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 $s<1.$



Endpoint Calderon-Zygmund inequality of nonlocal fractional laplacian
I think the estimate is false, at least for $n=1$ and $0 < s < 1/2$, due to the non-locality you mention (I imagine similar arguments would work in other non-local cases). If it held, then one would have


$$ \|v\|\_{L^p([1,2])} \leq C \| v \|\_{L^1([-4,4])} \quad (1)$$
for some fixed constant $C$ whenever $(-\Delta)^s v$ vanished on $[-3,3]$. In particular the estimate (1) should hold when $v$ is a translated Riesz potential $(x-x\_0)^{2s-1}$ for any $x\_0$ outside of say $[-5,5]$ (one can mollify first and take limits if one wants $f$ to be a nice function rather than a Dirac mass). By taking difference quotients and passing to the limit, the same estimate (1) must then hold for $\frac{\partial}{\partial x\_0} (x-x\_0)^{2s-1}$, and similarly for higher derivatives. Thus (1) holds for all functions of the form $P \left(\frac{1}{x-10} \right) (x-10)^{2s-1}$ (say) for any polynomial $P$, hence by the Weierstrass approximation theorem it holds for all $v \in C([-4,4])$ (say). But one can easily contradict (1) in this class by considering a $v$ that is concentrated in a small subinterval of $[1,2]$.


