The Boundedness of the Bilinear Fractional Integrals along Curves
Fuente:
arXiv
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| Autori principali: | , , |
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| Natura: | Preprint |
| Pubblicazione: |
2024
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| _version_ | 1866915463913537536 |
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| author | Li, Junfeng Yu, Haixia Zhao, Minqun |
| author_facet | Li, Junfeng Yu, Haixia Zhao, Minqun |
| contents | In this paper, for general curves $(t,γ(t))$ satisfying some suitable curvature conditions, we obtain some $L^p(\mathbb{R})\times L^q(\mathbb{R}) \rightarrow L^r(\mathbb{R})$ estimates for the bilinear fractional integrals $H_{α,γ}$ along the curves $(t,γ(t))$, where
$$H_{α,γ}(f,g)(x):=\int_{0}^{\infty}f(x-t)g(x-γ(t))\,\frac{\textrm{d}t}{t^{1-α}}$$
and $α\in (0,1)$. At the same time, we also establish an almost sharp Hardy-Littlewood-Sobolev inequality, i.e., the $L^p(\mathbb{R})\rightarrow L^q(\mathbb{R})$ estimate, for the fractional integral operators $I_{α,γ}$ along the curves $(t,γ(t))$, where $$I_{α,γ}f(x):=\int_{0}^{\infty}\left|f(x-γ(t))\right|\,\frac{\textrm{d}t}{t^{1-α}}.$$ |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2411_14830 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | The Boundedness of the Bilinear Fractional Integrals along Curves Li, Junfeng Yu, Haixia Zhao, Minqun Classical Analysis and ODEs In this paper, for general curves $(t,γ(t))$ satisfying some suitable curvature conditions, we obtain some $L^p(\mathbb{R})\times L^q(\mathbb{R}) \rightarrow L^r(\mathbb{R})$ estimates for the bilinear fractional integrals $H_{α,γ}$ along the curves $(t,γ(t))$, where $$H_{α,γ}(f,g)(x):=\int_{0}^{\infty}f(x-t)g(x-γ(t))\,\frac{\textrm{d}t}{t^{1-α}}$$ and $α\in (0,1)$. At the same time, we also establish an almost sharp Hardy-Littlewood-Sobolev inequality, i.e., the $L^p(\mathbb{R})\rightarrow L^q(\mathbb{R})$ estimate, for the fractional integral operators $I_{α,γ}$ along the curves $(t,γ(t))$, where $$I_{α,γ}f(x):=\int_{0}^{\infty}\left|f(x-γ(t))\right|\,\frac{\textrm{d}t}{t^{1-α}}.$$ |
| title | The Boundedness of the Bilinear Fractional Integrals along Curves |
| topic | Classical Analysis and ODEs |
| url | https://arxiv.org/abs/2411.14830 |