The Boundedness of Hausdorff Operators on Function Spaces

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.