Boundedness of Hausdorff operators on the power weighted Hardy spaces

Abstract

In this paper, we consider the two-dimensional Hausdorff operators on the power weighted Hardy space $H_{{{\left| X \right|}^\alpha }}^1({R^2})( - 1 \leqslant \alpha \leqslant 0)$ , defined by

$${H_{\Phi ,A}}f(x) = \int {_{{R^2}}} \Phi (u)f(A(u)x)du,$$

, where Φ ∈ L _loc ¹(R ²), A(u) = (a _ij (u)) _i,j=1 ² is a 2 × 2 matrix, and each a _i,j is a measurable function. We obtain that H _Φ,A is bounded from $H_{{{\left| X \right|}^\alpha }}^1({R^2})( - 1 \leqslant \alpha \leqslant 0)$ to itself, if

$$\int {_{{R^2}}} \left| {\Phi (u)} \right|\left| {\det \;{A^{ - 1}}(u)} \right|{\left\| {A(u)} \right\|^{ - \alpha }}\;\ln \;(1 + \frac{{{{\left\| {{A^{ - 1}}(u)} \right\|}^2}}}{{\left| {\det \;{A^{ - 1}}(u)} \right|}})du < \infty .$$

. This result improves some known theorems, and in some sense it is sharp.