$L^q$-Extensions of $L^p$-spaces by fractional diffusion equations
Der-Chen Chang Jie Xiao
Discrete & Continuous Dynamical Systems - A 2015, 35(5): 1905-1920 doi: 10.3934/dcds.2015.35.1905
Based on the geometric-measure-theoretic analysis of a new $L^p$-type capacity defined in the upper-half Euclidean space, this note characterizes a nonnegative Randon measure $\mu$ on $\mathbb R^{1+n}_+$ such that the extension $R_\alpha L^p(\mathbb R^n)\subseteq L^q(\mathbb R^{1+n}_+,\mu)$ holds for $(\alpha,p,q)\in (0,1)\times(1,\infty)\times(1,\infty)$ where $u=R_\alpha f$ is the weak solution of the fractional diffusion equation $(\partial_t + (-\Delta_x)^\alpha)u(t, x) = 0$ in $\mathbb R^{1+n}_+$ subject to $u(0,x)=f(x)$ in $\mathbb R^n$.
keywords: $L^p$-type capacities fractional diffusion equations heat kernel. $L^q$-extensions
Boundedness of second order Riesz transforms associated to Schrödinger operators on Musielak-Orlicz-Hardy spaces
Jun Cao Der-Chen Chang Dachun Yang Sibei Yang
Communications on Pure & Applied Analysis 2014, 13(4): 1435-1463 doi: 10.3934/cpaa.2014.13.1435
Let $L:=-\Delta+V$ be a Schrödinger operator with the nonnegative potential $V$ belonging to the reverse Hölder class $RH_{q_0}(R^n)$ for some $q_0\in[n,\infty)$ with $n\geq 3$, and $\varphi: R^n\times[0,\infty)\to[0,\infty)$ a function such that $\varphi(x,\cdot)$ is an Orlicz function, $\varphi(\cdot,t)\in A_{\infty}(R^n)$ (the class of uniformly Muckenhoupt weights) and its uniformly critical lower type index $i(\varphi)\in (\frac{n}{n+1},1]$. In this article, the authors prove that the second order Riesz transform $\nabla^2L^{-1}$ associated with $L$ is bounded from the Musielak-Orlicz-Hardy space associated with $L$, $H_{\varphi,L}(R^n)$, to the Musielak-Orlicz-Hardy space $H_{\varphi}(R^n)$, via establishing an atomic characterization of $H_{\varphi,L}(R^n)$. As an application, the authors prove that the operator $VL^{-1}$ is bounded on the Musielak-Orlicz-Hardy space $H_{\varphi,L}(R^n)$, which further gives the maximal inequality associated with $L$ in $H_{\varphi,L}(R^n)$. All these results are new even when $\varphi(x,t):=t^p$, with $p\in(\frac{n}{n+1},1]$, for all $x\in R^n$ and $t\in[0,\infty)$.
keywords: Musielak-Orlicz-Hardy space atom Lusin area function Schrödinger operator second order Riesz transform.

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