NDLI: Regularity for Inhomogeneous Dirichlet Problems of Some Schrödinger Equations on Domains

Content Provider	Springer Nature Link
Author	Yang, Dachun Yang, Sibei
Copyright Year	2015
Abstract	Let $$n\ge 3, \Omega $$ be a bounded, simply connected and semiconvex domain in $${\mathbb {R}}^n$$ and $$L_{\Omega }:=-\Delta +V$$ a Schrödinger operator on $$L^2 (\Omega )$$ with the Dirichlet boundary condition, where $$\Delta $$ denotes the Laplace operator and the potential $$0\le V$$ belongs to the reverse Hölder class $$RH_{q_0}({\mathbb {R}}^n)$$ for some $$q_0\in (\max \{n/2,2\},\infty ]$$ . Assume that the growth function $$\varphi :\,{\mathbb {R}}^n\times [0,\infty ) \rightarrow [0,\infty )$$ satisfies that $$\varphi (x,\cdot )$$ is an Orlicz function and $$\varphi (\cdot ,t)\in {\mathbb {A}}_{\infty }({\mathbb {R}}^n)$$ (the class of uniformly Muckenhoupt weights). Let $$H_{\varphi ,\,L_{{\mathbb {R}}^n},\,r}(\Omega )$$ be the Musielak–Orlicz–Hardy space whose elements are restrictions of elements of the Musielak–Orlicz–Hardy space, associated with $$L_{{\mathbb {R}}^n}:=-\Delta +V$$ on $${\mathbb {R}}^n$$ , to $$\Omega $$ . In this article, the authors show that the operators $$VL^{-1}_\Omega $$ and $$\nabla ^2L^{-1}_\Omega $$ are bounded from $$L^1(\Omega )$$ to weak- $$L^1(\Omega )$$ , from $$L^p(\Omega )$$ to itself, with $$p\in (1,2]$$ , and also from $$H_{\varphi ,\,L_{{\mathbb {R}}^n},\,r}(\Omega )$$ to the Musielak–Orlicz space $$L^\varphi (\Omega )$$ or to $$H_{\varphi ,\,L_{{\mathbb {R}}^n},\,r}(\Omega )$$ itself. As applications, the boundedness of $$\nabla ^2{\mathbb {G}}_D$$ on $$L^p(\Omega )$$ , with $$p\in (1,2]$$ , and from $$H_{\varphi ,\,L_{{\mathbb {R}}^n},\,r}(\Omega )$$ to $$L^\varphi (\Omega )$$ or to $$H_{\varphi ,\,L_{{\mathbb {R}}^n},\,r}(\Omega )$$ itself is obtained, where $${\mathbb {G}}_D$$ denotes the Dirichlet Green operator associated with $$L_\Omega $$ . All these results are new even for the Hardy space $$H^1_{L_{{\mathbb {R}}^n},\,r}(\Omega )$$ , which is just $$H_{\varphi ,\,L_{{\mathbb {R}}^n},\,r}(\Omega )$$ with $$\varphi (x,t):=t$$ for all $$x\in {\mathbb {R}}^n$$ and $$t\in [0,\infty )$$ .
Starting Page	2097
Ending Page	2129
Page Count	33
File Format	PDF
ISSN	10506926
Journal	Journal of Geometric Analysis
Volume Number	26
Issue Number	3
e-ISSN	1559002X
Language	English
Publisher	Springer US
Publisher Date	2015-06-03
Publisher Place	New York
Access Restriction	One Nation One Subscription (ONOS)
Subject Keyword	Schrödinger equation Musielak–Orlicz–Hardy space Semiconvex domain Dirichlet boundary condition Schrödinger operator Function spaces arising in harmonic analysis $H^p$-spaces Singular and oscillatory integrals Harmonic analysis and PDE Spaces of measurable functions Differential Geometry Convex and Discrete Geometry Fourier Analysis Abstract Harmonic Analysis Dynamical Systems and Ergodic Theory Global Analysis and Analysis on Manifolds
Content Type	Text
Resource Type	Article
Subject	Geometry and Topology

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