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Inverses of Product Kernels and Flag Kernels on Graded Lie Groups

Amelia Stokolosa

TL;DR

The paper proves that for a direct product of graded Lie groups G=G_1×...×G_ν, a left-invariant singular integral T(f)=f*K with K a product or flag kernel is invertible on L^2(G) only if its inverse is convolution with a kernel of the same type, i.e., T^{-1}(g)=g*L where L is a product or flag kernel accordingly. The authors develop a multi-parameter version of the Christ–Geller a priori estimate by employing right-invariant Sobolev spaces and commuting differential operators to relate T and T^{-1}. Central to the argument are commutator estimates, a Littlewood–Paley decomposition, and a careful analysis of growth and cancellation conditions, enabling the transfer of kernel-class regularity to the inverse. The results extend classical single-parameter inversion theorems to multi-parameter (ν-parameter) settings and broaden the applicability to a larger class of kernels on graded Lie groups, with a parallel treatment for flag kernels via dilations and kernel-cancellation arguments.

Abstract

Let $T(f) = f * K$, where $K$ is a product kernel or a flag kernel on a direct product of graded Lie groups $G= G_1 \times \cdots \times G_ν$. Suppose $T$ is invertible on $L^2(G)$. We prove that its inverse is given by $T^{-1}(g) = g*L$, where $L$ is a product kernel or a flag kernel accordingly.

Inverses of Product Kernels and Flag Kernels on Graded Lie Groups

TL;DR

The paper proves that for a direct product of graded Lie groups G=G_1×...×G_ν, a left-invariant singular integral T(f)=f*K with K a product or flag kernel is invertible on L^2(G) only if its inverse is convolution with a kernel of the same type, i.e., T^{-1}(g)=g*L where L is a product or flag kernel accordingly. The authors develop a multi-parameter version of the Christ–Geller a priori estimate by employing right-invariant Sobolev spaces and commuting differential operators to relate T and T^{-1}. Central to the argument are commutator estimates, a Littlewood–Paley decomposition, and a careful analysis of growth and cancellation conditions, enabling the transfer of kernel-class regularity to the inverse. The results extend classical single-parameter inversion theorems to multi-parameter (ν-parameter) settings and broaden the applicability to a larger class of kernels on graded Lie groups, with a parallel treatment for flag kernels via dilations and kernel-cancellation arguments.

Abstract

Let , where is a product kernel or a flag kernel on a direct product of graded Lie groups . Suppose is invertible on . We prove that its inverse is given by , where is a product kernel or a flag kernel accordingly.
Paper Structure (7 sections, 13 theorems, 130 equations)

This paper contains 7 sections, 13 theorems, 130 equations.

Table of Contents

  1. Introduction
  2. Background and Notation
  3. Inversion Theorem for Product Kernels
  4. A multi-parameter a priori estimate
  5. Proof of the A Priori Estimate for Product Kernels
  6. Proof of the Inversion Theorem for Product Kernels
  7. Inversion Theorem for Flag Kernels

Key Result

Theorem 1.1

Let $T$ be a left-invariant singular integral operator given by $T(f) = f*K$, where $K$ is a product kernel (respectively a flag kernel) on a direct product of graded Lie groups $G= G_1 \times \cdots \times G_{\nu}$. If $T$ is invertible as an operator on $L^2(G)$, then its inverse is also of the fo

Theorems & Definitions (40)

  • Theorem 1.1
  • Remark 1.2
  • Definition 2.1
  • Definition 3.1
  • Remark 3.2
  • Definition 3.3
  • Definition 3.4
  • Definition 3.5
  • Remark 3.6
  • Definition 3.7
  • ...and 30 more