AI 中文总结
本文研究多参数马尔钦维奇乘法子的端点估计，证明其最佳性并扩展了一维结果到任意多参数。
AI 中文摘要
本文证明了多参数马尔钦维奇乘法算子的端点估计的精确性。更具体地说，这一结果是更一般定理的推论，该定理适用于多参数$\mathcal R_{2,n}$-乘子，这一类包含所有具有有界$\mathcal{V}_q(\mathbb{R}^{\otimes n})$-变差的乘子，其中$1\le q<2$。$\mathcal R_{2,n}$是本文中引入的多参数推广，是Coifman、Rubio de Francia和Semmes所提出的$\mathcal R_2$-乘子的推广。我们证明了$\mathcal R_{2,n}$-乘子算子局部地将$L\log^{{3(n-1)}/{2}+{1}/{2}}L$映射到$L^{1,\infty}$，并且这个估计是最佳的，将Tao和Wright的一维结果推广到任意多参数。我们还建立了此类乘子算子的$L^p(\mathbb R^n)\to L^p(\mathbb R^n)$算子范数在$p \to 1^+$时的精确界$O((p')^{{3n}/{2}})$。我们对$L\log^{{3(n-1)}/{2}+{1}/{2}}L$到$L^{1,\infty}$结果的证明结合了乘子的向量值端点估计与通过对偶从Chang-Wilson-Wolff不等式获得的隐含的平方函数对$L\log^{\sigma/2}L$的刻画。后者在每一步迭代中产生辅助代理函数，这些函数被输入到中间的一维向量值弱$(1,1)$估计中。
英文摘要
In this paper we prove sharp endpoint estimates for multiparameter Marcinkiewicz multiplier operators. More precisely, this result is a consequence of a more general theorem for multiparameter $\mathcal R_{2,n}$-multipliers, a class that contains all multipliers of bounded $\mathcal{V}_q(\mathbb{R}^{\otimes n})$-variation for $1\le q<2$. The class $\mathcal R_{2,n}$ is a multiparameter generalization, introduced in this paper, of the $\mathcal R_2$-multipliers of Coifman, Rubio de Francia, and Semmes. We show that $\mathcal R_{2,n}$-multiplier operators locally map $L\log^{{3(n-1)}/{2}+{1}/{2}}L$ into $L^{1,\infty}$, and that this estimate is best possible, extending the corresponding one-parameter result of Tao and Wright to arbitrarily many parameters. We also establish the sharp bound $O((p')^{{3n}/{2}})$ for the $L^p(\mathbb R^n)\to L^p(\mathbb R^n)$ operator norms of such multiplier operators as $p \to 1^+$. The proof of our $L\log^{{3(n-1)}/{2}+{1}/{2}}L$-to-$L^{1,\infty}$ result combines a vector-valued endpoint estimate for the multipliers with an implicit square function characterization of $L\log^{σ/2}L$, obtained via duality from the Chang-Wilson-Wolff inequality. The latter produces, at each iterative step, auxiliary proxy functions that are fed into an intermediate one-parameter vector-valued weak-$(1,1)$ estimate.
Comments32 pages. Submitted for publication
