ACTA issues

Sufficient conditions for the Lebesgue integrability of double Fourier transforms

Ferenc Móricz, Antal Veres

Acta Sci. Math. (Szeged) 79:1-2(2013), 175-190

Abstract. We consider complex-valued functions $f\in L^p({\msbm R}^2)$ for some $1< p\le2$ and give sufficient conditions for its Fourier transform $\hat f$ to belong to $L^r (\{(u,v) \in{\msbm R}^2: |u|\ge1$ and $|v|\ge1\} )$, where $0< r< q$ and $1/p+1/q=1$. Under additional conditions, we also give sufficient conditions, under which we have $\hat f\in L^r({\msbm R}^2)$. These sufficient conditions are in terms of the $L^p$-integral modulus or the ordinary modulus of continuity of $f$. Our theorems apply for functions in the Lipschitz classes $\mathop{\rm Lip}(\alpha_1, \alpha_2)$, where $0< \alpha_1, \alpha_2 \le1$ as well as for functions of bounded $s$-variation on ${\msbm R}^2$, where $0< s< p$. 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.

AMS Subject Classification (1991): 42A38, 42B10; 28A35

Keyword(s): double Fourier transform of functions $f\in L^p ({\msbm R}^2)$, $1\le p\le2$, inversion formula, Hausdorff--Young inequality, Lebesgue integrability of $\hat f$, $L^p$-integral modulus of continuity, integral Lipschitz classes $\mathop{\rm Lip}(\alpha_1, \alpha_2)_p$ ($0< \alpha_1$, $\alpha_2\le1$), functions of bounded $s$-variation, $s>0$

Received August 1, 2012, and in revised form November 23, 2012. (Registered under 85/2012.)