Date of Award
August 2013
Degree Type
Dissertation
Degree Name
Doctor of Philosophy
Department
Mathematics
First Advisor
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.
Recommended Citation
Lin, Xiaoying, "The Boundedness of Hausdorff Operators on Function Spaces" (2013). Theses and Dissertations. 256.
https://dc.uwm.edu/etd/256