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(ℝⁿ) to L ¹(ℝⁿ) and from the weak Hardy space H ^1,∞(ℝⁿ) to L ^1,∞(ℝⁿ), 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.