The authors were supported by the Academy of Finland through project Nos. 314829 (all), 336323 (Sinko) and 346314 (Hytönen), as well as by the Jenny and Antti Wihuri Foundation (Oikari).1 For notation see Section 2.2 below.
Let T be a non-degenerate Calderón–Zygmund operator and let b : ℝ d → ℂ {b:\mathbb{R}^{d}\to\mathbb{C}} be locally integrable. Let 1 < p ≤ q < ∞ {1<p\leq q<\infty} and let μ p ∈ A p {\mu^{p}\in A_{p}} and λ q ∈ A q {\lambda^{q}\in A_{q}} , where A p {A_{p}} denotes the usual class of Muckenhoupt weights. We show that ∥ [ b , T ] ∥ L μ p → L λ q ∼ ∥ b ∥ BMO ν α , [ b , T ] ∈ 𝒦 ( L μ p , L λ q ) iff b ∈ VMO ν α , \lVert[b,T]\rVert_{L^{p}_{\mu}\to L^{q}_{\lambda}}\sim\lVert b\rVert_{% \operatorname{BMO}_{\nu}^{\alpha}},\quad[b,T]\in\mathcal{K}(L^{p}_{\mu},L^{q}_% {\lambda})\quad\text{iff}\quad b\in\operatorname{VMO}_{\nu}^{\alpha}, where L μ p = L p ( μ p ) {L^{p}_{\mu}=L^{p}(\mu^{p})} and α / d = 1 / p - 1 / q {\alpha/d=1/p-1/q} , the symbol 𝒦 {\mathcal{K}} stands for the class of compact operators between the given spaces, and the fractional weighted BMO ν α {\operatorname{BMO}_{\nu}^{\alpha}} and VMO ν α {\operatorname{VMO}_{\nu}^{\alpha}} spaces are defined through the following fractional oscillation and Bloom weight: 𝒪 ν α ( b ; Q ) = ν ( Q ) - α / d ( 1 ν ( Q ) ∫ Q | b - 〈 b 〉 Q | ) , ν = ( μ λ ) β , β = ( 1 + α / d ) - 1 . \mathcal{O}_{\nu}^{\alpha}(b;Q)=\nu(Q)^{-\alpha/d}\biggl{(}\frac{1}{\nu(Q)}% \int_{Q}\lvert b-\langle b\rangle_{Q}\rvert\biggr{)},\quad\nu=\biggl{(}\frac{% \mu}{\lambda}\biggr{)}^{\beta},\quad\beta=(1+\alpha/d)^{-1}. The key novelty is dealing with the off-diagonal range p < q {p<q} , whereas the case p = q {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.
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