arXiv 1107.43472011-07-21math.FAmath.CA

非积分算子的一个T（1）定理

A T(1)-Theorem for non-integral operators

Dorothee Frey, Peer Christian Kunstmann

备注：: 51 pages

摘要：

设$X$是齐型空间，$L$是$L^2(X)$上具有有界全纯泛函演算的扇形算子.我们假设半群$\{e^{-tL}\}_{t>0}$满足Davies-Gaffney估计。与$L$相关联的是身份的某些近似。我们称一个算子$T$为非整算子，如果包含$T$的复合算子和这些近似满足一定的加权范数估计。Davies-Gaffney估计和加权范数估计一起代替了Calderón-Zygmund理论中$T$上通常的核估计。本文证明了在附加的假设下，与$L$相关的垂直Littlewood-Paley-Stein平方函数在$L^2(X)$上有界，非积分算子$T$在$L^2(X)$上有界当且仅当$T(1) \in BMO_L(X)$和$T^{\ast}(1) \in BMO_{L^{\ast}}(X)$ .这里，$BMO_L(X)$和$BMO_{L^{\ast}}(X)$表示最近定义的与$L$相关联的$BMO(X)$空间，其推广了John和Nirenberg的空间$BMO(X)$。推广了F. Bernicot提出的T（1）-定理的第二个版本，在较弱的非对角估计下，给出了他提出的一个问题的肯定答案。作为应用，我们证明了与$L$相关的仿积算子的$L^2(X)$ -有界性.此外，我们还研究了一个$T(b)$ -定理成立的判据。

英文摘要：

Let $X$ be a space of homogeneous type and let $L$ be a sectorial operator with bounded holomorphic functional calculus on $L^2(X)$. We assume that the semigroup $\{e^{-tL}\}_{t>0}$ satisfies Davies-Gaffney estimates. Associated to $L$ are certain approximations of the identity. We call an operator $T$ a non-integral operator if compositions involving $T$ and these approximations satisfy certain weighted norm estimates. The Davies-Gaffney and the weighted norm estimates are together a substitute for the usual kernel estimates on $T$ in Calderón-Zygmund theory. In this paper, we show, under the additional assumption that a vertical Littlewood-Paley-Stein square function associated to $L$ is bounded on $L^2(X)$, that a non-integral operator $T$ is bounded on $L^2(X)$ if and only if $T(1) \in BMO_L(X)$ and $T^{\ast}(1) \in BMO_{L^{\ast}}(X)$. Here, $BMO_L(X)$ and $BMO_{L^{\ast}}(X)$ denote the recently defined $BMO(X)$ spaces associated to $L$ that generalize the space $BMO(X)$ of John and Nirenberg. Generalizing a recent result due to F. Bernicot, we show a second version of a T(1)-Theorem under weaker off-diagonal estimates, which gives a positive answer to a question raised by him. As an application, we prove $L^2(X)$-boundedness of a paraproduct operator associated to $L$. We moreover study criterions for a $T(b)$-Theorem to be valid.

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。

摘要：

英文摘要：

相关论文