbb {R}}^d)\) where \(p'\), \(q'\) and \(\alpha '\) are the conjugate exponents of p, q and \(\alpha \) respectively. We give sufficient conditions, in terms of a modulus of continuity of f, for \(\widehat{f}\) to be in a Lebesgue space \(L^{\beta }({\mathbb {R}}^{d})\) or in an amalgam space \((L^{p'},l^{s})^{\beta }({\mathbb {R}}^d)\) other than \((L^{p'},l^{q'})^{\alpha '}({\mathbb {R}}^d)\). As an application, we obtain a sufficient condition for the solvability in \([L^{2}({\mathbb {R}}^d)]^d\) of the equation \(\hbox {div}{F}=f\) .