Convergence of Vilenkin–Fourier series in variable Hardy spaces

Abstract

AbstractLet be a variable exponent function satisfying the log‐Hölder condition and . We introduce the variable Hardy and Hardy–Lorentz spaces and containing Vilenkin martingales. We prove that the partial sums of the Vilenkin–Fourier series converge to the original function in the ‐ and ‐norm if . We generalize this result for smaller as well. We show that the maximal operator of the Fejér means of the Vilenkin–Fourier series is bounded from to and from to if , and . This last condition is surprising because the corresponding results for Fourier series or Fourier transforms hold without this condition. This implies some norm and almost everywhere convergence results for the Fejér means of the Vilenkin–Fourier series.