## Theses and Dissertations

August 2013

Dissertation

#### Degree Name

Doctor of Philosophy

Mathematics

Dashan Fan

#### Committee Members

Gabriella Pinter, Lijing Sun, Hans Volkmer, Chao Zhu

#### Keywords

Analysis, Hardy Space, Harmonic Analysis, Hausdorff Operator, High-Dimensional Hardy Space, Operator Theory

#### Abstract

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)$. Our work shows that for$\Phi(t)\geq 0$, the Hausdorff operator$\hphi$is bounded on the Hardy space$H^{1}$if and only if$\Phi$is a Lebesgue integrable function; and$\hphi$is bounded on the local Hardy space$h^{1}(\bbR)$if and only if the functions$\Phi(t)\chi_{(1,\infty)}(t)$and$\Phi(t)\chi_{(0,1)(t)}\log(\recip{t})$are Lebesgue integrable. These results solve an open question posed by the Israeli mathematician Liflyand. We also establish an$H^{1}(\bbR)\rightarrow H^{1,\infty}(\bbR)$boundedness theorem for$\hphi$. As applications, we obtain many decent properties for the Hardy operator and the$k$th order Hardy operators. For instance, we know that the Hardy operator$\scrH$is bounded from$H^{1}(\bbR)\rightarrow H^{1,\infty}(\bbR)$, bounded on the atomic space$H_{A}^{1}(\bbR_{+})$, but it is not bounded on both$H^{1}(\bbR)$and the local Hardy space$h^{1}(\bbR).$We also extend part of these results to the high dimensional Hausdorff operators. Here, we study two high dimensional extentions on the Hausdorff operator$\hphi$: $\tilde{H}_{\Phi,\beta}(f)(x)=\int_{\bbR^{n}}\frac{\Phi(y)}{\Abs{y}^{n-\beta}}f(\frac{x}{\Abs{y}})\dy,\quad n\geq \beta\geq 0,$ and $H_{\Phi,\beta}(f)(x)=\int_{\bbR^{n}}\frac{\Phi(\frac{x}{\Abs{y}})}{\Abs{y}^{n-\beta}}f(y)\dy, \quad n\geq \beta\geq 0,$ where$\Phi$is a local integrable function. For$0

Additionally, we study boundedness of Hausdorff operators on some Herz type spaces, and some bilinear Hausdorff operators and fractional Hausdorff operators.

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