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Surjectivity of convolution operators on harmonic $NA$ groups

Effie Papageorgiou

Abstract

Let $μ$ be a radial compactly supported distribution on a harmonic $NA$ group. We prove that the right convolution operator $c_μ:f \mapsto f* μ$ maps the space of smooth $\mathfrak{v}$-radial functions onto itself if and only if the spherical Fourier transform $\widetildeμ(λ)$, $λ\in \mathbb{C}$, is slowly decreasing. As an application, we prove that certain averages over spheres are surjective on the space of smooth $\mathfrak{v}$-radial functions.

Surjectivity of convolution operators on harmonic $NA$ groups

Abstract

Let be a radial compactly supported distribution on a harmonic group. We prove that the right convolution operator maps the space of smooth -radial functions onto itself if and only if the spherical Fourier transform , , is slowly decreasing. As an application, we prove that certain averages over spheres are surjective on the space of smooth -radial functions.

Paper Structure

This paper contains 12 sections, 149 equations.

Table of Contents

  1. Introduction and statement of the results
  2. Preliminaries
  3. $\mathfrak{v}$-radial functions.
  4. Derivatives.
  5. Fourier analysis
  6. Radon transform
  7. Paley-Wiener type results
  8. Paley-Wiener type results for functions.
  9. Paley-Wiener type theorems for distributions
  10. Slow decrease implies surjectivity
  11. Surjectivity implies slow decrease
  12. Mean value operators

Theorems & Definitions (8)

  • proof
  • proof
  • proof
  • proof
  • proof
  • proof : Proof of Theorem \ref{['thm: converse']}
  • proof : Proof of Proposition \ref{['prop: surj MVO']}
  • proof