Boundedness of Hausdorff operators on the power weighted Hardy spaces

In this paper, we consider the two-dimensional Hausdorff operators on the power weighted Hardy space H|X|α1(R2)(−1≤α≤0)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $$H_{{{\left| X \right|}^\alpha }}^1({R^2})( - 1 \leqslant \alpha \leqslant 0)$$ \end{document} , defined by HΦ,Af(x)=∫R2Φ(u)f(A(u)x)du,\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $${H_{\Phi ,A}}f(x) = \int {_{{R^2}}} \Phi (u)f(A(u)x)du,$$ \end{document}, where Φ ∈ Lloc1(R2), A(u) = (aij(u))i,j=12 is a 2 × 2 matrix, and each ai,j is a measurable function. We obtain that HΦ,A is bounded from H|X|α1(R2)(−1≤α≤0)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $$H_{{{\left| X \right|}^\alpha }}^1({R^2})( - 1 \leqslant \alpha \leqslant 0)$$ \end{document} to itself, if ∫R2|Φ(u)||detA−1(u)|‖A(u)‖−αln(1+‖A−1(u)‖2|detA−1(u)|)du<∞.\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $$\int {_{{R^2}}} \left| {\Phi (u)} \right|\left| {\det \;{A^{ - 1}}(u)} \right|{\left\| {A(u)} \right\|^{ - \alpha }}\;\ln \;(1 + \frac{{{{\left\| {{A^{ - 1}}(u)} \right\|}^2}}}{{\left| {\det \;{A^{ - 1}}(u)} \right|}})du < \infty .$$ \end{document}. This result improves some known theorems, and in some sense it is sharp.