# Endpoint Calderon-Zygmund inequality of nonlocal fractional laplacian

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 $$s<1.$$

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 $$[1,2]$$.