Fractional Bloom boundedness and compactness of commutators

Let $T$ be a non-degenerate Calder\'on-Zygmund operator and let $b:\mathbb{R}^d\to\mathbb{C}$ be locally integrable. Let $1<p\leq q<\infty$ 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_{\mu}\to L^q_{\lambda}}\sim \|b\|_{\operatorname{BMO}_{\nu}^{\alpha}},\qquad [b,T]\in \mathcal{K}(L^p_{\mu}, L^q_{\lambda})\quad\mbox{iff}\quad b\in \operatorname{VMO}_{\nu}^{\alpha}, \end{align*} where $L^p_\mu=L^p(\mu^p)$ and $\alpha/d = 1/p-1/q,$ , the symbol $\mathcal{K}$ stands for the class of compact operators between the given spaces, and the fractional weighted $\operatorname{BMO}_{\nu}^{\alpha}$ and $\operatorname{VMO}_{\nu}^{\alpha}$ spaces are defined through the following fractional oscillation and Bloom weight \begin{align*} \mathcal{O}_{\nu}^{\alpha}(b;Q) = \nu^{-\alpha/d}(Q)\Big(\frac{1}{\nu(Q)}\int_Q |b-\langle b\rangle_Q|\Big),\qquad \nu = \big(\frac{\mu}{\lambda}\big)^{\beta},\quad \beta = (1+\alpha/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.