Lp estimates for degenerate non-local Kolmogorov operators

Let z = (x, y) is an element of R-d x RN-d , with 1 <= d <= N. We prove a priori estimates of the following type: parallel to Delta(alpha/2)(x) v parallel to(Lp(RN)) <= c(p)parallel to L(x)v + Sigma(N)(i,j=1) a(ij)z(i)partial derivative(zj) v parallel to(Lp(RN)), 1<p<infinity, for v is an element of C-0(infinity) (R-N), where L-x is a non-local operator comparable with the R-d-fractional Laplacian Delta(alpha/2)(x) in terms of symbols, alpha is an element of (0, 2). We require that when L-x is replaced by the classical R-d-Laplacian Delta(x), i.e., in the limit local case alpha = 2, the operator Delta(x)+ Sigma(N)(i,j=1) a(ij)z(i)partial derivative(zj), satisfy a weak type Hormander condition with invariance by suitable dilations. Such estimates were only known for alpha = 2. This is one of the first results on L-p estimates for degenerate non-local operators under Hormander type conditions. We complete our result on L-p-regularity for L-x + Sigma(N)(i,j=1) a(ij)z(i)partial derivative(zj), by proving estimates like parallel to Delta(alpha i/2)(yi) v parallel to(Lp(RN)) <= c(p)parallel to L(x)v + Sigma(N)(i,j=1) a(ij)z(i)partial derivative(zj) v parallel to(Lp(RN)), involving fractional Laplacians in the degenerate directions y(i) (here alpha(i )is an element of (0,1 Lambda alpha) depends on alpha and on the numbers of commutators needed to obtain the yi-direction). The last estimates are new even in the local limit case alpha = 2 which is also considered. (C) 2017 Elsevier Masson SAS. All rights reserved.