On almost everywhere convergence of Marcinkiewicz sums of double Fourier series

Abstract

Let f be a function summable on the two-dimensional torus with Fourier series\(\sum\limits_{k \in \mathbb{Z}^2 } {\hat f_K e^{2\pi i(k,x)} } \). The Marcinkiewicz means\(\delta _{\varphi ,n} (f,x) = \sum\limits_{K \in \mathbb{Z}^2 } {\varphi \left( {\frac{{\max \{ \left| {K_1 } \right|,\left| {K_2 } \right|\} }}{n}} \right)} \hat f_K ^{e^{2\pi i(K,X)} } \). where ϕ is a function defined on [0, 1], are considered. The following theorem is proved. Let α > 0 and assume that the function ϕ, concave on [0, 1], is such that ϕ(0)=1, ϕ(1)=0 and its modulus of continuity satisfies the relation ω(ϕ,h)=0 (log−2−α(1+1/k)). Then Open image in new window for almost all x.