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Subelliptic sharp Gårding inequality on compact Lie groups

Duván Cardona, Serena Federico, Michael Ruzhansky

Abstract

In this work we establish a subelliptic sharp Gårding inequality on compact Lie groups for pseudo-differential operators with symbols belonging to global subelliptic Hörmander classes. In order for the inequality to hold we require the global matrix-valued symbol to satisfy the suitable classical nonnegativity condition in our setting. Our result extends to $\mathscr{S}^m_{ρ,δ}(G)$-classes, $0\leq δ<ρ$, the one in [26] about the validity of the sharp Gårding inequality for the class $\mathscr{S}^m_{1,0}(G)$. We remark that the result we prove here is already new and sharp in the case of the torus.

Subelliptic sharp Gårding inequality on compact Lie groups

Abstract

In this work we establish a subelliptic sharp Gårding inequality on compact Lie groups for pseudo-differential operators with symbols belonging to global subelliptic Hörmander classes. In order for the inequality to hold we require the global matrix-valued symbol to satisfy the suitable classical nonnegativity condition in our setting. Our result extends to -classes, , the one in [26] about the validity of the sharp Gårding inequality for the class . We remark that the result we prove here is already new and sharp in the case of the torus.

Paper Structure

This paper contains 10 sections, 14 theorems, 165 equations.

Table of Contents

  1. Introduction
  2. Outline and historical remarks
  3. The subelliptic sharp Gårding inequality
  4. Sub-Laplacians and pseudo-differential operators on compact Lie groups
  5. Pseudo-differential operators via localisations
  6. The positive sub-Laplacian and pseudo-differential operators via global symbols
  7. Odd and even functions on compact Lie groups
  8. Subelliptic Hörmander classes on compact Lie groups
  9. Proof of the main Theorem \ref{['MainTheorem']}
  10. Final remarks

Key Result

Theorem \oldthetheorem

Let $G$ be a compact Lie group and let $\mathcal{L}=\mathcal{L}_X$\mathcal{L}_X:=-X_1^2-X_2^2-\cdots - X_{k}^2.$$ be the (positive) sub-Laplacian associated with a system $X=\{X_{i}\}_{i=1}^{k}$ of left-invariant vector fields satisfying Hörmander's condition of step $\kappa$which means that the vec be a continuous linear operator with global symbol $a\in {S}^{m,\,\mathcal{L}}_{\rho,\delta}( G\tim

Theorems & Definitions (33)

  • Theorem \oldthetheorem: Subelliptic sharp Gårding inequality
  • Corollary \oldthetheorem: Elliptic sharp Gårding Inequality
  • Remark \oldthetheorem
  • Remark \oldthetheorem
  • Remark \oldthetheorem
  • Proposition \oldthetheorem: Leibniz rule for difference operators
  • Definition \oldthetheorem
  • Theorem \oldthetheorem: Equivalence of classes, Fischer2015RuzRuzhanskyTurunenWirth2014
  • Definition \oldthetheorem
  • Proposition \oldthetheorem
  • ...and 23 more