On BMO and Hardy regularity estimates for a class of non-local elliptic equations

Abstract

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 $p=\infty$.