Cesàro-summability of higher-dimensional Fourier series

Abstract. The triangular and cubic Cesàro summability of higher dimensional Fourier series is investigated. It is proved that the maximal operator of the Cesàro means of a \(d\)-dimensional Fourier series is bounded from the Hardy space \(H_p({\mathbb T}^d)\) to \(L_p({\mathbb T}^d)\) for all \(d/(d+\alpha\wedge 1) < p \leq \infty\) and, consequently, is of weak type (1,1). As a consequence we obtain that the Cesàro means of a function \(f \in L_p({\mathbb T}^d)\) converge a.e. and in \(L_p\)-norm \((1\leq p < \infty)\) to \(f\). Moreover, we prove for the endpoint \(p=d/(d+\alpha\wedge 1)\) that the maximal operator is bounded from \(H_p({\mathbb T}^d)\) to the weak \(L_p({\mathbb T}^d)\) space.