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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 $s<1.$

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1 Answer 1

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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 $[1,2]$.

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