An application of a global lifting method for homogeneous Hörmander vector fields to the Gibbons conjecture

2019 ◽  
Vol 26 (6) ◽  
Author(s):  
Stefano Biagi
Keyword(s):  
Vector Fields ◽  
Hörmander Vector Fields

A completeness result for time-dependent vector fields and applications

2015 ◽  
Vol 17 (04) ◽  
pp. 1450040 ◽  
Author(s):  
Stefano Biagi ◽  
Andrea Bonfiglioli
Keyword(s):  
Vector Field ◽  
Lie Groups ◽  
Vector Fields ◽  
Time Dependent ◽  
Sufficient Condition ◽  
Invariance Condition ◽  
The Third ◽  
Completeness Result ◽  
Dependent Vector ◽  
Hörmander Vector Fields

We provide a sufficient condition for the completeness of a time-dependent vector field in ℝN, generalizing the well-known left-invariance condition on Lie groups. This result can be applied to the construction of Lie groups associated to suitable families X of Hörmander vector fields, without the need to use the Third Fundamental Theorem of Lie. Further applications are given to the control-theoretic distance related to X, and to the existence of the relevant geodesics.

scholarly journals Space regularity for evolution operators modeled on Hörmander vector fields with time dependent measurable coefficients

2020 ◽  
Author(s):  
Marco Bramanti
Keyword(s):  
Vector Fields ◽  
Evolution Operators ◽  
Invariant Vector ◽  
Measurable Coefficients ◽  
Regularity Estimates ◽  
Hörmander Vector Fields ◽  
Derivatives Of ◽  
Homogeneous Vector Fields ◽  
Quantitative Regularity ◽  
Continuity Estimate

Abstract We consider a heat-type operator $$\mathcal {L}$$ L structured on the left invariant 1-homogeneous vector fields which are generators of a Carnot group, with a uniformly positive matrix of bounded measurable coefficients depending only on time. We prove that if $$\mathcal {L}u$$ L u is smooth with respect to the space variables, the same is true for u, with quantitative regularity estimates in the scale of Sobolev spaces defined by right invariant vector fields. Moreover, the solution and its space derivatives of every order satisfy a 1/2-Hölder continuity estimate with respect to time. The result is proved both for weak solutions and for distributional solutions, in a suitable sense.

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