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On bounded energy of convolution of fractal measures

Guangzeng Yi

Abstract

For all $s\in[0,1]$ and $t\in(0,s]\cup [2-s,2)$, we find the supremum of numbers $ω\in(0,2)$ such that $\text{I}_ω(μ\astσ) \lesssim 1$, where $μ$ is any Borel measure on $B(1)$ with $\text{I}_t(μ)\leq 1$ and $σ$ is any $(s,1)$-Frostman measure on a $C^2$-graph with non-zero curvature. As an application, we use this to show the sharp $L^6$-decay of Fourier transform of $σ$ when $s\in [\frac{2}{3}, 1]$.

On bounded energy of convolution of fractal measures

Abstract

For all and , we find the supremum of numbers such that , where is any Borel measure on with and is any -Frostman measure on a -graph with non-zero curvature. As an application, we use this to show the sharp -decay of Fourier transform of when .

Paper Structure

This paper contains 13 sections, 15 theorems, 107 equations.

Table of Contents

  1. Introduction
  2. Examples
  3. Application
  4. Proof ideas
  5. Outline of the paper
  6. Preliminary
  7. Notations and definitions
  8. Reduction to incidence estimates
  9. Proof of Theorem \ref{['main']} for $\mathbf{t\in[2-s,s+1]}$
  10. Sobolev estimates
  11. Estimating incidences
  12. Proof of Theorem \ref{['main']} for $\mathbf{t\in(0,s]}$
  13. An incidence estimate

Key Result

Theorem 1.4

For $s\in[0,1]$ and $t\in(0,2)$, we have Moreover, if $\textup{I}_t(\mu)<+\infty$ and $\sigma$ is an $(s,C_\sigma)$-Frostman measure on $\Gamma$, then for $\mathfrak{f}(s,t)=s+t$ or $\mathfrak{f}(s,t)=s+1$, there exists a constant $C=C(\psi, s,t,\epsilon)>0$ such that

Theorems & Definitions (42)

  • Definition 1.3
  • Theorem 1.4
  • Example 1.8: Case $\mathbf{t\in(0, s]}$ and $\mathbf{s\in(0, 1]}$
  • Example 1.9: Case $\mathbf{t\in[2-s, s+1]}$ and $\mathbf{s\in[\tfrac{1}{2}, 1]}$
  • Example 1.10: Case $\mathbf{t\in[3s, s+1]}$ and $\mathbf{s\in [0,\tfrac{1}{2}]}$
  • Example 1.14: Case $\mathbf{t\in[s+1, 2)}$ and $\mathbf{s\in[0, 1)}$
  • Theorem 1.15
  • proof : Proof of Theorem \ref{['main2']}
  • Remark 1.19
  • Theorem 1.21
  • ...and 32 more