Abstract
The aim of this paper is to prove the existence and several selected properties of a global fundamental Heat kernel Γ for the parabolic operators \({\cal H} = \sum\nolimits_{j = 1}^m {X_j^2 - {\partial _t}} \), where X1,…,Xm are smooth vector fields on ℝn satisfying Hörmander’s rank condition, and enjoying a suitable homogeneity assumption with respect to a family of non-isotropic dilations. The proof of the existence of Γ is based on a (algebraic) global lifting technique, together with a representation of Γ in terms of the integral (performed over the lifting variables) of the Heat kernel for the Heat operator associated with a suitable sub-Laplacian on a homogeneous Carnot group. Among the features of Γ we prove: homogeneity and symmetry properties; summability properties; its vanishing at infinity; the uniqueness of the bounded solutions of the related Cauchy problem; reproduction and density properties; an integral representation for the higher-order derivatives.
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Acknowledgements
We wish to thank Marco Bramanti for useful discussions. Some of the results of this paper have been presented by the first-named author during the Conference “New trends in PDEs” (May 29–30, 2018 - University of Catania, Italy). We thank the anonymous Referee for his careful reading of the paper and his suggestions.
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Biagi, S., Bonfiglioli, A. Global heat kernels for parabolic homogeneous Hörmander operators. Isr. J. Math. 259, 89–127 (2024). https://doi.org/10.1007/s11856-023-2482-z
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DOI: https://doi.org/10.1007/s11856-023-2482-z