Two - Dimensional Walsh

The main aim of this paper is to prove that for the boundedness of the maximal operator σ ∗ from the Hardy space Hp ( I ) to the space Lp ( I ) the assumption p > max {1/ (α + 1) , 1/ (β + 1)} is essential. We denote the set of non-negative integers by N. For a set X 6= ∅ let X be its Cartesian product X×X taken with itself. By a dyadic interval in I := [0, 1) we mean one of the form [ l2−k, (l + 1) 2−k ) for some k ∈ N, 0 ≤ l < 2. Given k ∈ N and x ∈ [0, 1), let Ik(x) denote the dyadic interval of length 2−k which contains the point x. The Cartesian product of two dyadic intervals is said to be a rectangle. Clearly, the dyadic rectangle of area 2−n × 2−m containing (x, x) ∈ I is given by In,m (x, x) := In (x) × Im (x). We also use the notation mes (A) for the Lebesgue measure of any measurable set A. Let r0 (x) be a function defined by r0 (x) = { 1, if x ∈ [0, 1/2), −1, if x ∈ [1/2, 1), r0 (x + 1) = r0 (x) . The Rademacher system is defined by rn (x) = r0 (2 x) , n ≥ 1 and x ∈ [0, 1). Let w0, w1, . . . represent the Walsh functions, i.e. w0 (x) = 1 and if n = 21 + · · ·+ 2r is a positive integer with n1 > n2 > · · · > nr then wn (x) = rn1 (x) · · · rnr (x) . 2000 Mathematics Subject Classification. 42C10.