On the Fourier transform of measures in Besov spaces
Fuente:
arXiv
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| Autori principali: | , , |
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| Natura: | Preprint |
| Pubblicazione: |
2025
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| _version_ | 1866916663549493248 |
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| author | Basak, Riju Spector, Daniel Stolyarov, Dmitriy |
| author_facet | Basak, Riju Spector, Daniel Stolyarov, Dmitriy |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2503_21075 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | On the Fourier transform of measures in Besov spaces Basak, Riju Spector, Daniel Stolyarov, Dmitriy Functional Analysis Classical Analysis and ODEs 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. |
| title | On the Fourier transform of measures in Besov spaces |
| topic | Functional Analysis Classical Analysis and ODEs |
| url | https://arxiv.org/abs/2503.21075 |