Calderon-Zygmund type estimates for nonlocal PDE with Hölder continuous kernel

Abstract

We study interior $L^p$-regularity theory, also known as Calderon-Zygmund theory, of the equation \[ \int_{\mathbb{R}^n} \int_{\mathbb{R}^n} \frac{K(x,y)\ (u(x)-u(y))\, (φ(x)-φ(y))}{|x-y|^{n+2s}}\, dx\, dy = \langle f, φ\rangle \quad φ\in C_c^\infty(\mathbb{R}^n). \] For $s\in (0,1)$, $t\in [s,2s]$, $p\in [2,\infty )$, $K$ an elliptic, symmetric, Hölder continuous kernel, if $f\in {\left(H_{00}^{t,p^{\prime }}(\Omega )\right)}^{\ast }$, then the solution $u$ belongs to $H_{loc}^{2s-t,p}(\Omega )$ as long as $2s-t<1$. The increase in differentiability is independent of the Hölder coefficient of $K$. For example, our result shows that if $f\in L_{loc}^p$ then $u\in H_{loc}^{2s-\delta ,p}$ for any $\delta \in (0,s]$ as long as $2s-\delta <1$. This is different than the classical analogue of divergence-form equations $\mathrm{div}(\bar{K}\nabla u)=f$ (i.e. $s=1$) where a $C^{\gamma }$-Hölder continuous coefficient $\bar{K}$ only allows for estimates of order $H^{1+\gamma }$. In fact, it is another appearance of the differential stability effect observed in many forms by many authors for this kind of nonlocal equations -- only that in our case we do not get a "small" differentiability improvement, but all the way up to $\min \{2s-t,1\}$. The proof argues by comparison with the (much simpler) equation \[ \int_{\mathbb{R}^n} K(z,z) (-Δ)^{\frac{t}{2}} u(z) \, (-Δ)^{\frac{2s-t}{2}} φ(z)\, dz = \langle g,φ\rangle \quad φ\in C_c^\infty(\mathbb{R}^n). \] and showing that as long as $K$ is Hölder continuous and $s,t,2s-t\in (0,1)$ then the "commutator" \[ \int_{\mathbb{R}^n} K(z,z) (-Δ)^{\frac{t}{2}} u(z) \, (-Δ)^{\frac{2s-t}{2}} φ(z)\, dz - c\int_{\mathbb{R}^n} \int_{\mathbb{R}^n} \frac{K(x,y)\ (u(x)-u(y))\, (φ(x)-φ(y))}{|x-y|^{n+2s}}\, dx\, dy \] behaves like a lower order operator.