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arXiv 0903.4725math.CAmath.FA
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次线性算子在乘积Hardy空间上的有界性及其应用

Boundedness of Sublinear Operators on Product Hardy Spaces and Its Application

Der-Chen Chang, Dachun Yang, Yuan Zhou

备注：
J. Math. Soc. Japan (to appear)
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摘要：

令$p\in(0, 1]$。本文证明了一个次线性算子$T$（它最初定义在具有紧支撑的光滑函数上）可以从乘积Hardy空间$H^p({{\mathbb R}^n\times{\mathbb R}^m})$推广到某个拟Banach空间${\mathcal B}$，当且仅当$T$将所有$(p, 2, s_1, s_2)$ -原子映射到${\mathcal B}$的一致有界元上.这里是$s_1\ge\lfloor n(1/p-1)\rfloor$和$s_2\ge\lfloor m(1/p-1)\rfloor$。通常，$\lfloor n(1/p-1)\rfloor$表示不大于$n(1/p-1)$的最大整数。应用这一结果，证明了由Calderón-Zygmund算子和Lipschitz函数生成的算子从具有某些$p>1$的Lebesgue空间$L^p({{\mathbb R}^n\times{\mathbb R}^m})$或具有某些$p\le1$但接近1的Hardy空间$H^p({{\mathbb R}^n\times{\mathbb R}^m})$到具有某些$q>1$的Lebesgue空间$L^q({{\mathbb R}^n\times{\mathbb R}^m})$的有界性.

英文摘要：

Let $p\in(0, 1]$. In this paper, the authors prove that a sublinear operator $T$ (which is originally defined on smooth functions with compact support) can be extended as a bounded sublinear operator from product Hardy spaces $H^p({{\mathbb R}^n\times{\mathbb R}^m})$ to some quasi-Banach space ${\mathcal B}$ if and only if $T$ maps all $(p, 2, s_1, s_2)$-atoms into uniformly bounded elements of ${\mathcal B}$. Here $s_1\ge\lfloor n(1/p-1)\rfloor$ and $s_2\ge\lfloor m(1/p-1)\rfloor$. As usual, $\lfloor n(1/p-1)\rfloor$ denotes the maximal integer no more than $n(1/p-1)$. Applying this result, the authors establish the boundedness of the commutators generated by Calderón-Zygmund operators and Lipschitz functions from the Lebesgue space $L^p({{\mathbb R}^n\times{\mathbb R}^m})$ with some $p>1$ or the Hardy space $H^p({{\mathbb R}^n\times{\mathbb R}^m})$ with some $p\le1$ but near 1 to the Lebesgue space $L^q({{\mathbb R}^n\times{\mathbb R}^m})$ with some $q>1$.

机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。

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