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The Boundedness of the (Sub)bilinear Maximal Function Along “Non-flat” Smooth Curves
Journal of Fourier Analysis and Applications ( IF 1.2 ) Pub Date : 2020-08-03 , DOI: 10.1007/s00041-020-09770-6
Alejandra Gaitan , Victor Lie

Let \(\mathcal {NF}\) be the class of smooth non-flat curves near the origin and near infinity introduced in Lie (Am J Math 137(2):313–363, 2015) and let \(\gamma \in \mathcal {NF}\). We show—via a unifying approach relative to the corresponding bilinear Hilbert transform \(H_{\Gamma }\)—that the (sub)bilinear maximal function along curves \(\Gamma =(t,-\gamma (t))\) defined as$$\begin{aligned} M_\Gamma (f,g)(x):=\sup \limits _{\varepsilon >0} \frac{1}{2\varepsilon } \int _{-\varepsilon }^\varepsilon |f(x-t)g(x+\gamma (t))|dt \end{aligned}$$is bounded from \(L^p(\mathbb {R})\times L^{q}(\mathbb {R})\rightarrow L^r(\mathbb {R})\) for all pq and r Hölder indices, i.e. \(\frac{1}{p}+\frac{1}{q}=\frac{1}{r}\), with \(1<p,\,q\le \infty \) and \(1\le r\le \infty \). This is the maximal boundedness range for \(M_{\Gamma }\), that is, our result is sharp.

中文翻译:

(亚)双线性极大函数沿“非平坦”光滑曲线的有界性

\(\ mathcal {NF} \)为Lie(Am J Math 137(2):313–363,2015)中引入的接近原点和接近无穷大的平滑非平坦曲线的类,并设\(\ gamma \在\ mathcal {NF} \)中。通过相对于相应的双线性希尔伯特变换\(H _ {\ Gamma} \)的统一方法,我们证明了沿着曲线\(\ Gamma =(t,-\ gamma(t))\ )定义为$$ \ begin {aligned} M_ \ Gamma(f,g)(x):= \ sup \ limits _ {\ varepsilon> 0} \ frac {1} {2 \ varepsilon} \ int _ {-\ varepsilon} ^ \ varepsilon | f(xt)g(x + \ gamma(t))| dt \ end {aligned} $$\(L ^ p(\ mathbb {R})\ times L ^ {q}的限制(\ mathbb {R})\ rightarrow L ^ r(\ mathbb {R})\)对于所有p,  q- [R赫尔德指数,即\(\压裂{1} {P} + \压裂{1} {Q} = \压裂{1} {R} \),用\(1 <P,\,Q \文件\ infty \)\(1 \ le r \ le \ infty \)。这是\(M _ {\ Gamma} \)的最大有界范围,也就是说,我们的结果是Sharp
更新日期:2020-08-03
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