Local Lipschitz continuity of the inverse of the fractional pp-Laplacian, Hölder type continuity and continuous dependence of solutions to associated parabolic equations on bounded domains

Let p∈(1,∞)p∈(1,∞), s∈(0,1)s∈(0,1) and Ω⊂RNΩ⊂RN an arbitrary bounded open set. In the first part we consider the inverse Φs,p:=[(−Δ)p,Ωs]−1 of the fractional pp-Laplace operator (−Δ)p,Ωs with the Dirichlet boundary condition. We show that in the singular case p∈(1,2)p∈(1,2), the operator Φs,pΦs,p is locally Lipschitz continuous on L∞(Ω)L∞(Ω) and that global Lipschitz continuity cannot be achieved. We use this result to show that in the case N>spN>sp, if 2NN+2s0C>0 such that for every t>0t>0 and u,v∈Lq(Ω)u,v∈Lq(Ω) (q≥2q≥2), we have ‖Ss,p(t)u−Ss,p(t)v‖L∞(Ω)≤C∣Ω∣β(s)t−δ(s)‖u−v‖Lq(Ω)γ(s) where β(s),δ(s)β(s),δ(s) and γ(s)γ(s) are explicit constants depending only on N,s,p,qN,s,p,q.