论文标题
通过波数据包估计的弱型Carleson定理
The weak-type Carleson theorem via wave packet estimates
论文作者
论文摘要
我们证明了Carleson Maximal Partial founier sum运算符的稀疏$（p，1）$ - norms实际上是弱$ l^{p} $ norms，实际上是$ \ sillsim（p-1）^{ - 1} $ as $ p \ to 1^+$。这是对Carleson-Hunt定理的改进，在该定理中，对于受限制的弱$ $ l^p $类型规范，获得了增长顺序相同的上限，并且在我们结果之前是最强的定量界限。此外，我们稀疏的$（P，1）$ - 规范限制了端点$ p = 1 $的新结果。特别是，我们获得了来自加权Arias de Reyna空间的傅立叶级系列功能$ QA _ {\ infty}（w）$，其中包含加权的Antonov空间$ l \ log l \ log l \ log \ log \ log \ log \ log \ log l（\ mathbb t; w）几乎在a_1 $中的$ w \ in a_1 $中的$ w \ novery novery everyeshia这是Antonov和Arias de Reyna的结果的扩展，那里必须是Lebesgue的衡量标准。 我们治疗的骨干是一种新的，较清晰的接近$ l^1 $ carleson，将定理嵌入了调制不变的波数据包变换。 Carleson嵌入的证据依赖于新开发的平滑多频分解，该分解在端点$ p = 1 $附近，胜过过去作品的抽象希尔伯特太空方法，包括纳扎罗夫（Nazarov），奥伯林（Oberlin）和蒂（Thiele）的开创性作品。作为应用程序的另一个示例，我们获得了由于Culiuc，OU和第一作者而导致双线性希尔伯特转换的稀疏界限的量化版本。
We prove that the weak-$L^{p}$ norms, and in fact the sparse $(p,1)$-norms, of the Carleson maximal partial Fourier sum operator are $\lesssim (p-1)^{-1}$ as $p\to 1^+$. This is an improvement on the Carleson-Hunt theorem, where the same upper bound on the growth order is obtained for the restricted weak-$L^p$ type norm, and which was the strongest quantitative bound prior to our result. Furthermore, our sparse $(p,1)$-norms bound imply new and stronger results at the endpoint $p=1$. In particular, we obtain that the Fourier series of functions from the weighted Arias de Reyna space $ QA_{\infty}(w) $, which contains the weighted Antonov space $L\log L\log\log\log L(\mathbb T; w)$, converge almost everywhere whenever $w\in A_1$. This is an extension of the results of Antonov and Arias De Reyna, where $w$ must be Lebesgue measure. The backbone of our treatment is a new, sharply quantified near-$L^1$ Carleson embedding theorem for the modulation-invariant wave packet transform. The proof of the Carleson embedding relies on a newly developed smooth multi-frequency decomposition which, near the endpoint $p=1$, outperforms the abstract Hilbert space approach of past works, including the seminal one by Nazarov, Oberlin and Thiele. As a further example of application, we obtain a quantified version of the family of sparse bounds for the bilinear Hilbert transforms due to Culiuc, Ou and the first author.
