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| Formato: | Preprint |
| Publicado: |
2024
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| Acceso en línea: | https://arxiv.org/abs/2401.17867 |
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| author | Orponen, Tuomas Puliatti, Carmelo Pyörälä, Aleksi |
| author_facet | Orponen, Tuomas Puliatti, Carmelo Pyörälä, Aleksi |
| contents | Let $s \in [0,1]$ and $t \in [0,\min\{3s,s + 1\})$. Let $σ$ be a Borel measure supported on the parabola $\mathbb{P} = \{(x,x^{2}) : x \in [-1,1]\}$ satisfying the $s$-dimensional Frostman condition $σ(B(x,r)) \leq r^{s}$. Answering a question of the first author, we show that there exists an exponent $p = p(s,t) \geq 1$ such that $$\|\hatσ\|_{L^{p}(B(R))} \leq C_{s,t}R^{(2 - t)/p}, \qquad R \geq 1.$$ Moreover, when $s \geq 2/3$ and $t \in [0,s + 1)$, the previous inequality is true for $p \geq 6$.
We also obtain the following fractal geometric counterpart of the previous results. If $K \subset \mathbb{P}$ is a Borel set with $\dim_{\mathrm{H}} K = s \in [0,1]$, and $n \geq 1$ is an integer, then $$ \dim_{\mathrm{H}}(nK) \geq \min\{3s - s \cdot 2^{-(n - 2)},s + 1\}.$$ |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2401_17867 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | On Fourier transforms of fractal measures on the parabola Orponen, Tuomas Puliatti, Carmelo Pyörälä, Aleksi Classical Analysis and ODEs Metric Geometry 28A80, 42B10, 11B30 Let $s \in [0,1]$ and $t \in [0,\min\{3s,s + 1\})$. Let $σ$ be a Borel measure supported on the parabola $\mathbb{P} = \{(x,x^{2}) : x \in [-1,1]\}$ satisfying the $s$-dimensional Frostman condition $σ(B(x,r)) \leq r^{s}$. Answering a question of the first author, we show that there exists an exponent $p = p(s,t) \geq 1$ such that $$\|\hatσ\|_{L^{p}(B(R))} \leq C_{s,t}R^{(2 - t)/p}, \qquad R \geq 1.$$ Moreover, when $s \geq 2/3$ and $t \in [0,s + 1)$, the previous inequality is true for $p \geq 6$. We also obtain the following fractal geometric counterpart of the previous results. If $K \subset \mathbb{P}$ is a Borel set with $\dim_{\mathrm{H}} K = s \in [0,1]$, and $n \geq 1$ is an integer, then $$ \dim_{\mathrm{H}}(nK) \geq \min\{3s - s \cdot 2^{-(n - 2)},s + 1\}.$$ |
| title | On Fourier transforms of fractal measures on the parabola |
| topic | Classical Analysis and ODEs Metric Geometry 28A80, 42B10, 11B30 |
| url | https://arxiv.org/abs/2401.17867 |