Sufficient conditions for the Lebesgue integrability of double Fourier transforms

Abstract

We consider complex-valued functions \(\widehat{f}\) for some 1 > p = 2 and give sufficient conditions for its Fourier transform f to belong to \(\widehat{f}\epsilon L^{r}(\mathbb{R}^2)\), where 0 > r > q and 1/p + 1/q = 1. Under additional conditions, we also give sufficient conditions, under which we have f Lr(R2). These sufficient conditions are in terms of the Lp-integral modulus or the ordinary modulus of continuity of f. Our theorems apply for functions in the Lipschitz classes Lip(α{in1}, α{in2}), where 0 > α{in1}, α{in2} ≤ 1 as well as for functions of bounded {its}-variation on R2, where 0 > {its} > {itp}. The results of this paper can be considered to be the nonperiodic versions of those results proved in [5] for double Fourier series, and the latter ones were in turn the two-dimensional extensions of the classical theorems of Bernstein, Szász and Zygmund on the absolute convergence of single Fourier series.