Parametrized Area Integrals on Hardy Spaces and Weak Hardy Spaces

Abstract

In this paper, the authors prove that if Ω satisfies a class of the integral Dini condition, then the parametrized area integral \( \mu ^{\rho }_{{\Omega ,S}} \) is a bounded operator from the Hardy space H 1(ℝ n ) to L 1(ℝ n ) and from the weak Hardy space H 1,∞(ℝ n ) to L 1,∞(ℝ n ), respectively. As corollaries of the above results, it is shown that \( \mu ^{\rho }_{{\Omega ,S}} \) is also an operator of weak type (1, 1) and of type (p, p) for 1 < p < 2, respectively. These conclusions are substantial improvement and extension of some known results.