Nonlocal self-improving properties

Solutions to nonlocal equations with measurable coefficients are higher differentiable.¶, we consider nonlocal integrodifferential equations with measurable whose model is given by¶∫ℝn ∫ℝn[u(x)−u(y)][η(x)−η(y)]K(x,y)<mspace class="thinspace" width="0.3em"/>dx<mspace class="thinspace" width="0.3em"/>dy= ∫ℝnfη<mspace class="thinspace" width="0.3em"/>dx<mspace class="quad" width="1em"/>for all<mspace class="nbsp" width="1em"/>η∈ Cc∞(ℝn),¶ the kernel(<mspace class="thinspace" width="0.3em"/>⋅<mspace class="thinspace" width="0.3em"/>) a measurable function and satisfies the bounds¶1Λ|x−|n+2α≤(x,y)≤ Λ|x−|n+2α¶0<α< 1,Λ> 1, while∈ Llocq(ℝn) for> 2n∕(n+ 2α). main result states that there exists a positive, universal exponentδ≡δ(n,α,Λ,q) such that for every solution[math] self-improving property¶ν∈ Wα,2(ℝn)<mspace class="quad" width="1em"/>⇒<mspace class="quad" width="1em"/>u∈ Wα+δ,2+δ(ℝn)¶. This differentiability improvement is a genuinely nonlocal phenomenon and not appear in the local case, where solutions to linear equations in divergence with measurable coefficients are known to be higher integrable but are not, in, higher differentiable.¶ result is achieved by proving a new version of the Gehring lemma certain families of lifted reverse Hölder-type inequalities in[math] and is implied by delicate covering and exit-time arguments. In turn, such reverseölder inequalities are based on the concept of dual pairs, that is, pairs[math] of measures and in[math] are canonically associated to solutions. We also allow for more general equations involving a source term an integrodifferential operator whose kernel does not necessarily have to be order [math] .