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

Let (p) over bar := (p1,..., p(n)), (r) over bar := (r1,..., r(n)). [1,8)(n), L-(r) over bar(R-n) be the mixed-norm Lebesgue space, and theta an integrable function. In this paper, via establishing the boundedness of the mixed centered Hardy-Littlewood maximal operator M-(p) over bar from L-(r) over bar(R-n) to itself or to the weak mixed-norm Lebesgue space WL(r) over bar(R-n) under some sharp assumptions on (p) over bar and (r) over bar, the authors show that the theta-mean of f is an element of L-(r) over bar(R-n) converges to f almost everywhere over the diagonal if the Fourier transform (sic) of theta belongs to some mixed-norm homogeneous Herz space. E-(p) over bar(R-n) with (p) over bar being the conjugate index of (p) over bar. 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 theta-means to the unrestricted case. Finally, the authors show that the theta-mean of f is an element of L-(r) over bar(R-n) converges over the diagonal to f at all its (p) over bar -Lebesgue points if and only if (sic) belongs to E-(p) over bar(R-n), and a similar conclusion also holds true for the unrestricted convergence at strong (p) over bar -Lebesgue points. Observe that, in all these results, those Herz spaces to which (sic) belongs prove to be the best choice in some sense.