On a class of hypoelliptic operators with unbounded coefficients in \matbb R^N

2008/03/04 by Bálint Farkas, B. Farkas, Farkas, B. +3
Mathematics · #35B65 #35J70 #35K15 #35K65 #Analysis of PDEs (math.AP) #FOS: Mathematics #Numerical methods in inverse problems #Spectral Theory in Mathematical Physics #advanced mathematical theories #math.AP #msc:35B65 #msc:35J70 #msc:35K15 #msc:35K65

paper · pdf · doi:10.48550/arxiv.0803.0509

arxiv created 2008/03/04 · openalex publication_date 2008/03/04 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We consider a class of non-trivial perturbations \mathscr A of the degenerate Ornstein-Uhlenbeck operator in \mathbb R^N. In fact we perturb both the diffusion and the drift part of the operator (say Q and B) allowing the diffusion part to be unbounded in \mathbb R^N. Assuming that the kernel of the matrix Q(x) is invariant with respect to x∈ \mathbb R^N and the Kalman rank condition is satisfied at any x∈\mathbb R^N by the same m<N, and developing a revised version of Bernstein's method we prove that we can associate a semigroup \T(t)\ of bounded operators (in the space of bounded and continuous functions) with the operator \mathscr A. Moreover, we provide several uniform estimates for the spatial derivatives of the semigroup \T(t)\ both in isotropic and anisotropic spaces of (Hölder-) continuous functions. Finally, we prove Schauder estimates for some elliptic and parabolic problems associated with the operator \mathscr A.

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