Transference method for cone-like restricted summability of the two-dimensional Walsh-like systems

Abstract

. In the present paper we investigate the boundedness of the maximal operator of some d -dimensional means, provided that the set of the indeces is inside a cone-like set L . Applying some assumptions on the summation kernels P n 1 ,..., n d we state that the cone-like restricted maximal operator T γ CLR is bounded from the Hardy space H γ p to the Lebesgue space L p for p > p 0 . In the end point p 0 assuming some natural conditions on one-dimensional kernels we show that the maximal operator T γ CLR is not bounded from the Hardy space H γ p 0 to the Lebesgue space L p 0 .