Summability of Fourier transforms on mixed-norm Lebesgue spaces via associated Herz spaces

Abstract

Let [Formula: see text], [Formula: see text] be the mixed-norm Lebesgue space, and [Formula: see text] an integrable function. In this paper, via establishing the boundedness of the mixed centered Hardy–Littlewood maximal operator [Formula: see text] from [Formula: see text] to itself or to the weak mixed-norm Lebesgue space [Formula: see text] under some sharp assumptions on [Formula: see text] and [Formula: see text], the authors show that the [Formula: see text]-mean of [Formula: see text] converges to [Formula: see text] almost everywhere over the diagonal if the Fourier transform [Formula: see text] of [Formula: see text] belongs to some mixed-norm homogeneous Herz space [Formula: see text] with [Formula: see text] being the conjugate index of [Formula: see text]. Furthermore, by introducing another mixed-norm homogeneous Herz space and establishing a characterization of this Herz space, the authors then extend the above almost everywhere convergence of [Formula: see text]-means to the unrestricted case. Finally, the authors show that the [Formula: see text]-mean of [Formula: see text] converges over the diagonal to [Formula: see text] at all its [Formula: see text]-Lebesgue points if and only if [Formula: see text] belongs to [Formula: see text], and a similar conclusion also holds true for the unrestricted convergence at strong [Formula: see text]-Lebesgue points. Observe that, in all these results, those Herz spaces to which [Formula: see text] belongs prove to be the best choice in some sense.