与散度型椭圆算子相关的广义消失平均振动空间
Generalized Vanishing Mean Oscillation Spaces Associated with Divergence Form Elliptic Operators
摘要：
设$L$是一个具有复有界可测系数的散度型椭圆算子，$ω$是$(0,\infty)$上严格临界下型的正凹函数，$p_ω\in (0, 1]$和$ρ(t)={t^{-1}}/ω^{-1}(t^{-1})$是$t\in (0,\infty).$的正凹函数.本文引入了与$L$相联系的广义VMO空间${\mathop\mathrm{VMO}_ {ρ, L}({\mathbb R}^n)}$，并利用帐篷空间刻画了它们.作为应用，作者证明了$(\mathrm{VMO}_{ρ,L} ({\mathbb R}^n))^\ast=B_{ω,L^\ast}({\mathbb R}^n)$，其中$L^\ast$表示$L$在$L^2({\mathbb R}^n)$中的伴随算子，$B_{ω,L^\ast}({\mathbb R}^n)$表示Orlicz-Hardy空间$H_{ω,L^\ast}({\mathbb R}^n)$的Banach完备化.请注意，对于所有$t\in (0,\infty)$和$p\in (0,1]$的$ω(t)=t^p$是满足假设的正凹函数的典型例子。特别地，当$p=1$时，则$ρ(t)\equiv 1$和$({\mathop\mathrm{VMO}_{1, L}({\mathbb R}^n)})^\ast=H_{L^\ast}^1({\mathbb R}^n)$，其中$H_{L^\ast}^1({\mathbb R}^n)$是Hofmann和Mayboroda引入的Hardy空间。
英文摘要：
Let $L$ be a divergence form elliptic operator with complex bounded measurable coefficients, $ω$ a positive concave function on $(0,\infty)$ of strictly critical lower type $p_ω\in (0, 1]$ and $ρ(t)={t^{-1}}/ω^{-1}(t^{-1})$ for $t\in (0,\infty).$ In this paper, the authors introduce the generalized VMO spaces ${\mathop\mathrm{VMO}_ {ρ, L}({\mathbb R}^n)}$ associated with $L$, and characterize them via tent spaces. As applications, the authors show that $(\mathrm{VMO}_{ρ,L} ({\mathbb R}^n))^\ast=B_{ω,L^\ast}({\mathbb R}^n)$, where $L^\ast$ denotes the adjoint operator of $L$ in $L^2({\mathbb R}^n)$ and $B_{ω,L^\ast}({\mathbb R}^n)$ the Banach completion of the Orlicz-Hardy space $H_{ω,L^\ast}({\mathbb R}^n)$. Notice that $ω(t)=t^p$ for all $t\in (0,\infty)$ and $p\in (0,1]$ is a typical example of positive concave functions satisfying the assumptions. In particular, when $p=1$, then $ρ(t)\equiv 1$ and $({\mathop\mathrm{VMO}_{1, L}({\mathbb R}^n)})^\ast=H_{L^\ast}^1({\mathbb R}^n)$, where $H_{L^\ast}^1({\mathbb R}^n)$ was the Hardy space introduced by Hofmann and Mayboroda.
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