Some new results on H summability of fourier series

In this paper we shall be concerned with Hα summability, for 0 < α ≤ 2 of the Fourier series of arbitrary L1([-π, π]) functions. The methods employed here are a modification of the real variable ones introduced by J. Marcinkiewicz. The needed modications give direct proofs of maximal theorems with respect to A1 weights. We also give a counter-example of a measure such that there is no convergence a.e. to the density of the measure. Finally, we present a Kakutani type of theorem, proving the w*- density, in the space of of probability measures defined on [-π, π] of Borel measures for which there is no H2 summability a.e.Mathematics Subject Classification (2010): Primary 42B08; Secondary 26A24.Keywords: Fourier series, strong summability, Marcinkiewicz function, A1-weight