On the Fourier transform of measures in Besov spaces

TL;DR

This work addresses the decay of the Fourier transform of Riesz potentials $I_\alpha\mu$ when the measure $\mu$ lies in negative Besov spaces, specifically $\mu\in M_b(\mathbb{R}^d)\cap \dot{B}^{\beta-d}_{\infty,\infty}(\mathbb{R}^d)$ with $\beta\in(0,d]$ and $\alpha$ in $(\frac{d-\beta}{2}, d-\frac{\beta}{2})$. The authors develop a unified Besov–Morrey framework showing a quantitative weak-$L^p$ bound for $\widehat{I_\alpha\mu}$ with $p=\frac{2d}{2\alpha+\beta}$, namely $\|\widehat{I_\alpha\mu}\|_{L^{p,\infty}} \le C \|\mu\|_{M_b}^{1/2} (\sup_{t>0} t^{(d-\beta)/2}\|p_t*\mu\|_\infty)^{1/2}$. This includes Frostman measures and recovers BV/sobolev-related corollaries, extending Herz–Ko–Lee. The paper proves sharpness of the exponents via probabilistic and explicit constructions, underscoring the optimality of the endpoint estimates and highlighting the natural Morrey–Besov perspective on Fourier decay problems.

Abstract

We prove quantitative estimates for the decay of the Fourier transform of the Riesz potential of measures that are in homogeneous Besov spaces of negative exponent: \begin{align*} \|\widehat{I_αμ}\|_{L^{p, \infty}} \leq C \|μ\|_{M_b}^{\frac{1}{2}}\left(\sup_{t>0} t^{\frac{d-β}{2}}\|p_{t}\ast μ\|_{\infty}\right)^{\frac{1}{2}}, \end{align*} where $p=\frac{2d}{2α+β}$ with $β\in (0,d)$ and $I_αμ$ is the Riesz potential of $μ$ of order $α\in ((d-β)/2,d-β/2)$. Our results are naturally applicable to the Morrey space $\mathcal{M}^β$, including for example the Frostman measure $μ_K$ of any compact set $K$ with $0<\mathcal{H}^β(K)<+\infty$ for some $β\in (0,d]$. When $μ=Dχ_E$ for $χ_E \in \operatorname*{BV}(\mathbb{R}^d)$, $α=1$, and $β=d-1$, our results extend the work of Herz and Ko--Lee. We provide examples which show the sharpness of our results.

Theorems & Definitions (20)

• Theorem 1.1: Ko--Lee
• Theorem 1.2
• Remark 1.3
• Corollary 1.4
• Corollary 1.5
• Theorem 1.6
• Theorem 1.7
• Lemma 2.1
• proof
• Lemma 2.2