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- Title
Global L estimates for degenerate Ornstein-Uhlenbeck operators.
- Authors
Bramanti, Marco; Cupini, Giovanni; Lanconelli, Ermanno; Priola, Enrico
- Abstract
We consider a class of degenerate Ornstein-Uhlenbeck operators in $${\mathbb{R}^{N}}$$ , of the kindwhere ( a), ( b) are constant matrices, ( a) is symmetric positive definite on $${\mathbb{R} ^{p_{0}}}$$ ( p ≤ N), and ( b) is such that $${\mathcal{A}}$$ is hypoelliptic. For this class of operators we prove global L estimates (1 < p < ∞) of the kind:and corresponding weak type (1,1) estimates. This result seems to be the first case of global estimates, in Lebesgue L spaces, for complete Hörmander's operatorsproved 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:for any $${u \in C_{0}^{\infty} \left(S\right)}$$ , where S is the strip $${\mathbb{R}^{N} \times \left[-1, 1\right]}$$ and L is the Kolmogorov-Fokker-Planck operator $${\mathcal{A} - \partial_{t}}$$ . To get this estimate we use in a crucial way the left invariance of L with respect to a Lie group structure in $${\mathbb{R}^{N+1}}$$ and some results on singular integrals on nonhomogeneous spaces recently proved in Bramanti (Revista Matematica Iberoamericana, 2009, in press).
- Subjects
GLOBAL analysis (Mathematics); DEGENERATE differential equations; ORNSTEIN-Uhlenbeck process; OPERATOR theory; MATHEMATICAL symmetry; MATHEMATICAL constants; FOKKER-Planck equation
- Publication
Mathematische Zeitschrift, 2010, Vol 266, Issue 4, p789
- ISSN
0025-5874
- Publication type
Article
- DOI
10.1007/s00209-009-0599-3