Summability in mixed-norm Hardy spaces

Abstract. The mixed norm Hardy spaces \(H_{\vec{p}}(\mathbb{R}^d)\) is investigated, where \(\vec{p}=(p_1,\ldots,p_d) \in (0,\infty]^{d}\). A general summability method, the so called \(\theta\)-summability is considered for multi-dimensional Fourier transforms. Under some conditions on \(\theta\), it is proved that the maximal operator of the
\(\theta\)-means is bounded from \(H_{\vec{p}}(\mathbb{R}^d)\) to \(L_{\vec{p}}(\mathbb{R}^d)\). This implies some norm and almost everywhere convergence results for the \(\theta\)-means, amongst others the generalization of the well known Lebesgue's theorem.