1 results
On BMO and Hardy regularity estimates for a class of non-local elliptic equations
- Part of
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- Journal:
- Proceedings of the Royal Society of Edinburgh. Section A: Mathematics / Volume 153 / Issue 6 / December 2023
- Published online by Cambridge University Press:
- 14 December 2022, pp. 2025-2052
- Print publication:
- December 2023
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- Article
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Let $\sigma \in (0,\,2)$
, $\chi ^{(\sigma )}(y):={\mathbf 1}_{\sigma \in (1,2)}+{\mathbf 1}_{\sigma =1} {\mathbf 1}_{y\in B(\mathbf {0},\,1)}$
, where $\mathbf {0}$
denotes the origin of $\mathbb {R}^n$
, and $a$
be a non-negative and bounded measurable function on $\mathbb {R}^n$
. In this paper, we obtain the boundedness of the non-local elliptic operator\[ Lu(x):=\int_{\mathbb{R}^n}\left[u(x+y)-u(x)-\chi^{(\sigma)}(y)y\cdot\nabla u(x)\right]a(y)\,\frac{{\rm d}y}{|y|^{n+\sigma}} \]from the Sobolev space based on $\mathrm {BMO}(\mathbb {R}^n)\cap (\bigcup _{p\in (1,\infty )}L^p(\mathbb {R}^n))$
to the space $\mathrm {BMO}(\mathbb {R}^n)$
, and from the Sobolev space based on the Hardy space $H^1(\mathbb {R}^n)$
to $H^1(\mathbb {R}^n)$
. Moreover, for any $\lambda \in (0,\,\infty )$
, we also obtain the unique solvability of the non-local elliptic equation $Lu-\lambda u=f$
in $\mathbb {R}^n$
, with $f\in \mathrm {BMO}(\mathbb {R}^n)\cap (\bigcup _{p\in (1,\infty )}L^p(\mathbb {R}^n))$
or $H^1(\mathbb {R}^n)$
, in the Sobolev space based on $\mathrm {BMO}(\mathbb {R}^n)$
or $H^1(\mathbb {R}^n)$
. The boundedness and unique solvability results given in this paper are further devolvement for the corresponding results in the scale of the Lebesgue space $L^p(\mathbb {R}^n)$
with $p\in (1,\,\infty )$
, established by H. Dong and D. Kim [J. Funct. Anal. 262 (2012), 1166–1199], in the endpoint cases of $p=1$
and $p=\infty$
.
