On bounded energy of convolution of fractal measures

TL;DR

This paper investigates the boundedness of convolution measures involving fractal measures and establishes sharp decay rates for their Fourier transforms, advancing understanding in fractal harmonic analysis.

Contribution

It determines the supremum of energy bounds for convolutions of fractal measures on graphs with curvature, revealing new bounds and decay properties.

Findings

Identifies the supremum of erivatives for convolutions of fractal measures.

Establishes sharp L^6-decay rates for Fourier transforms of measures on curved fractal sets.

Provides new bounds for energy integrals of convoluted fractal measures.

Abstract

For all $s\in[0,1]$ and $t\in(0,s]\cup [2-s,2)$, we find the supremum of numbers $\omega\in(0,2)$ such that $\text{I}_\omega(\mu\ast\sigma) \lesssim 1$, where $\mu$ is any Borel measure on $B(1)$ with $\text{I}_t(\mu)\leq 1$ and $\sigma$ 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 $\sigma$ when $s\in [\frac{2}{3}, 1]$.