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Tame algebra estimates for product and flag kernels on graded Lie groups

Amelia Stokolosa

Abstract

We prove that product kernels and flag kernels on a direct product of graded Lie groups $G_1 \times \cdots \times G_ν$ satisfy so-called \emph{tame algebra estimates}. Tame algebra estimates are central to the study of nonlinear partial differential equations via, for instance, the Nash-Moser inverse function theorem. In addition, the special structure of these estimates generates a new Banach-algebraic proof of an inversion theorem for product kernels and flag kernels.

Tame algebra estimates for product and flag kernels on graded Lie groups

Abstract

We prove that product kernels and flag kernels on a direct product of graded Lie groups satisfy so-called \emph{tame algebra estimates}. Tame algebra estimates are central to the study of nonlinear partial differential equations via, for instance, the Nash-Moser inverse function theorem. In addition, the special structure of these estimates generates a new Banach-algebraic proof of an inversion theorem for product kernels and flag kernels.

Paper Structure

This paper contains 16 sections, 17 theorems, 154 equations.

Table of Contents

  1. Introduction
  2. Main results
  3. Acknowledgments
  4. Background
  5. Product kernels
  6. A Fréchet topology on product kernels
  7. Multi-parameter tame algebra estimate for $\mathcal{P}^{\vec{k}}$
  8. Proof of the tame algebra estimate for $\mathcal{P}^{(k_1, k_2)}$
  9. Single-parameter tame algebra estimate
  10. Multi-parameter tame algebra estimate
  11. Flag kernels
  12. A Fréchet topology on flag kernels
  13. Multi-parameter tame algebra estimate for $\mathcal{F}^{\vec{k}}$
  14. Proof of the tame algebra estimate for $\mathcal{F}^{(k_1, k_2)}$
  15. New proof of an inversion theorem for product kernels and flag kernels
  16. ...and 1 more sections

Key Result

Theorem 1.1

Let $(k_1, k_2) \in \mathbb{Z}^2_{\geq 0}$. Suppose $K, L \in \mathcal{P}^{(k_1, k_2)} (G_1 \times G_2)$. Then, we have where $\mathop{\mathrm{Op}}\nolimits(K)$ denotes the right-invariant operator given by group convolution $\mathop{\mathrm{Op}}\nolimits(K)f= K*f$. The implicit constant depends on $(k_1, k_2) \in \mathbb{Z}^2_{\geq 0}$.

Theorems & Definitions (50)

  • Theorem 1.1: Tame algebra estimate for product kernels
  • Theorem 1.2: Tame algebra estimate for flag kernels
  • Definition 2.1
  • Definition 3.1
  • Remark 3.2
  • Remark 3.3
  • Definition 3.4
  • Remark 3.5
  • Remark 3.6
  • Definition 3.7
  • ...and 40 more