论文标题
换向器的分数花的界限和紧凑性
Fractional Bloom boundedness and compactness of commutators
论文作者
论文摘要
令$ t $为非脱位calderón-zygmund操作员，让$ b：\ mathbb {r}^d \ to \ mathbb {c} $是本地集成的。令$ 1 <p \ leq q <\ infty $，让$μ^p \在a_p $和a_q中的$λ^q \中，其中$ a_ {p} $表示通常的muckenhoupt权重。我们表明 \ begin {align*} \ | [B，T] l^q_λ）\ quad \ mbox {iff} \ quad b \ in \ operatatorName {vmo}_ν^α， \ end {align*}其中$ l^p_μ= l^p（μ^p）$和$α/d = 1/p-1/q，$，符号$ \ mathcal {k} $代表给定空间之间的紧凑型操作员，以及分数加权的$ \ \ perperOteD $ \ \ operate $ \ operate $ \ operate { $ \ operatatorName {vmo}_ν^α$空间是通过以下分数振荡定义的 \ begin {align*} \ Mathcal {o}_ν^α（b; q）=ν^{ - α/d}（q）（q）\ big（\ frac {1} {1} {ν（q）} \ int_q | b- \ langle b \ langle b \ rangle_q | \ big） = \ big（\fracμλ\ big）^β，\ quadβ=（1+α/d）^{ - 1}。 \ end {align*}主要新颖性是处理偏离范围$ p <q $，而案例$ p = q $先前是由莱西和李研究的。但是，在这两种情况下，另一种新颖性是我们的方法允许复杂值函数$ b $，而基于$ b $的中位数的其他论点本质上是真实价值的。
Let $T$ be a non-degenerate Calderón-Zygmund operator and let $b:\mathbb{R}^d\to\mathbb{C}$ be locally integrable. Let $1<p\leq q<\infty$ and let $μ^p\in A_p$ and $λ^q\in A_q,$ where $A_{p}$ denotes the usual class of Muckenhoupt weights. We show that \begin{align*} \|[b,T]\|_{L^p_μ\to L^q_λ}\sim \|b\|_{\operatorname{BMO}_ν^α},\qquad [b,T]\in \mathcal{K}(L^p_μ, L^q_λ)\quad\mbox{iff}\quad b\in \operatorname{VMO}_ν^α, \end{align*} where $L^p_μ=L^p(μ^p)$ and $α/d = 1/p-1/q,$ , the symbol $\mathcal{K}$ stands for the class of compact operators between the given spaces, and the fractional weighted $\operatorname{BMO}_ν^α$ and $\operatorname{VMO}_ν^α$ spaces are defined through the following fractional oscillation and Bloom weight \begin{align*} \mathcal{O}_ν^α(b;Q) = ν^{-α/d}(Q)\Big(\frac{1}{ν(Q)}\int_Q |b-\langle b\rangle_Q|\Big),\qquad ν = \big(\fracμλ\big)^β,\quad β= (1+α/d)^{-1}. \end{align*} The key novelty is dealing with the off-diagonal range $p<q$, whereas the case $p=q$ was previously studied by Lacey and Li. However, another novelty in both cases is that our approach allows complex-valued functions $b$, while other arguments based on the median of $b$ on a set are inherently real-valued.
