论文标题
一类Littlewood-Paley操作员的尖锐渐近估计值
Sharp asymptotic estimates for a class of Littlewood-Paley operators
论文作者
论文摘要
众所周知，针对所有$ 1 <p <\ iffty $的$ l^p（\ mathbb {r}）$，在$ l^p（\ mathbb {r}）$上形成的利特伍德 - 帕利操作员有限。在此注释中，显示出$$ \ | s _ {\ Mathcal {i} _ {e_2}}} \ | _ {l^p（\ Mathbb {r}）\ rightArrow l^p（\ Mathbb {r}）} \ sim（p-1） $ s _ {\ MATHCAL {i} _ {e_2}} $表示相对于二阶lace lacary SET $ e_2 = \ pm（2^k -2^l）形成的经典的Littlewood -paley操作员：
It is well-known that Littlewood-Paley operators formed with respect to lacunary sets of finite order are bounded on $L^p (\mathbb{R})$ for all $1<p<\infty$. In this note it is shown that $$ \| S_{\mathcal{I}_{E_2}} \|_{L^p (\mathbb{R}) \rightarrow L^p (\mathbb{R})} \sim (p-1)^{-2} \quad (p \rightarrow 1^+) ,$$ where $S_{\mathcal{I}_{E_2}}$ denotes the classical Littlewood-Paley operator formed with respect to the second order lacunary set $ E_2 = \{ \pm ( 2^k - 2^l ) : k,l \in \mathbb{Z} \text{ with } k>l \} $.
