Meyers inequality and strong stability for stable-like operators
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| Format: | Preprint |
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2012
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| _version_ | 1866910680796364800 |
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| author | Bass, Richard F. Ren, Hua |
| author_facet | Bass, Richard F. Ren, Hua |
| contents | Let $α\in (0,2)$, let $${\cal E}(u,u)=\int_{\Bbb R^d}\int_{\Bbb R^d} (u(y)-u(x))^2\frac{A(x,y)}{|x-y|^{d+α}}\, dy\, dx$$ be the Dirichlet form for a stable-like operator, let $$Γu(x)=\int_{\Bbb R^d} (u(y)-u(x))^2\frac{A(x,y)}{|x-y|^{d+α}}\, dy,$$ let $L$ be the associated infinitesimal generator, and suppose $A(x,y)$ is jointly measurable, symmetric, bounded, and bounded below by a positive constant. We prove that if $u$ is the weak solution to $Lu=h$, then $Γu\in L^p$ for some $p>2$. This is the analogue of an inequality of Meyers for solutions to divergence form elliptic equations. As an application, we prove strong stability results for stable-like operators. If $A$ is perturbed slightly, we give explicit bounds on how much the semigroup and fundamental solution are perturbed. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_1207_2715 |
| institution | arXiv |
| publishDate | 2012 |
| record_format | arxiv |
| spellingShingle | Meyers inequality and strong stability for stable-like operators Bass, Richard F. Ren, Hua Functional Analysis 45K05 Let $α\in (0,2)$, let $${\cal E}(u,u)=\int_{\Bbb R^d}\int_{\Bbb R^d} (u(y)-u(x))^2\frac{A(x,y)}{|x-y|^{d+α}}\, dy\, dx$$ be the Dirichlet form for a stable-like operator, let $$Γu(x)=\int_{\Bbb R^d} (u(y)-u(x))^2\frac{A(x,y)}{|x-y|^{d+α}}\, dy,$$ let $L$ be the associated infinitesimal generator, and suppose $A(x,y)$ is jointly measurable, symmetric, bounded, and bounded below by a positive constant. We prove that if $u$ is the weak solution to $Lu=h$, then $Γu\in L^p$ for some $p>2$. This is the analogue of an inequality of Meyers for solutions to divergence form elliptic equations. As an application, we prove strong stability results for stable-like operators. If $A$ is perturbed slightly, we give explicit bounds on how much the semigroup and fundamental solution are perturbed. |
| title | Meyers inequality and strong stability for stable-like operators |
| topic | Functional Analysis 45K05 |
| url | https://arxiv.org/abs/1207.2715 |