Fractional Bloom boundedness and compactness of commutators

Abstract

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.