BMO, H^1, and Calderon-Zygmund operators for non doubling measures
BMO, H^1, and Calderon-Zygmund operators for non doubling measures
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DOI：
10.1007/pl00004432
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发表时间：
2001
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影响因子：
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通讯作者：
X. Domènech
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文献类型：
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作者：
X. Domènech
给定${mathbb R}^d$上的Radon测度$\Mu$，它可能是非倍增的，我们引入了关于这个测度的BMO型空间。证明了经典空间$BMO(\MU)$当$\MU$是加倍测度时的许多性质对本文引入的TYPEBMO空间仍然有效，而不假设$\MU$加倍.例如，在$L^2(\Mu)$上有界的Calderón-Zygmund算子也从$L^\inty(\Mu)$有界到新的BMO空间。此外，这个空间还满足John-Nirenberg不等式，并且它的前件是原子空间$H^1$。利用一个尖锐的极大算子，证明了从$L^\inty(Mu)$到新BMO空间和从它的前对偶$H^1(\Mu)$有界到$L^1(\Mu)$的算子一定在$L^p(\Mu)$，$1<；p<；inty$上有界.由此结果可以得到非加倍测度柯西变换Thet(1)定理的一个新的证明。最后证明了有界于$L^2(MU)$上的Calderón-Zygmund算子的交换子在$L^p(MU)，1<；p<；\inty$上是有界的。
Given a Radon measure $\mu$ on ${\mathbb R}^d$, which may be non doubling, we introduce a space of typeBMOwith respect to this measure. It is shown that many properties which hold for the classical space $BMO(\mu)$ when $\mu$ is a doubling measure remain valid for the space of typeBMOintroduced in this paper, without assuming $\mu$ doubling. For instance, Calderón-Zygmund operators which are bounded on $L^2(\mu)$ are also bounded from $L^\infty(\mu)$ into the newBMOspace. Moreover, this space also satisfies a John-Nirenberg inequality, and its predual is an atomic space $H^1$. Using a sharp maximal operator it is shown that operators which are bounded from $L^\infty(\mu)$ into the newBMOspace and from its predual $H^1$ into $L^1(\mu)$ must be bounded on $L^p(\mu)$, $1< p< infty$. From this result one can obtain a new proof of theT(1) theorem for the Cauchy transform for non doubling measures. Finally, it is proved that commutators of Calderón-Zygmund operators bounded on $L^2(\mu)$ with functions of the newBMOare bounded on $L^p(\mu), 1< p < \infty$.