Sobolev spaces on Lie groups: embedding theorems and algebra properties
The authors introduce a Sobolev space which is adapted to a family of vector fields and a positive character of a noncompact Lie group. Let \(\mathbb G\) be a noncompact connected Lie group and \(\{X_1,\dots, X_l\}\) be a family of left-invariant vector fields satisfying Hörmander's condition, \(\mu_\chi\) be a measure such that \(\textrm{d}\mu_\chi = \chi\textrm{d}\rho\), where \(\rho\) is a right Haar measure and \(\chi\) is a continuous positive character of \(\mathbb G\). Then the following differential operator is defined: \[ \Delta_{\chi} = \sum_{j=1}^{l}(X_j^2 - c_jX_j), \] where \(c_j = (X_j\chi)(e)\). For \(\alpha\geq 0\), \(1 < p < \infty\) the Sobolev space \(L^{p}_{\alpha}(\mu_\chi)\) is defined as a set of functions from \(L^{p}(\mu_\chi)\) such that \(\Delta_{\chi}^{\alpha/2}f \in L^{p}(\mu_\chi)\). The first main result (Theorem~1.1) establishes an embedding of \(L^{p}_{\alpha}(\mu_\chi)\) into some Lebesgue space \(L^q(\mu_{\tilde\chi})\), with \(q\) and \(\tilde\chi\) depending on \(\chi\), \(\alpha\), \(p\), and \(d\), the local dimension of \(\mathbb G\). Then, as the second main result (Theorem~1.2) the authors provide algebraic properties of \(L^{p}_{\alpha}(\mu_\chi)\), in particular, they prove that \(L^{p}_{\alpha}(\mu_\chi)\cap L^{\infty}\) is an algebra for every \(p\in(1,\infty)\). Moreover, some applications to PDEs are provided: local well-posedness and regularity results of solutions of some nonlinear heat equations associated with \(\Delta_\chi\), and some nonlinear Schrödinger equations.
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