Cesàro summability of one- and two-dimensional trigonometric-Fourier series

Ferenc Weisz¹

¹

Colloquium Mathematicum, Tome 74 (1997) no. 1, pp. 123-133.

Résumé

We introduce p-quasilocal operators and prove that if a sublinear operator T is p-quasilocal and bounded from $L_∞$ to $L_∞$ then it is also bounded from the classical Hardy space $H_p(T)$ to $L_p$ (0 p ≤ 1). As an application it is shown that the maximal operator of the one-parameter Cesàro means of a distribution is bounded from $H_p(T)$ to $L_p$ (3/4 p ≤ ∞) and is of weak type $(L_1,L_1)$. We define the two-dimensional dyadic hybrid Hardy space $H_{1}^{♯}(T^2)$ and verify that the maximal operator of the Cesàro means of a two-dimensional function is of weak type $(H_{1}^{♯}(T^2),L_1)$. So we deduce that the two-parameter Cesàro means of a function $f ∈ H_1^{♯}(T^2) ⊃ Llog L$ converge a.e. to the function in question.

Détail

DOI : 10.4064/cm-74-1-123-133

Keywords: p-atom, Hardy spaces, Cesàro summability, atomic decomposition, p-quasilocal operator, interpolation