L^p estimates for degenerate non-local Kolmogorov operators

Let z=(x,y)∈R^d×R^N−d, with 1≤d<N. We prove a priori estimates of the following type: ‖Δ_x ^{[Formula presented]}v‖_L^p(R^N)≤c_p‖L_xv+∑i,j=1Na_ijz_i∂_zj v‖_L^p(R^N),1<p<∞ for v∈C₀ ^∞(R^N), where L_x is a non-local operator comparable with the R^d-fractional Laplacian Δ_x ^{[Formula presented]} in terms of symbols, α∈(0,2). We require that when L_x is replaced by the classical R^d-Laplacian Δ_x, i.e., in the limit local case α=2, the operator Δ_x+∑_i,j=1 ^Na_ijz_i∂_zj satisfy a weak type Hörmander condition with invariance by suitable dilations. Such estimates were only known for α=2. This is one of the first results on L^p estimates for degenerate non-local operators under Hörmander type conditions. We complete our result on L^p-regularity for L_x+∑_i,j=1 ^Na_ijz_i∂_zj by proving estimates like ‖Δ_yi ^{[Formula presented]}v‖_L^p(R^N)≤c_p‖L_xv+∑i,j=1Na_ijz_i∂_zj v‖_L^p(R^N), involving fractional Laplacians in the degenerate directions y_i (here α_i∈(0,1∧α) depends on α and on the numbers of commutators needed to obtain the y_i-direction). The last estimates are new even in the local limit case α=2 which is also considered.