$L^p\to L^q$ estimates for Stein's spherical maximal operators
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arXiv
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| Autori principali: | , , , |
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| Natura: | Preprint |
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2025
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| _version_ | 1866910824547745792 |
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| author | Liu, Naijia Shen, Minxing Song, Liang Yan, Lixin |
| author_facet | Liu, Naijia Shen, Minxing Song, Liang Yan, Lixin |
| contents | In this article we consider a modification of the Stein's spherical maximal operator of complex order $α$ on ${\mathbb R^n}$: $$ {\mathfrak M}^α_{[1,2]} f(x) =\sup\limits_{t\in [1,2]} \big| {1\over Γ(α) } \int_{|y|\leq 1} \left(1-|y|^2 \right)^{α-1} f(x-ty) dy\big|. $$ We show that when $n\geq 2$, suppose $\|{\mathfrak M}^α_{[1,2]} f \|_{L^q({\mathbb R^n})} \leq C\|f \|_{L^p({\mathbb R^n})}$ holds for some $α\in \mathbb{C}$, $p,q\geq1$, then we must have that $q\geq p$ and $${\rm Re}\,α\geq σ_n(p,q):=\max\left\{\frac{1}{p}-\frac{n}{q},\ \frac{n+1}{2p}-\frac{n-1}{2}\left(\frac{1}{q}+1\right),\frac{n}{p}-n+1\right\}.$$
Conversely, we show that ${\mathfrak M}^α_{[1,2]}$ is bounded from $L^p({\mathbb R^n})$ to $L^q({\mathbb R^n})$ provided that $q\geq p$ and ${\rm Re}\,α>σ_2(p,q)$ for $n=2$; and ${\rm Re}\,α>\max\left\{σ_n(p,q), 1/(2p)- (n-2)/(2q) -(n-1)/4\right\}$ for $n>2$. The range of $α,p$ and $q$ is almost optimal in the case either $n=2$, or $α=0$, or $(p,q)$ lies in some regions for $n>2$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2502_09030 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | $L^p\to L^q$ estimates for Stein's spherical maximal operators Liu, Naijia Shen, Minxing Song, Liang Yan, Lixin Classical Analysis and ODEs In this article we consider a modification of the Stein's spherical maximal operator of complex order $α$ on ${\mathbb R^n}$: $$ {\mathfrak M}^α_{[1,2]} f(x) =\sup\limits_{t\in [1,2]} \big| {1\over Γ(α) } \int_{|y|\leq 1} \left(1-|y|^2 \right)^{α-1} f(x-ty) dy\big|. $$ We show that when $n\geq 2$, suppose $\|{\mathfrak M}^α_{[1,2]} f \|_{L^q({\mathbb R^n})} \leq C\|f \|_{L^p({\mathbb R^n})}$ holds for some $α\in \mathbb{C}$, $p,q\geq1$, then we must have that $q\geq p$ and $${\rm Re}\,α\geq σ_n(p,q):=\max\left\{\frac{1}{p}-\frac{n}{q},\ \frac{n+1}{2p}-\frac{n-1}{2}\left(\frac{1}{q}+1\right),\frac{n}{p}-n+1\right\}.$$ Conversely, we show that ${\mathfrak M}^α_{[1,2]}$ is bounded from $L^p({\mathbb R^n})$ to $L^q({\mathbb R^n})$ provided that $q\geq p$ and ${\rm Re}\,α>σ_2(p,q)$ for $n=2$; and ${\rm Re}\,α>\max\left\{σ_n(p,q), 1/(2p)- (n-2)/(2q) -(n-1)/4\right\}$ for $n>2$. The range of $α,p$ and $q$ is almost optimal in the case either $n=2$, or $α=0$, or $(p,q)$ lies in some regions for $n>2$. |
| title | $L^p\to L^q$ estimates for Stein's spherical maximal operators |
| topic | Classical Analysis and ODEs |
| url | https://arxiv.org/abs/2502.09030 |