论文标题
双线性希尔伯特（Hilbert
Bilinear Hilbert Transforms and (Sub)Bilinear Maximal Functions along Convex Curves
论文作者
论文摘要
在本文中，我们确定$ l^p（\ mathbb {r}）\ times l^q（\ mathbb {r}）\ rightarrow l^r（\ mathbb {r}）$ the birinear hilbert hilbert transform transform $h_γ（f，g）沿CONVEX curve $γ$γ$γ$γ$γ$γ$γ$γ$γ$γ $h_γ（f，g）（x）：= \ mathrm {p。\，v。} \ int _ { - \ infty}^{\ infty} f（x-t）g（x-γ（x-γ（t）） $ \ frac {1} {p}+\ frac {1} {q} = \ frac {1} {r} $和$ r> \ frac {1} {2} {2} $，$ p> 1 $和$ q> 1 $。此外，相同的$ l^p（\ mathbb {r}）\ times l^q（\ mathbb {r}）\ rightArrow l^r（\ Mathbb {r}）$界属性具有相应的（sub）双线性功能的相应（sub）最大函数$m_γ（f，g）$m_γ$ $m_γ$ $ $m_γ（ $M_γ（f，g）（x）：= \ sup _ {\ varepsilon> 0} \ frac {1} {2 \ varepsilon} \ int \ int _ { - \ \ varepsilon}^{\ varepsilon}^{\ varepsilon} \，\ textrm {d}t。$$
In this paper, we determine the $L^p(\mathbb{R})\times L^q(\mathbb{R})\rightarrow L^r(\mathbb{R})$ boundedness of the bilinear Hilbert transform $H_γ(f,g)$ along a convex curve $γ$ $$H_γ(f,g)(x):=\mathrm{p.\,v.}\int_{-\infty}^{\infty}f(x-t)g(x-γ(t)) \,\frac{\textrm{d}t}{t},$$ where $p$, $q$, and $r$ satisfy $\frac{1}{p}+\frac{1}{q}=\frac{1}{r}$, and $r>\frac{1}{2}$, $p>1$, and $q>1$. Moreover, the same $L^p(\mathbb{R})\times L^q(\mathbb{R})\rightarrow L^r(\mathbb{R})$ boundedness property holds for the corresponding (sub)bilinear maximal function $M_γ(f,g)$ along a convex curve $γ$ $$M_γ(f,g)(x):=\sup_{\varepsilon>0}\frac{1}{2\varepsilon}\int_{-\varepsilon}^{\varepsilon}|f(x-t)g(x-γ(t))| \,\textrm{d}t.$$
