Maximal operators on variable Lebesgue and Hardy spaces and applications in Fourier analysis

We summarize some results about the variable Lebesgue, Hardy and Hardy-Lorentz spaces Lp(·)(􏱄), Hp(·)(􏱄) and Hp(·),q(􏱄), where 􏱄 = 􏰹 or 􏱄 = 􏰓 or 􏱄 = [0,1). Besides the usual Hardy-Littlewood and Doob maximal functions, we introduce new dyadic maximal functions and show that they are bounded on Lp(·)(􏱄) provided that 1 < p− ≤ ∞. We present the atomic decompositions of the spaces Hp(·)(􏱄) and Hp(·),q(􏱄). As application in Fourier analysis, we consider the Fejér, Cesàro and Riesz summability of trigonometric and Walsh-Fourier series and Fourier transforms. Under some conditions, we prove that the maximal operators of the corresponding means are bounded from Hp(·)(􏱄) to Lp(·)(􏱄) and from Hp(·),q(􏱄) to Lp(·),q(􏱄). This implies some norm and almost everywhere convergence results of the Fejér, Cesàro and Riesz means.