Nonlocal self-improving properties

Abstract

Solutions to nonlocal equations with measurable coefficients are higher differentiable.

Specifically, we consider nonlocal integrodifferential equations with measurable coefficients whose model is given by

$${\int }_{{ℝ}^n}{\int }_{{ℝ}^n}\left[u\left(x\right)-u\left(y\right)\right]\left[\eta \left(x\right)-\eta \left(y\right)\right]K\left(x,y\right)dxdy={\int }_{{ℝ}^n}f\eta dxfor\; all\eta \in C_c^{\infty }\left({ℝ}^n\right),$$

where the kernel $K\left(\cdot \right)$ is a measurable function and satisfies the bounds

$$\frac{1}{\Lambda |x-y{|}^{n+2\alpha }}\le K\left(x,y\right)\le \frac{\Lambda }{|x-y{|}^{n+2\alpha }}$$

with $0<\alpha <1$, $\Lambda >1$, while $f\in L_{loc}^q\left({ℝ}^n\right)$ for some $q>2n∕\left(n+2\alpha \right)$. The main result states that there exists a positive, universal exponent $\delta \equiv \delta \left(n,\alpha ,\Lambda ,q\right)$ such that for every weak solution $u$ the self-improving property

$$u\in W^{\alpha ,2}\left({ℝ}^n\right)\Rightarrow u\in W_{loc}^{\alpha +\delta ,2+\delta }\left({ℝ}^n\right)$$

holds. This differentiability improvement is a genuinely nonlocal phenomenon and does not appear in the local case, where solutions to linear equations in divergence form with measurable coefficients are known to be higher integrable but are not, in general, higher differentiable.

The result is achieved by proving a new version of the Gehring lemma involving certain families of lifted reverse Hölder-type inequalities in ${ℝ}^{2n}$ and which is implied by delicate covering and exit-time arguments. In turn, such reverse Hölder inequalities are based on the concept of dual pairs, that is, pairs $\left(\mu ,U\right)$ of measures and functions in ${ℝ}^{2n}$ which are canonically associated to solutions. We also allow for more general equations involving as a source term an integrodifferential operator whose kernel does not necessarily have to be of order $\alpha$.