Parametrized Area Integrals on Hardy Spaces and Weak Hardy Spaces

In this paper, the authors prove that if . satisfies a class of the integral Dini condition, then the parametrized area integral ...,S is a bounded operator from the Hardy space H 1(.n) to L 1(.n) and from the weak Hardy space H 1,.(.n) to L 1,.(.n), respectively. As corollaries of the above results, it is shown that ...,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.