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Kernel estimates and weak (1,1)-boundedness of pseudo-differential operators on compact Lie groups

Duván Cardona, Rafik Yeghoyan, Michael Ruzhansky

Abstract

Given a compact Lie group $G$ and its unitary dual $\widehat{G}$, we establish the weak (1,1) continuity for pseudo-differential operators in the global Hörmander classes of order $-n(1-ρ)/2$ on $G\times \widehat{G}$. Our approach consists of proving suitable estimates for the kernel of such operators. Furthermore, we use these kernel estimates to give an alternative proof for the $H^1(G)$-$L^1(G)$-continuity of these classes now allowing the full range $0\leqδ\leqρ\leq1, \;ρ\neq0,\;δ\neq1$. The conditions for the operators are formulated using the Hörmander classes $S^m_{ρ,δ}(G):=S^m_{ρ,δ}(G\times \widehat{G})$ of symbols in the non-commutative phase space $G\times \widehat{G}$, which are extensions of the well-known $(ρ,δ)$-classes in the Euclidean space. Our results are formulated in the complete range $0\leq δ\leq ρ\leq 1,$ $ρ\neq0,\;$$δ\neq 1$. As an application of this boundedness result we provide end-point a-priori $L^1$-estimates for the sub-Laplacian $\mathcal{L}_{sub}=X^2+Y^2,$ and for the heat type operator $T=Z-X^2-Y^2$ on $SU(2)\cong \mathbb{S}^3$ that cannot be obtained by application of the standard pseudo-differential calculus due to Hörmander. More precisely, we prove that if one considers the subelliptic problem, \begin{equation}\label{IVP:abstract} \begin{cases}Tu=f ,& \text{ } \\u,f\in \mathscr{D}'(SU(2)):=(C^\infty(SU(2)))', & \text{ } \end{cases} \end{equation} then, for $f\in W^{1,-\frac{1}{4}}(SU(2)),$ one has that $u\in L^{1,\infty}(SU(2)).$

Kernel estimates and weak (1,1)-boundedness of pseudo-differential operators on compact Lie groups

Abstract

Given a compact Lie group and its unitary dual , we establish the weak (1,1) continuity for pseudo-differential operators in the global Hörmander classes of order on . Our approach consists of proving suitable estimates for the kernel of such operators. Furthermore, we use these kernel estimates to give an alternative proof for the --continuity of these classes now allowing the full range . The conditions for the operators are formulated using the Hörmander classes of symbols in the non-commutative phase space , which are extensions of the well-known -classes in the Euclidean space. Our results are formulated in the complete range . As an application of this boundedness result we provide end-point a-priori -estimates for the sub-Laplacian and for the heat type operator on that cannot be obtained by application of the standard pseudo-differential calculus due to Hörmander. More precisely, we prove that if one considers the subelliptic problem, \begin{equation}\label{IVP:abstract} \begin{cases}Tu=f ,& \text{ } \\u,f\in \mathscr{D}'(SU(2)):=(C^\infty(SU(2)))', & \text{ } \end{cases} \end{equation} then, for one has that
Paper Structure (12 sections, 17 theorems, 161 equations)

This paper contains 12 sections, 17 theorems, 161 equations.

Table of Contents

  1. Introduction
  2. Overview
  3. Methodology
  4. Main results
  5. Preliminaries
  6. The group Fourier transform
  7. Global pseudo-differential operators
  8. The spaces $BMO$ and $H^1$ on compact Lie groups
  9. Proof of the main theorems
  10. Local estimates for kernels
  11. $L^1$-theory of pseudo-differential operators
  12. Applications to subelliptic estimates on $SU(2)\cong \mathbb{S}^3$

Key Result

Theorem 1.1

Let $G$ be a compact Lie group of topological dimension $n\geq 1$. Consider $T\in\Psi^{m}_{\rho,\delta}(G)$, with For $R>0$, we have that the kernel of $T$ satisfies the following. For $y\in\overline{B(z, R)}$ we have Here the constant $C>0,$ in the above estimates is independent of $R>0.$

Theorems & Definitions (36)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Remark 1.5
  • Definition 2.1
  • Definition 2.2
  • Proposition 2.3
  • proof
  • Theorem 2.4
  • ...and 26 more