论文标题
关于Banach功能空间的BMO
BMO with respect to Banach function spaces
论文作者
论文摘要
对于每个Cube $ q \ subset \ mathbb {r}^n $，我们让$ x_q $是$ q $ $ q $的quasi-banach函数空间，以至于$ \ |> |> |> | c_q \ | _ {x_q} \ simeq 1美元\ | f \ | _ {\ mathrm {bmo} _x}＆：= \ sup_q \ \，\ | f - {\ textStyle \ frac {1} {| q | |} \ | f \ | _ {\ mathrm {bmo} _x^*}＆：= \ sup_q \，\ fig_c \，\ | f-c \ | ____ {x_q}。 \ end {align*}我们在$ x $上研究必要的足够条件，以便$$ \ mathrm {bmo} = \ mathrm {bmo} _x = \ mathrm {bmo} _ {x}}^**。尤其是$$，我们将嵌入$ \ mathrm {bmo} \ hookrightArrow \ mathrm {bmo} _x $在所谓的稀疏集合中进行全面表征\ mathrm {bmo} $。我们的主要定理在该领域恢复并改善了所有以前已知的结果。
For every cube $Q \subset \mathbb{R}^n$ we let $X_Q$ be a quasi-Banach function space over $Q$ such that $\|χ_Q\|_{X_Q} \simeq 1$, and for $X= \{X_Q\}$ define \begin{align*} \|f\|_{\mathrm{BMO}_X} &:=\sup_Q \,\|f-{\textstyle\frac{1}{|Q|}\int_Qf} \|_{X_Q},\\ \|f\|_{\mathrm{BMO}_X^*} &:=\sup_Q \,\inf_c\, \|f-c\|_{X_Q}. \end{align*} We study necessary and sufficient conditions on $X$ such that $$ \mathrm{BMO} = \mathrm{BMO}_X = \mathrm{BMO}_{X}^*. $$ In particular, we give a full characterization of the embedding $\mathrm{BMO} \hookrightarrow \mathrm{BMO}_X$ in terms of so-called sparse collections of cubes and we give easily checkable and rather weak sufficient conditions for the embedding $\mathrm{BMO}_X^* \hookrightarrow \mathrm{BMO}$. Our main theorems recover and improve all previously known results in this area.
