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arXiv · 1804.10154

Sobolev spaces on Lie groups: embedding theorems and algebra properties

Tommaso Bruno·Marco M. Peloso·Anita Tabacco·Maria Vallarino
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Abstract

Let $G$ be a noncompact connected Lie group, denote with $ρ$ a right Haar measure and choose a family of linearly independent left-invariant vector fields $\mathbf{X}$ on $G$ satisfying Hörmander's condition. Let $χ$ be a positive character of $G$ and consider the measure $μ_χ$ whose density with respect to $ρ$ is $χ$. In this paper, we introduce Sobolev spaces $L^p_α(μ_χ)$ adapted to $\mathbf{X}$ and $μ_χ$ ($1<p<\infty$, $α\geq 0$) and study embedding theorems and algebra properties of these spaces. As an application, we prove local well-posedness and regularity results of solutions of some nonlinear heat and Schrödinger equations on the group.

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Tommaso Bruno, Marco M. Peloso, Anita Tabacco, Maria Vallarino. 2018-04-26. Sobolev spaces on Lie groups: embedding theorems and algebra properties. https://arxiv.org/abs/1804.10154

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