Cesàro summability of two-parameter trigonometric-Fourier series
Denote by \(\sigma^*f\) the maximal operator of the unrestricted Cesàro means of the double Fourier series of a distribution \(f\) on \(\mathbb{T}^2\), where \(\mathbb{T}:=[-\pi,\pi)\). The author proves that \(\sigma^*\) is bounded from the Hardy-Lorentz space \(H_{p,q}(\mathbb{T}\times\mathbb{T})\) into the Lorentz space \(L_{p,q}(\mathbb{T}^2)\), provided \(3/4<p\leq\infty\) and \(0<q\leq \infty\). Reviewer's remark: The special case that \(\sigma^*\) is bounded from the hybrid Hardy space \(H^\#_1(\mathbb{T}\times\mathbb{T})\) into weak-\(L^1(\mathbb{T}^2)\) as well as its consequence that the Cesàro means of a function \(f\in H^\#_1(\mathbb{T}\times\mathbb{T})\supset L\log^+L(\mathbb{T}^2)\) converge unrestrictedly to \(f\) almost everywhere, was proved by the reviewer [Stud. Math. 116, No. 1, 89-100 (1995; Zbl 0842.47020)].
scientific article; zbMATH DE number 1042512
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