Some remarks on Riesz transforms on exterior Lipschitz domains
Let $n\ge 2$ and $\mathcal {L}=-\mathrm {div}(A\nabla \cdot )$ be an elliptic operator on $\mathbb {R}^n$ . Given an exterior Lipschitz domain $\Omega $ , let $\mathcal {L}_D$ be the elliptic operator $\mathcal {L}$ on $\Omega $ subject to the Dirichlet...
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Cambridge University Press
2025-01-01
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| Series: | Forum of Mathematics, Sigma |
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| Online Access: | https://www.cambridge.org/core/product/identifier/S2050509425000192/type/journal_article |
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| author | Renjin Jiang Sibei Yang |
| author_facet | Renjin Jiang Sibei Yang |
| author_sort | Renjin Jiang |
| collection | DOAJ |
| description | Let
$n\ge 2$
and
$\mathcal {L}=-\mathrm {div}(A\nabla \cdot )$
be an elliptic operator on
$\mathbb {R}^n$
. Given an exterior Lipschitz domain
$\Omega $
, let
$\mathcal {L}_D$
be the elliptic operator
$\mathcal {L}$
on
$\Omega $
subject to the Dirichlet boundary condition. Previously, it was known that the Riesz transform
$\nabla \mathcal {L}_D^{-1/2}$
is not bounded for
$p>2$
and
$p\ge n$
, even if
$\mathcal {L}=\Delta $
is the Laplace operator and
$\Omega $
is a domain outside a ball. Suppose that A are CMO coefficients or VMO coefficients satisfying certain perturbation property, and
$\partial \Omega $
is
$C^1$
. We prove that for
$p>2$
and
$p\in [n,\infty )$
, it holds that
$$ \begin{align*}\inf_{\phi\in\mathcal{K}_p(\mathcal{L}_D^{1/2})}\left\|\nabla (f-\phi)\right\|_{L^p(\Omega)}\sim \left\|\mathcal{L}^{1/2}_D f\right\|_{L^p(\Omega)} \end{align*} $$
|
| format | Article |
| id | doaj-art-033f7755870e4f149e5f5784cee80a3f |
| institution | OA Journals |
| issn | 2050-5094 |
| language | English |
| publishDate | 2025-01-01 |
| publisher | Cambridge University Press |
| record_format | Article |
| series | Forum of Mathematics, Sigma |
| spelling | doaj-art-033f7755870e4f149e5f5784cee80a3f2025-08-20T02:07:40ZengCambridge University PressForum of Mathematics, Sigma2050-50942025-01-011310.1017/fms.2025.19Some remarks on Riesz transforms on exterior Lipschitz domainsRenjin Jiang0https://orcid.org/0000-0003-3458-9610Sibei Yang1https://orcid.org/0000-0002-6910-9958Academy for Multidisciplinary Studies, Capital Normal University, Beijing, 100048, ChinaSchool of Mathematics and Statistics, Gansu Key Laboratory of Applied Mathematics and Complex Systems, Lanzhou University, Lanzhou, 730000, China; E-mail:Let $n\ge 2$ and $\mathcal {L}=-\mathrm {div}(A\nabla \cdot )$ be an elliptic operator on $\mathbb {R}^n$ . Given an exterior Lipschitz domain $\Omega $ , let $\mathcal {L}_D$ be the elliptic operator $\mathcal {L}$ on $\Omega $ subject to the Dirichlet boundary condition. Previously, it was known that the Riesz transform $\nabla \mathcal {L}_D^{-1/2}$ is not bounded for $p>2$ and $p\ge n$ , even if $\mathcal {L}=\Delta $ is the Laplace operator and $\Omega $ is a domain outside a ball. Suppose that A are CMO coefficients or VMO coefficients satisfying certain perturbation property, and $\partial \Omega $ is $C^1$ . We prove that for $p>2$ and $p\in [n,\infty )$ , it holds that $$ \begin{align*}\inf_{\phi\in\mathcal{K}_p(\mathcal{L}_D^{1/2})}\left\|\nabla (f-\phi)\right\|_{L^p(\Omega)}\sim \left\|\mathcal{L}^{1/2}_D f\right\|_{L^p(\Omega)} \end{align*} $$ https://www.cambridge.org/core/product/identifier/S2050509425000192/type/journal_article35J2535B6542B3542B37 |
| spellingShingle | Renjin Jiang Sibei Yang Some remarks on Riesz transforms on exterior Lipschitz domains Forum of Mathematics, Sigma 35J25 35B65 42B35 42B37 |
| title | Some remarks on Riesz transforms on exterior Lipschitz domains |
| title_full | Some remarks on Riesz transforms on exterior Lipschitz domains |
| title_fullStr | Some remarks on Riesz transforms on exterior Lipschitz domains |
| title_full_unstemmed | Some remarks on Riesz transforms on exterior Lipschitz domains |
| title_short | Some remarks on Riesz transforms on exterior Lipschitz domains |
| title_sort | some remarks on riesz transforms on exterior lipschitz domains |
| topic | 35J25 35B65 42B35 42B37 |
| url | https://www.cambridge.org/core/product/identifier/S2050509425000192/type/journal_article |
| work_keys_str_mv | AT renjinjiang someremarksonriesztransformsonexteriorlipschitzdomains AT sibeiyang someremarksonriesztransformsonexteriorlipschitzdomains |