Fractional Bloom boundedness and compactness of commutators

Abstract

Abstract 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.