Maximal estimates for the Riesz summability of several-dimensional Fourier transforms and Fourier series (Q2717141)

Convergence of Riesz means \(\sigma^{\alpha,\gamma}_Tf\) of the Fourier transform of an integrable function or a tempered distribution \(f\) on \(\mathbb{R}^d\) has been studied by many authors. In the case \(d= 1\) and \(\gamma=1\) or \(2\), \textit{E. M. Stein} and \textit{G. Weiss} [``Introduction to Fourier analysis on Euclidean spaces'' (1971; Zbl 0294.42003)] and \textit{P. L. Butzer} and \textit{N. R. Nessel} [``Fourier analysis and approximation'' (1971; Zbl 0217.42603)] proved that the Riesz means of an integrable \(f\) converge to \(f\) almost everywhere as \(T\to\infty\). In the case \(d=2\) and \(\gamma\geq 1\), Stein and Weiss (loc. cit.) proved that the Riesz means of an \(f\in L^p\), \(1\leq p<\infty\), converge to \(f\) in \(L^p\) norm, as \(T\to\infty\).NEWLINENEWLINENEWLINEThe author [to appear in Trans. Am. Math. Soc.] has had a recent breakthrough concerning boundedness of operators from Hardy spaces to \(L^p\) spaces on \(\mathbb{R}^d\) by extending the concept of quasi-local to operators which take tempered distributions to measurable functions which enables him to circumvent the impediment that Hardy spaces on \(\mathbb{R}^d\) cannot, in general, be decomposed into rectangular \(p\)-atoms. Using this breakthrough, he generalizes some of the classical results about Riesz means to the case \(d>2\). Namely, he shows under mild conditions that the maximal operator associated with the Riesz means is bounded from a Hardy-Lorentz space \(H^{p,q}\) (defined by a nontangential maximal function) to a mixed norm space \(L^{p,q}\). It follows that the Riesz means of an \(f\), which belongs to one of these Hardy-Lorentz spaces (in particular, any \(f\in L(\log L)^{d- 1}\)), converge to \(f\) almost everywhere. He also obtains an analogue of this result for conjugate functions and their transforms, and a compact version (i.e., one valid for Fourier series) as well.NEWLINENEWLINEFor the entire collection see [Zbl 0944.00034].