A generalization for Fourier transforms of a theorem due to Marcinkiewicz
The author defines properly the Marcinkiewicz means of the Fourier transforms of tempered distributions and proves that its maximal operator is bounded from the Hardy-Lorentz space \(H_{p,q}({\mathbb{R}}^2)\) to the Lorentz space \(L_{p,q}({\mathbb{R}}^2)\) for all \(p_0<p\leq\infty\) and \(0<q\leq \infty,\) where \(p_0<1\) depends on a parameter in the definition of the means, and is also of weak type \((1,1).\) As a consequence, the author proves that the Marcinkiewicz means of a function \(f\in L_1({\mathbb{R}}^2)\) converge a.e. to the function in question, which is a generalization of a summability result due to Marcinkiewicz and to Zhizhiashvili. Then the author shows that the Marcinkiewicz means are uniformly bounded on the Hardy-Lorentz space \(H_{p,q}({\mathbb{R}}^2)\) for the same \(p\) and \(q,\) and so they converge in the \(H_{p,q}({\mathbb{R}}^2)\) norm. Similar results for the Riesz transforms are also given. The author uses the same techniques as in his previous papers.
scientific article; zbMATH DE number 1563086