Improved Hölder regularity of fractional (p, q) -Poisson equation with regular data

We prove a quantitative Hölder continuity result for viscosity solutions to the equation (-Δₚ) ^su (x) + PV ₑ䂞 |u (x) -u (x+z) |^q-2 (u (x) -u (x+z) ) ξ (x, z) |z|^{n+ tq} dz=f in\; B₂, where t, s (0, 1), 12, \\ \1, sp+αβ{p-1\} p (1, 2]. array. \] Moreover, if \sp+αβ{p-1, spp-2\}>1 when p>2, or sp+αβp-1>1 when p (1, 2], the solution is locally Lipschitz. This extends the result of 20 to the case of Hölder continuous modulating coefficients. Additionally, due to the equivalence between viscosity and weak solutions, our result provides a local Lipschitz estimate for weak solutions of (-Δₚ) ^su (x) =0 provided either p (1, 2] or sp>p-2 when p>2, thereby improving recent works 9, 10, 24.