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The Boundedness Of Hausdorff Operators On Function Spaces, Xiaoying Lin

Theses and Dissertations

For a fixed kernel function $\Phi$, the one dimensional Hausdorff operator is defined in the integral form by

\[

\hphi (f)(x)=\int_{0}^{\infty}\frac{\Phi(t)}{t}f(\frac{x}{t})\dt.

\]

By the Minkowski inequality, it is easy to check that the Hausdorff operator is bounded on the Lebesgue spaces $L^{p}$ when $p\geq 1$, with some size condition assumed on the kernel functions $\Phi$. However, people discovered that the above boundedness property is quite different on the Hardy space $H^{p}$ when $0

In this thesis, we first study the boundedness of $\hphi$ on the Hardy space $H^{1}$, and on the local Hardy space $h^{1}(\bbR)$. Our work shows that for …