L^p-maximal hypoelliptic regularity of nonlocal kinetic Fokker–Planck operators

For p∈(1,∞), let u(t,x,v) and f(t,x,v) be in L^p(R×R^d×R^d) and satisfy the following nonlocal kinetic Fokker–Plank equation on R^1+2d in the weak sense: ∂_tu+v⋅∇_xu=Δ^α/2 _vu+f, where α∈(0,2) and Δ_v ^α/2 is the usual fractional Laplacian applied to v-variable. We show that there is a constant C=C(p,α,d)>0 such that for any f(t,x,v)∈L^p(R×R^d×R^d)=L^p(R^1+2d), ‖Δ_x ^α/(2(1+α))u‖_p+‖Δ_v ^α/2u‖_p⩽C‖f‖_p, where ‖⋅‖_p is the usual L^p-norm in L^p(R^1+2d;dz). In fact, in this paper the above inequality is established for a large class of time-dependent non-local kinetic Fokker–Plank equations on R^1+2d, with U_tv⋅∇_x and L_σt ^ν_t in place of v⋅∇_x and Δ_v ^α/2. See Theorem 3.3 for details.