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)), (varphi(x)-varphi(y))}{|x-y|^{n+2s}}, dx, dy = langle f, varphi rangle quad varphi in C_c^infty(mathbb{R}^n). ] For $s in (0,1)$, $t in [s,2s]$, $p in [2,infty)$, $K$ an elliptic, symmetric, Holder continuous kernel, if $f in left (H^{t,p}_{00}(Omega)right )^ast$, then the solution $u$ belongs to $H^{2s-t,p}_{loc}(Omega)$ as long as $2s-t < 1$. The increase in differentiability is independent of the Holder coefficient of $K$. For example, our result shows that if $fin L^{p}_{loc}$ then $uin H^{2s-delta,p}_{loc}$ for any $deltain (0, s]$ as long as $2s-delta < 1$. This is different than the classical analogue of divergence-form equations ${rm div}(bar{K} abla u) = f$ (i.e. $s=1$) where a $C^gamma$-Holder 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) (-Delta)^{frac{t}{2}} u(z) , (-Delta)^{frac{2s-t}{2}} varphi(z), dz = langle g,varphirangle quad varphi in C_c^infty(mathbb{R}^n). ] and showing that as long as $K$ is Holder continuous and $s,t, 2s-t in (0,1)$ then the commutator [ int_{mathbb{R}^n} K(z,z) (-Delta)^{frac{t}{2}} u(z) , (-Delta)^{frac{2s-t}{2}} varphi(z), dz - cint_{mathbb{R}^n} int_{mathbb{R}^n} frac{K(x,y) (u(x)-u(y)), (varphi(x)-varphi(y))}{|x-y|^{n+2s}}, dx, dy ] behaves like a lower order operator.
Download