Fractional Integral Operators on Anisotropic Hardy Spaces

Abstract

In this paper, we introduce the fractional integral operator T of degree α of order m with respect to a dilation A for 0 < α < 1 and \(m \in {\mathbb{N}}\). First we establish the Hardy-Littlewood-Sobolev inequalities for T on anisotropic Hardy spaces associated with dilation A, which show that T is bounded from H p to H q , or from H p to L q , where 0 < p ≤ 1/(1 + α) and 1/q = 1/p − α. Then we give anisotropic Hardy spaces estimates for a class of multilinear operators formed by fractional integrals or Calderon-Zygmund singular integrals. Finally, we apply the above results to give the boundedness of the commutators of T and a BMO function.