# Sobolev Spaces On Lie Groups

• Maria Vallarino (Politecnico di Torino, Italy)
• Tuesday 20 March 2018, 14:00-15:00
• Strathcona LT2.

Let $G$ be a noncompact nonunimodular Lie group and $\{X_1, ...,X_q\}$ a set of left-invariant vector felds which satisfy the Hormander’s condition. Defne the subLaplacian $\Delta = – \sum_{i=1}q X_i2$, which is selfadjoint with respect to a right Haar measure $\rho$ of $G$. We study various properties of the Sobolev spaces associated with the operator $\Delta$, i.e. the spaces $Lp_\alpha$.

In particular, we shall discuss Sobolev embeddings and algebra properties of the spaces £Lp_\alpha\$. Such properties are useful for applications to well-posedness results for semilinear differential equations associated with the subLaplacian on the group.

The results discussed in the talk are joint work with Tommaso Bruno, Marco M. Peloso and Anita Tabacco [1, 3].

The counterpart of our results on unimodular Lie groups were obtained by Coulhon, Russ and Tardivel-Nachef [2].

References

[1] T. Bruno, M.M. Peloso, A. Tabacco and M. Vallarino, Weighted Sobolev spaces on nonuni- modular Lie groups, preprint

[2] T. Coulhon, E. Russ and V. Tardivel-Nachef, Sobolev algebras on Lie groups and Rie- mannian manifolds, Amer. J. Math. 123 (2001), no. 2, 283-342

[3] M.M. Peloso and M. Vallarino, Sobolev algebras on nonunimodular Lie groups, arXiv:1710.07566

This talk is part of the Analysis Seminar series.