Abstract. In this paper we consider integrability conditions for dyadic maximal Walsh series. Namely, we give a condition on the coefficients of a Walsh series which is sufficient for the series being the Walsh--Fourier series of a function belonging to the dyadic Hardy space. In the classical trigonometric case the analogous question involves the real periodic Hardy space. Then the problem leads to integrability conditions on both the trigonometric series and its conjugate, which in fact can be reduced to integrability conditions for cosine and sine series.

DOI: 10.14232/actasm-015-032-x

AMS Subject Classification (1991): 42C10; 30H10

Keyword(s): Walsh--Paley system, Hardy spaces, integrability conditions, dyadic maximal function

Received May 4, 2015, and in revised form August 14, 2015. (Registered under 32/2015.)