Endpoint estimates for some maximal operators associated to the circular conical surface

For psi is an element of C(0)(infinity)(R(d)) and m > 0 we consider the maximal operator given by [GRAPHICS] It is well known that M(m) is a L(p)-bounded operator for 1 < p <= infinity. Also A. Seeger and T. Tao proved that M(m) is of weak-type L log log L if m not equal 1. In this paper we consider the case m = 1 and prove M(1) maps the standard Hardy space H(1) to weak L(1) if d >= 4. (c) 2008 Elsevier Inc. All rights reserved.