On the dimension-free control of higher order truncated Riesz transforms by higher order Riesz transforms
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| Format: | Preprint |
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2025
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| _version_ | 1866916034093514752 |
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| author | Kucharski, Maciej Kwaśnicki, Mateusz Wróbel, Błażej |
| author_facet | Kucharski, Maciej Kwaśnicki, Mateusz Wróbel, Błażej |
| contents | Fix a positive integer $k$. Let $R_k$ be a higher order Riesz transform of order $k$ on $\mathbb{R}^d$ and let $R_k^t,$ $t>0,$ be the corresponding truncated Riesz transform. We study the relation between $\|R_k f\|_{L^p(\mathbb{R}^d)}$ and $\|R_k^t f\|_{L^p(\mathbb{R}^d)}$ for $p=1$, $p=\infty,$ and $p=2.$ We do this by analyzing the factorization operator $M_k^t$ defined by the relation $R_k^t=M_k^t R_k.$ The operator $M_k^t$ is a convolution operator associated with an $L^1$ radial kernel $b_{k,d}^t(x)=t^{-d}b_{k,d}(x/t),$ where $b_{k,d}(x):=b_{k,d}^1(x).$
We prove that $b_{k,d} \ge 0$ only for $k=1,2.$ We also show that for fixed $k\ge 3$, \[ \lim_{d\to \infty}\|b_{k,d}\|_{L^1(\mathbb{R}^d)}=\infty. \]
This contrasts with the cases $k=1,2$, where it is known that $\|b_{k,d}\|_{L^1(\mathbb{R}^d)}=1$. Finally, we show that for any positive integer $k$, the Fourier transform of $b_{k,d}$ is bounded in absolute value by $1.$ This implies the contractive estimate \[ \|R_k^t f\|_{L^2(\mathbb{R}^d)}\le \|R_k f\|_{L^2(\mathbb{R}^d)} \] and an analogous estimate for general singular integrals with smooth kernels for radial input functions $f.$ |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2507_17510 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | On the dimension-free control of higher order truncated Riesz transforms by higher order Riesz transforms Kucharski, Maciej Kwaśnicki, Mateusz Wróbel, Błażej Classical Analysis and ODEs Functional Analysis 42B25, 42B20, 42B15 Fix a positive integer $k$. Let $R_k$ be a higher order Riesz transform of order $k$ on $\mathbb{R}^d$ and let $R_k^t,$ $t>0,$ be the corresponding truncated Riesz transform. We study the relation between $\|R_k f\|_{L^p(\mathbb{R}^d)}$ and $\|R_k^t f\|_{L^p(\mathbb{R}^d)}$ for $p=1$, $p=\infty,$ and $p=2.$ We do this by analyzing the factorization operator $M_k^t$ defined by the relation $R_k^t=M_k^t R_k.$ The operator $M_k^t$ is a convolution operator associated with an $L^1$ radial kernel $b_{k,d}^t(x)=t^{-d}b_{k,d}(x/t),$ where $b_{k,d}(x):=b_{k,d}^1(x).$ We prove that $b_{k,d} \ge 0$ only for $k=1,2.$ We also show that for fixed $k\ge 3$, \[ \lim_{d\to \infty}\|b_{k,d}\|_{L^1(\mathbb{R}^d)}=\infty. \] This contrasts with the cases $k=1,2$, where it is known that $\|b_{k,d}\|_{L^1(\mathbb{R}^d)}=1$. Finally, we show that for any positive integer $k$, the Fourier transform of $b_{k,d}$ is bounded in absolute value by $1.$ This implies the contractive estimate \[ \|R_k^t f\|_{L^2(\mathbb{R}^d)}\le \|R_k f\|_{L^2(\mathbb{R}^d)} \] and an analogous estimate for general singular integrals with smooth kernels for radial input functions $f.$ |
| title | On the dimension-free control of higher order truncated Riesz transforms by higher order Riesz transforms |
| topic | Classical Analysis and ODEs Functional Analysis 42B25, 42B20, 42B15 |
| url | https://arxiv.org/abs/2507.17510 |