Global L^p estimates for degenerate Ornstein-Uhlenbeck operators

2008/07/25 by Marco Bramanti, M. Bramanti, Giovanni Cupini +9
Mathematics · #35H10 (Prrimary) #35K70 #42B20 (Secondary) #5 #Advanced Harmonic Analysis Research #Advanced Mathematical Physics Problems #Analysis of PDEs (math.AP) #FOS: Mathematics #Spectral Theory in Mathematical Physics #math.AP #msc:35H10 #msc:35K70 #msc:42B20 #msc:5

paper · pdf · doi:10.48550/arxiv.0807.4020

arxiv created 2008/07/25 · openalex publication_date 2008/07/25 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We consider a class of degenerate Ornstein-Uhlenbeck operators in ℝ^N, of the kind A≡∑_i,j=1^p₀a_ij∂_x_ix_j² +∑_i,j=1^Nb_ijx_i∂_x_j% where (a_ij) ,(b_ij) are constant matrices, (a_ij) is symmetric positive definite on ℝ ^p₀ (p₀≤ N), and (b_ij) is such that A is hypoelliptic. For this class of operators we prove global L^p estimates (1<p<∞) of the kind:% \Vert ∂_x_ix_j²u\Vert_{L^p(ℝ% ^N)}≤ c\\Vert Au\Vert_L^p(ℝ^N)+\Vert u\Vert_{L^p(ℝ% ^N)}\ fori,j=1,2,...,p₀% and corresponding weak (1,1) estimates. This result seems to be the first case of global estimates, in Lebesgue L^p spaces, for complete Hörmander's operators ∑ X_i²+X₀, proved in absence of a structure of homogeneous group. We obtain the previous estimates as a byproduct of the following one, which is of interest in its own:% \Vert ∂_x_ix_j²u\Vert_L^p(S)≤ c\Vert Lu\Vert_L^p(S)% for any u∈ C₀^∞(S) , where S is the strip ℝ^N×[ -1,1] and L is the Kolmogorov-Fokker-Planck operator A-∂_t.

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