Solutions to nonlocal equations with measurable coefficients are higher differentiable.
Specifically, we consider nonlocal integrodifferential equations with measurable
coefficients whose model is given by
where the kernel
is a measurable function and satisfies the bounds
The main result states that there exists a positive, universal exponent
such that for every
the self-improving property
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
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
of measures and
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 .