Sobolev regularity for the nonlocal $(1,p)$-Laplace equations in the superquadratic case

We investigate the interior Sobolev regularity of weak solutions to the nonlocal $(1,p)$-Laplace equations in the superquadratic case $p\ge 2$. As a product, the explicit Hölder continuity estimates of weak solutions are derived. The proof relies on a detailed analysis of the structural characteristics of $(1,p)$-growth in the nonlocal setting, combined with the finite difference quotient method, tail estimates, refined energy estimates, and a Moser-type iteration scheme.