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Vector Valued Gårding Inequality for pseudo-differential operators on compact homogeneous manifolds

André Kowacs, Michael Ruzhansky

Abstract

We prove sufficient conditions in order to obtain a sharp Gårding inequality for pseudo-differential operators acting on vector-valued functions on compact Lie groups. As a consequence, we obtain a sharp Gårding inequality for compact homogeneous vector bundles and compact homogeneous manifolds. The sharp Gårding inequality is the strongest lower bound estimate known to hold for systems on $\mathbb{R}^n$, and the aim of this paper is to extend this property to systems on compact Lie groups and compact homogeneous manifolds. Our results extend previous works by Lax and Nirenberg [Comm. Pure Appl. Math., Vol. 8, 129-209, (1966)], and by Ruzhansky and Turunen [J. Funct. Anal., Vol. 267, 144-172, (2011)]. As an application, we establish existence and uniqueness of solution to a class of systems of initial value problems of pseudo-differential equations on compact Lie groups and compact homogeneous manifolds.

Vector Valued Gårding Inequality for pseudo-differential operators on compact homogeneous manifolds

Abstract

We prove sufficient conditions in order to obtain a sharp Gårding inequality for pseudo-differential operators acting on vector-valued functions on compact Lie groups. As a consequence, we obtain a sharp Gårding inequality for compact homogeneous vector bundles and compact homogeneous manifolds. The sharp Gårding inequality is the strongest lower bound estimate known to hold for systems on , and the aim of this paper is to extend this property to systems on compact Lie groups and compact homogeneous manifolds. Our results extend previous works by Lax and Nirenberg [Comm. Pure Appl. Math., Vol. 8, 129-209, (1966)], and by Ruzhansky and Turunen [J. Funct. Anal., Vol. 267, 144-172, (2011)]. As an application, we establish existence and uniqueness of solution to a class of systems of initial value problems of pseudo-differential equations on compact Lie groups and compact homogeneous manifolds.
Paper Structure (10 sections, 18 theorems, 143 equations)

This paper contains 10 sections, 18 theorems, 143 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. The group Fourier transform
  4. The quantisation on compact Lie groups
  5. The vector-valued quantisation on compact Lie groups
  6. Even and odd functions on compact Lie groups
  7. Main Results
  8. Sharp Gå rding inequality on compact Lie groups
  9. Sharp Gå rding inequality on compact homogeneous vector bundles
  10. Application for vector-valued diffusion problems

Key Result

Proposition 2.1

For any multi-index $\alpha$ there exist constants $C_{\lambda,\mu}\geq 0$ such that for any $f,g\in\mathcal{D}'(G)$, and all $[\xi]\in\widehat{G}$.

Theorems & Definitions (31)

  • Proposition 2.1: Leibniz's like formula for Difference Operators
  • Definition 2.2
  • Theorem 2.3: Equivalence of classes, RuzPseudo2010,VerIntri
  • Lemma 2.4: Taylor Expansion in Compact Lie Groups
  • Definition 2.5
  • Theorem 2.6
  • Definition 2.7
  • Proposition 2.8
  • proof
  • Proposition 2.9
  • ...and 21 more