与满足Davies-Gaffney估计的算子相关的消失平均振动空间
Vanishing Mean Oscillation Spaces Associated with Operators Satisfying Davies-Gaffney Estimates
摘要：
设$(\mathcal{X}, d, μ)$是度量测度空间，$L$是具有有界$H_\infty$泛函演算且满足Davies-Gaffney估计的线性算子，$Φ$是$(0,\infty)$上的凹函数，其临界下型为$p_Φ^-\in(0,1]$，且对所有$t\in(0,\infty)$均为$ρ(t)\equiv t^{-1}/Φ^{-1}(t^{-1})$.本文引入了与$L$相联系的广义VMO空间${\mathrm {VMO}}_{ρ,L}({\mathcal X})$，并通过帐篷空间建立了它的刻画.作为应用，作者证明了$({\mathrm {VMO}}_{ρ,L}({\mathcal X}))^*=B_{Φ,L^*}({\mathcal X})$，其中$L^*$表示$L$在$L^2({\mathcal X})$中的伴随算子，$B_{Φ,L^*}({\mathcal X})$表示Orlicz-Hardy空间$H_{Φ,L^*}({\mathcal X})$的Banach完备化.
英文摘要：
Let $(\mathcal{X}, d, μ)$ be a metric measure space, $L$ a linear operator which has a bounded $H_\infty$ functional calculus and satisfies the Davies-Gaffney estimate, $Φ$ a concave function on $(0,\infty)$ of critical lower type $p_Φ^-\in(0,1]$ and $ρ(t)\equiv t^{-1}/Φ^{-1}(t^{-1})$ for all $t\in(0,\infty)$. In this paper, the authors introduce the generalized VMO space ${\mathrm {VMO}}_{ρ,L}({\mathcal X})$ associated with $L$, and establish its characterization via the tent space. As applications, the authors show that $({\mathrm {VMO}}_{ρ,L}({\mathcal X}))^*=B_{Φ,L^*}({\mathcal X})$, where $L^*$ denotes the adjoint operator of $L$ in $L^2({\mathcal X})$ and $B_{Φ,L^*}({\mathcal X})$ the Banach completion of the Orlicz-Hardy space $H_{Φ,L^*}({\mathcal X})$.
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