Self-improving properties for the fractional $p$-Laplacian via nonlinear commutators

We investigate a class of nonlocal equations whose leading operator is modeled on either the fractional $p$-Laplacian or the regional fractional $p$-Laplacian, $p\in (1,\mathrm{\infty })$. We prove local self-improving properties of weak solutions to the fractional $p$-Laplacian in the case $p\in (1,\mathrm{\infty })$ with non-integrable right-hand side, as well as to the regional fractional $p$-Laplacian in the subquadratic case $1<p<2$, by extending the nonlinear commutator estimates developed by Schikorra (Math. Ann. 366 (1-2):695–720, 2016).