Some new results on H summability of Fourier series

Abstract

In this paper we shall be concerned with H_α 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 modifications 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 ω*-density, in the space of of probability measures defined on [−π, π] of Borel measures for which there is no H₂ summability a.e.