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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| Main Authors: | , |
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| Format: | Article |
| Language: | English |
| Published: |
Cambridge University Press
2025-01-01
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| Series: | Forum of Mathematics, Sigma |
| Subjects: | |
| Online Access: | https://www.cambridge.org/core/product/identifier/S2050509425000192/type/journal_article |
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| Summary: | 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
$$ \begin{align*}\inf_{\phi\in\mathcal{K}_p(\mathcal{L}_D^{1/2})}\left\|\nabla (f-\phi)\right\|_{L^p(\Omega)}\sim \left\|\mathcal{L}^{1/2}_D f\right\|_{L^p(\Omega)} \end{align*} $$
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| ISSN: | 2050-5094 |