Riesz means associated with certain product type convex domain

Abstract

In this paper we study the maximal operators T(*)(delta) and the convolution operators T(delta) associated with multipliers of the form (1 - max, {vertical bar xi(0)vertical bar, vertical bar xi(1)vertical bar,..., vertical bar xi(n-2)vertical bar}(+)(delta), (xi(0), xi(1),..., xi(n-2)) is an element of R(2) x R(n-2), n >= 3. We prove that T(*)(delta) satisfies the sharp weak type (p, p) inequality on H(p)(R(n)) when 2(n-1)/2n-1 < p <1 and delta = n(1/p - 1) + 1/2, or when p = 2(n-1)/2n-1 and delta > n(1/p - 1) + 1/2. We also obtain that T(delta) is bounded from L(p)(R(n)) to L(p)(R(n)) for delta > max{2 vertical bar 1/p - 1/2 vertical bar - 1/2, 0) and 1 <= p < infinity. The indicated ranges of parameters p and S cannot be improved. (C) 2011 Elsevier Inc. All rights reserved.