On high dimensional maximal functions associated to Gaussians, balls, and spheres
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arXiv
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| Formato: | Preprint |
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2025
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| _version_ | 1866914042350665728 |
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| author | Ciccone, Valentina Wróbel, Błażej |
| author_facet | Ciccone, Valentina Wróbel, Błażej |
| contents | We prove that for each $p\in (1,\infty),$ the norms on $L^p(\mathbb{R}^d)$ of the maximal functions associated to Gaussians (heat semigroup), balls (Hardy-Littlewood averages), and spheres (spherical averages) converge, as the dimension $d\to \infty,$ to the same quantity $λ(p)$. This is derived from the fact that the norms on $L^2(\mathbb{R}^d)$ of the maximal functions corresponding to the differences of Gaussian, ball, and spherical averages converge to zero with the dimension $d.$ The fact is proved with the aid of estimates for Fourier multiplier symbols corresponding to these averages, a general principle that allows us to control the norm of a maximal function corresponding to a Fourier multiplier operator by the norm of the multiplier operator itself, and concentration properties of high dimensional Gaussian random vectors. Moreover, relying on the properties of the $d$-dimensional maximal function for the heat semigroup $\mathcal{G}_\ast^d$, we show that $λ(p)$ satisfies $$ \frac25\frac{p}{p-1}\le\|\mathcal{G}_\ast^1\|_{L^p(\mathbb{R})\rightarrow L^p(\mathbb{R})}\le λ(p)\le \frac{p}{p-1}. $$ In particular, to obtain the middle inequality we show that the norms on $L^p(\mathbb{R}^d)$ of the maximal function for the heat semigroup are non-decreasing in $d.$ |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2509_13791 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | On high dimensional maximal functions associated to Gaussians, balls, and spheres Ciccone, Valentina Wróbel, Błażej Classical Analysis and ODEs 42B25 We prove that for each $p\in (1,\infty),$ the norms on $L^p(\mathbb{R}^d)$ of the maximal functions associated to Gaussians (heat semigroup), balls (Hardy-Littlewood averages), and spheres (spherical averages) converge, as the dimension $d\to \infty,$ to the same quantity $λ(p)$. This is derived from the fact that the norms on $L^2(\mathbb{R}^d)$ of the maximal functions corresponding to the differences of Gaussian, ball, and spherical averages converge to zero with the dimension $d.$ The fact is proved with the aid of estimates for Fourier multiplier symbols corresponding to these averages, a general principle that allows us to control the norm of a maximal function corresponding to a Fourier multiplier operator by the norm of the multiplier operator itself, and concentration properties of high dimensional Gaussian random vectors. Moreover, relying on the properties of the $d$-dimensional maximal function for the heat semigroup $\mathcal{G}_\ast^d$, we show that $λ(p)$ satisfies $$ \frac25\frac{p}{p-1}\le\|\mathcal{G}_\ast^1\|_{L^p(\mathbb{R})\rightarrow L^p(\mathbb{R})}\le λ(p)\le \frac{p}{p-1}. $$ In particular, to obtain the middle inequality we show that the norms on $L^p(\mathbb{R}^d)$ of the maximal function for the heat semigroup are non-decreasing in $d.$ |
| title | On high dimensional maximal functions associated to Gaussians, balls, and spheres |
| topic | Classical Analysis and ODEs 42B25 |
| url | https://arxiv.org/abs/2509.13791 |