值在一致凸空间中的BMO函数和Carleson测度
BMO functions and Carleson measures with values in uniformly convex spaces
摘要：
本文研究了向量值BMO函数与由其梯度定义的Carleson测度之间的关系。设$dA$和$dm$分别表示单位圆$D$和单位圆$\mathbb T$上的勒贝格测度.对于$1< q<\infty$和Banach空间$B$，我们证明了存在一个正的常数$c$，使得对于所有系数在$B$中的三角多项式$f$，$$\sup_{z_0\in D}\int_{D}(1-|z|)^{q-1}\|\nabla f(z)\|^q P_{z_0}(z) dA(z) \le c^q\sup_{z_0\in D}\int_{\T}\|f(z)-f(z_0)\|^qP_{z_0}(z) dm(z)$$成立当且仅当$B$允许一个等价的范数是$q$ -一致凸的，其中$$P_{z_0}(z)=\frac{1-|z_0|^2}{|1-\bar{z_0}z|^2} .$$逆不等式的有效性等价于存在一个等价的$q$ -一致光滑范数.
英文摘要：
This paper studies the relationship between vector-valued BMO functions and the Carleson measures defined by their gradients. Let $dA$ and $dm$ denote Lebesgue measures on the unit disc $D$ and the unit circle $\mathbb T$, respectively. For $1< q<\infty$ and a Banach space $B$ we prove that there exists a positive constant $c$ such that $$\sup_{z_0\in D}\int_{D}(1-|z|)^{q-1}\|\nabla f(z)\|^q P_{z_0}(z) dA(z) \le c^q\sup_{z_0\in D}\int_{\T}\|f(z)-f(z_0)\|^qP_{z_0}(z) dm(z)$$ holds for all trigonometric polynomials $f$ with coefficients in $B$ iff $B$ admits an equivalent norm which is $q$-uniformly convex, where $$P_{z_0}(z)=\frac{1-|z_0|^2}{|1-\bar{z_0}z|^2} .$$ The validity of the converse inequality is equivalent to the existence of an equivalent $q$-uniformly smooth norm.
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