A measure with small support and p-summable Fourier transform

TL;DR

This paper constructs a probability measure supported on a set with zero Hausdorff measure whose Fourier transform is p-summable, advancing understanding of measure support and Fourier analysis.

Contribution

It introduces a measure with small support and a p-summable Fourier transform, linking geometric measure properties with Fourier decay.

Findings

Constructed measure supported on zero Hausdorff measure set

Proved Fourier transform belongs to Lp space

Bridged geometric measure theory and harmonic analysis

Abstract

We construct a probability measure $\mu$ supported on a set of zero $2d/p$-Hausdorff measure such that $\hat{\mu}\in L_{p}(\mathbb{R}^d)$.