论文标题
Schatten-Von Neumann类中的Schur乘数
Schur multipliers in Schatten-von Neumann classes
论文作者
论文摘要
我们为Schatten $ P $ -Classes上的Schur乘数$ s_m $ $ S_M $ S_M建立了一个相当出乎意料且简单的标准，该标准解决了Mikael de La Salle提出的猜想。给定的$ 1 <p <\ infty $，一个简单的表格，我们的主要结果读取$ \ mathbf {r}^n \ times \ times \ mathbf {r}^n $矩阵，如下所示。 s_m：s_p \ to s_p \ big \ | _ {\ mathrm {cb}}} \ sillssim \ frac {p^2} {p-1} {p-1} \ sum_ {|γ| \ le [\ frac {n} {2}] +1} \ big \ | | x-y |^{|γ|} \ big \ {\ big | \ partial_x^γm（x，y）\ big | + \ big | \ partial_y^γm（x，y）\ big | \ big \} \ big \ | _ \ infty。$$，以这种形式，它是Hörmander-Mikhlin乘数定理的完整矩阵（非三吉型）扩增，该定理是较低的分数不同订单订单$σ> \ frac n n} $ n} $ n} $ n} $。它在毫无用处地包括了$ S_P $ -Multipliper的Arazy的猜想，并将其扩展到$α$划分的差异。这也导致了$ s_p $ norms的新的Littlewood-Paley特征，并在谐波分析中用于Nilpotent and Sight Simple Simple Lie Group代数。
We establish a rather unexpected and simple criterion for the boundedness of Schur multipliers $S_M$ on Schatten $p$-classes which solves a conjecture proposed by Mikael de la Salle. Given $1 < p < \infty$, a simple form our main result reads for $\mathbf{R}^n \times \mathbf{R}^n$ matrices as follows $$\big\| S_M: S_p \to S_p \big\|_{\mathrm{cb}} \lesssim \frac{p^2}{p-1} \sum_{|γ| \le [\frac{n}{2}] +1} \Big\| |x-y|^{|γ|} \Big\{ \big| \partial_x^γM(x,y) \big| + \big| \partial_y^γM(x,y) \big| \Big\} \Big\|_\infty.$$ In this form, it is a full matrix (nonToeplitz/nontrigonometric) amplification of the Hörmander-Mikhlin multiplier theorem, which admits lower fractional differentiability orders $σ> \frac{n}{2}$ as well. It trivially includes Arazy's conjecture for $S_p$-multipliers and extends it to $α$-divided differences. It also leads to new Littlewood-Paley characterizations of $S_p$-norms and strong applications in harmonic analysis for nilpotent and high rank simple Lie group algebras.
