Hausdorff operators on Hardy type spaces

During last 20 years, an essential part of the theory of Hausdorff operators is concentrated on their boundedness on the real Hardy space $H^1({\mathbb R}^d)$. The spaces introduced by Sweezy are, in many respects, natural extensions of this space. They are nested in full between $H^1({\mathbb R}^d)$ and $L_0^1({\mathbb R}^d)$. Contrary to $H^1({\mathbb R}^d)$, they are subject only to atomic characterization. For the estimates of Hausdorff operators on $H^1({\mathbb R}^d)$, other characterizations have always been applied. Since this option is excluded in the case of Sweezy spaces, in this paper an approach to the estimates of Hausdorff operators is elaborated, where only atomic decompositions are used. While on $H^1({\mathbb R}^d)$ this approach is applicable to the atoms of the same type, on the Sweezy spaces the same approach is not less effective for the sums of atoms of various types. For a single Hausdorff operator, the boundedness condition does not depend on the space but only on the parameters of the operator itself. The space on which this operator acts is characterized by the choice of atoms. An example is given (two-dimensional, for simplicity), where a matrix dilates the argument only in one variable.