Nonlocal self-improving properties
In this extremely interesting paper, the authors prove that solutions of nonlocal equations with measurable coefficients are higher differentiable. More precisely, consider nonlocal integrodifferential equations with measurable coefficients of the form \[ \int_{\mathbb{R}^n} \int_{\mathbb{R}^n} (u(x)-u(y)) (\eta(x)-\eta(y)) K(x,y)\,dx\,dy= \int_{\mathbb{R}^n} f\eta\,dx\quad \text{ for all } \eta \in C_{\text{c}}^\infty(\mathbb{R}^n), \] where the kernel \(K(\,\cdot\,,\cdot\,)\) is a measurable function and satisfies \[ \dfrac{1}{\Lambda |x-y|^{n+2\alpha}}\leq K(x,y)\leq \dfrac{\Lambda }{|x-y|^{n+2\alpha}} \] with \(\alpha\in(0,1),\) \(\Lambda>1,\) and \(f\in L^q_{\text{loc}}(\mathbb{R}^n)\) for some \(q>2n/(n+2\alpha)\). The main result of the paper states that there exists a positive, universal exponent \(\delta=\delta(n,\alpha,\Lambda,q)\) such that each weak solution \(u\) of the above nonlocal equation possesses the self-improving property \[ u\in W^{\alpha,2}(\mathbb{R}^n) \Rightarrow u\in W^{\alpha+\delta,2+\delta}_{\text{loc}}(\mathbb{R}^n). \] It is to be noted that this differentiability improvement is a genuinely nonlocal phenomenon and does not appear in the local case, where solutions to linear, divergence-form equations with measurable coefficients are known to be higher integrable but are not, in general, higher differentiable. The result of the paper is achieved by proving a fractional version of the Gehring lemma that involves certain families of lifted reverse Hölder-type inequalities in \(\mathbb{R}^{2n}\) 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 \((\mu,U)\) of measures and functions in \(\mathbb{R}^{2n}\) which are canonically associated to solutions. The authors allow also for more general equations that involve as a source term an integrodifferential operator with kernel which does not necessarily have to be of order \(\alpha\).
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