For $s\in(0,1],$ consider the following non-local fractional laplacian: $$(-\Delta)^sv= f ~~\text{on } \mathbb{R}^n.$$ Then how to use "the standard elliptic estimate" to obtain:

for $p\in[1, \frac{n}{n-2s}),$ $$\|v\|_{L^p(B_2\setminus B_1)} \lesssim \|f\|_{L^1(B_3\setminus B_{\frac{1}{2}})}+\|v\|_{L^1(B_4\setminus B_{\frac{1}{2}})}.$$ When I see the index $\frac{n}{n-2s}$, I immediately think it is due to the Sobolev inequality and the Calderon-Zygmund inequality, but the Calderon-Zygmund inequality fails at the endpoint index $1$ even in the case $s=1$. Any insights or references are appreciated!

Update:

When $s=1$, consider a cut-off function $\eta$ equals $1$ in $B_2\setminus B_1$, and $0$ out of $B_4\setminus B_{\frac{1}{2}}.$ Then we have $$-\Delta \eta v = f\eta -v\Delta \eta -2\nabla v. \nabla \eta$$ For any $\epsilon>0$, $h\in L^{\frac{n}{2}+\epsilon}(B_4\setminus B_{\frac{1}{2}})$, let $\phi\in H_0^1(B_4\setminus B_{\frac{1}{2}})$ be the solution of $$-\Delta \phi=h.$$ By Calderon-Zygmund inequality and Sobolev embedding we have $$\|\phi\|_{C^1(B_4\setminus B_{\frac{1}{2}})}\leq \|h\|_{L^{\frac{n}{2}+\epsilon}(B_4\setminus B_{\frac{1}{2}})}$$ Test $-\Delta \eta v = f\eta -v\Delta \eta -2\nabla v. \nabla \eta$ by $\phi,$ we obtain: $$\int\eta v h \lesssim (\|f\|_{L^1(B_3\setminus B_{\frac{1}{2}})}+\|v\|_{L^1(B_4\setminus B_{\frac{1}{2}})})\|\phi\|_{C^1(B_4\setminus B_{\frac{1}{2}})}.$$ Hence $$\int\eta v h \lesssim (\|f\|_{L^1(B_3\setminus B_{\frac{1}{2}})}+\|v\|_{L^1(B_4\setminus B_{\frac{1}{2}})}) \|h\|_{L^{\frac{n}{2}+\epsilon}(B_4\setminus B_{\frac{1}{2}})}.$$ By dual we obtain that for every $p\in[1, \frac{n}{n-2})$: $$\|\eta v\|_{L^p} \lesssim \|f\|_{L^1(B_3\setminus B_{\frac{1}{2}})}+\|v\|_{L^1(B_4\setminus B_{\frac{1}{2}})}.$$ But I have no idea to deal with the non-local case $s<1.$