UNIFORM SUBELLIPTICITY

We prove that uniform subellipticity of a positive symmetric second-order partial differential operator on L2(ℝd) is self-improving in the sense that it automatically extends to higher powers of the operator. The range of extension is governed by the degree of smoothness of the coefficients of the o...

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Bibliographic Details
Published in:Journal of operator theory 2009-06, Vol.62 (1), p.125-149
Main Authors: ELST, A.F.M. TER, ROBINSON, DEREK W.
Format: Article
Language:English
Subjects:
Coefficients
Commutators
Geometry
Hilbert spaces
Inner products
Mathematical inequalities
Mathematical theorems
Semigroups
Vector fields
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Summary:We prove that uniform subellipticity of a positive symmetric second-order partial differential operator on L2(ℝd) is self-improving in the sense that it automatically extends to higher powers of the operator. The range of extension is governed by the degree of smoothness of the coefficients of the operator. Secondly, if the operator is of the form $\sum _{\mathrm{i}=1}^{\mathrm{N}}{\mathrm{X}}_{\mathrm{i}}^{*}{\mathrm{X}}_{\mathrm{i}}$, where the Xi are vector fields on ℝd with coefficients in ${\mathrm{C}}_{\mathrm{b}}^{\mathrm{\infty }}\left({\mathrm{\mathbb{R}}}^{\mathrm{d}}\right)$ satisfying a uniform version of Hörmander's criterion for hypoellipticity, then we prove that it is uniformly subelliptic of order r–1, where r is the rank of the set of vector fields.
ISSN:0379-4024
1841-7744