Convergence of Vilenkin-Fourier series in variable Hardy spaces

摘要

Let p(.) : [0,1) -> (0, infinity) be a variable exponent function satisfying the log-Holder condition and 0 < q <= infinity. We introduce the variable Hardy and HardyLorentz spaces H-p(.) and H-p(.),H-q containing Vilenkin martingales. We prove that the partial sums of the Vilenkin-Fourier series converge to the original function in the L-p((.))- and L-p(.),L-q-norm if 1 < p_< infinity . We generalize this result for smaller p(.) as well. We show that the maximal operator of the Fejer means of the Vilenkin-Fourier series is bounded from H(p(.) )to L-p(.) and from H-p(.),H-q to L-p(i)q if 1/2 < p_ < infinity, 0 < q <= infinity and 1/p_ -1/p(+) < 1. 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 Fejer means of the Vilenkin-Fourier series.