Sobolev spaces on Lie groups: Embedding theorems and algebra properties |
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Authors: | Tommaso Bruno Marco M Peloso Anita Tabacco Maria Vallarino |
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Institution: | 1. Dipartimento di Scienze Matematiche “Giuseppe Luigi Lagrange”, Dipartimento di Eccellenza 2018–2022, Politecnico di Torino, Corso Duca degli Abruzzi 24, 10129 Torino, Italy;2. Dipartimento di Matematica, Università degli Studi di Milano, Via C. Saldini 50, 20133 Milano, Italy |
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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 X on G satisfying Hörmander's condition. Let χ be a positive character of G and consider the measure whose density with respect to ρ is χ. In this paper, we introduce Sobolev spaces adapted to X and (, ) 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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Keywords: | Sobolev embeddings Sobolev algebras Lie groups Riesz transforms |
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