论文标题
Hollenbeck-Verbitsky对分析和共分析投影的最佳恒定不平等的猜想
Hollenbeck-Verbitsky conjecture on best constant inequalities for analytic and co-analytic projections
论文作者
论文摘要
\ begin {摘要} 在本文中，我们解决了以这种形式不平等的最佳常数的问题： $ \ | \ big（| p_+f |^s+| p_-f |^s \ big）^{\ frac {\ frac {1} {s}}}} \ | _ {l^p（{\ mathbb {t}}}} \ | f \ | _ {l^p（{\ Mathbb {t}}）}，$$其中$ p_+f $和$ p_+f $和$ p_-f $ deote分析和复杂函数的co-Analyaltic投影$ f \ in l^p（{\ nathbb {t}} $ s $ p \ s $ p \ s $ s $ \ cite {hv.otaa}的Hollenbeck-Verbitsky猜想。我们还证明\\ $ 1 <p \ leq \ frac {4} {3} $和$ s \ s \ leq \ sec^2 \fracπ{2p} $，并确认$ s = \ sec^\ sec^2 \ sec^2 \ sec^2 \fracπ{2p} $是$ proir的pressiul and prestist prolist and pronist。在这个主题中似乎是新的“基本”不平等。 我们表明，这个结果意味着最佳常数不平等现象对$ \ Mathbb {r}^n $上的预测和半空间乘数的预测以及分析性martingales的类似物。还给出了单位磁盘中谐波功能的等值不平等的评论。 \ end {摘要}
\begin{abstract} In this paper we address the problem of finding the best constants in inequalities of the form: $$ \|\big(|P_+f|^s+|P_-f|^s\big)^{\frac{1}{s}}\|_{L^p({\mathbb{T}})}\leq A_{p,s} \|f\|_{L^p({\mathbb{T}})},$$ where $P_+f$ and $P_-f$ denote analytic and co-analytic projection of a complex-valued function $f \in L^p({\mathbb{T}}),$ for $p \geq 2$ and all $s>0$, thus proving Hollenbeck-Verbitsky conjecture from \cite{HV.OTAA}. We also prove the same inequalities for\\ $1<p\leq\frac{4}{3}$ and $s\leq \sec^2\fracπ{2p}$ and confirm that $s=\sec^2\fracπ{2p}$ is the sharp cutoff for $s.$ The proof uses a method of plurisubharmonic minorants and an approach of proving the appropriate "elementary" inequalities that seems to be new in this topic. We show that this result implies best constants inequalities for the projections on the real-line and half-space multipliers on $\mathbb{R}^n$ and an analog for analytic martingales. A remark on an isoperimetric inequality for harmonic functions in the unit disk is also given. \end{abstract}
