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Solvability of a class of evolution operators on compact Lie groups

Alexandre Kirilov, Wagner Augusto Almeida de Moraes, Pedro Meyer Tokoro

TL;DR

The work addresses global $C^{\infty}$-solvability of first-order Vekua-type evolution operators on $\mathbb{T}\times G$ for a compact Lie group $G$, namely $P u = \partial_t u - C(t) X u - A(t) u - B(t) \overline{u}$. It reduces to a normal form via a conjugation that diagonalizes the left-invariant drift, allowing a per-representation $2\times2$ system analysis and a robust solvability criterion in terms of averaged coefficients and the spectral data $\mu_m(\xi)$ of $\sigma_X(\xi)$. The main theorem imposes three explicit conditions: (I) a non-degeneracy of the diagonalization, (II) a global nonresonance (no exact obstructions), and (III) a Diophantine-type lower bound to control small denominators as $\langle\xi\rangle\to\infty$, ensuring existence of smooth solutions $u$ for all smooth right-hand sides $f$. The paper specializes to the case $G\cong\mathbb{S}^3$ (SU(2)) with explicit spectral data and extends the solvability theory to finite products of compact Lie groups, providing a practical framework for analyzing Vekua-type equations on compact homogeneous spaces in applications to geometry and mathematical physics.

Abstract

This paper provides sufficient conditions for the solvability of a class of first-order evolution operators of Vekua-type on the product of a one-dimensional torus and a compact Lie group. The conditions are expressed in terms of the time-dependent coefficients and the spectral behavior of a normalized left-invariant vector field on the group. The three-sphere case is discussed in detail, leading to more explicit criteria, and the main results are further extended to operators defined on finite products of compact Lie groups.

Solvability of a class of evolution operators on compact Lie groups

TL;DR

The work addresses global -solvability of first-order Vekua-type evolution operators on for a compact Lie group , namely . It reduces to a normal form via a conjugation that diagonalizes the left-invariant drift, allowing a per-representation system analysis and a robust solvability criterion in terms of averaged coefficients and the spectral data of . The main theorem imposes three explicit conditions: (I) a non-degeneracy of the diagonalization, (II) a global nonresonance (no exact obstructions), and (III) a Diophantine-type lower bound to control small denominators as , ensuring existence of smooth solutions for all smooth right-hand sides . The paper specializes to the case (SU(2)) with explicit spectral data and extends the solvability theory to finite products of compact Lie groups, providing a practical framework for analyzing Vekua-type equations on compact homogeneous spaces in applications to geometry and mathematical physics.

Abstract

This paper provides sufficient conditions for the solvability of a class of first-order evolution operators of Vekua-type on the product of a one-dimensional torus and a compact Lie group. The conditions are expressed in terms of the time-dependent coefficients and the spectral behavior of a normalized left-invariant vector field on the group. The three-sphere case is discussed in detail, leading to more explicit criteria, and the main results are further extended to operators defined on finite products of compact Lie groups.
Paper Structure (10 sections, 11 theorems, 131 equations)

This paper contains 10 sections, 11 theorems, 131 equations.

Table of Contents

  1. Introduction
  2. Overview of harmonic analysis and symbols on compact Lie groups
  3. Fourier analysis on compact Lie groups
  4. Symbols and left-invariant vector fields
  5. Partial Fourier series
  6. Reduction to a normal form
  7. Solvability of a Vekua-type evolution operator
  8. The case $\lambda=0$
  9. Example: the case $G=\mathbb{S}^3$
  10. Product of groups

Key Result

Theorem 2.1

A function $f$ belongs to $C^\infty(\mathbb{T}\times G)$ if and only if for every $[\xi]\in\widehat{G}$ and $1\le m,n\le d_\xi$, the coefficient $\widehat{f}(\cdot,\xi)_{mn}$ belongs to $C^\infty(\mathbb{T})$ and, for every $\beta\in\mathbb{N}_0$ and every $\ell>0$, there exists $C_{\beta,\ell}>0$ s

Theorems & Definitions (14)

  • Theorem 2.1: Characterization of $C^\infty(\mathbb{T}\times G)$
  • Theorem 2.2: Characterization of $\mathcal{D}'(\mathbb{T}\times G)$
  • Proposition 2.3: Faà di Bruno formula
  • Proposition 3.1: Kirilov, de Moraes, and Ruzhansky, 2021
  • proof
  • Theorem 4.1
  • Theorem 5.1
  • proof
  • Lemma 5.2
  • proof
  • ...and 4 more