Fractional Bloom boundedness and compactness of commutators

Let $T$ be a non-degenerate Calderón-Zygmund operator and let $b:{\mathbb{R}}^d\to \mathbb{C}$ be locally integrable. Let $1<p\le q<\mathrm{\infty }$ and let ${\mu }^p\in A_p$ and ${\lambda }^q\in A_q,$ 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 $L_{\mu }^p=L^p({\mu }^p)$ and $\alpha \mathrm{/}d=1\mathrm{/}p-1\mathrm{/}q,$ , the symbol $\mathcal{K}$ stands for the class of compact operators between the given spaces, and the fractional weighted ${\mathrm{BMO}}_{\nu }^{\alpha }$ and ${\mathrm{VMO}}_{\nu }^{\alpha }$ 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.