Weighted $L^p\to L^q$-boundedness of commutators and paraproducts in the Bloom setting

TL;DR

This paper characterizes the weighted boundedness of commutators of Calderón–Zygmund operators and paraproducts in the Bloom setting, introducing a new cancellative condition that is necessary for the boundedness in the case of different exponents.

Contribution

It provides the first complete characterization of weighted boundedness for these operators across all exponent ranges, including a new necessary cancellative condition.

Findings

Introduces a new cancellative condition for boundedness.

Provides a counterexample showing the failure of previous characterizations.

Completes the characterization of weighted boundedness for all p, q in (1, ∞).

Abstract

As our main result, we supply the missing characterization of the $L^p(\mu)\to L^q(\lambda)$ boundedness of the commutator of a non-degenerate Calder\'on--Zygmund operator $T$ and pointwise multiplication by $b$ for exponents $1<q<p<\infty$ and Muckenhoupt weights $\mu\in A_p$ and $\lambda\in A_q$. Namely, the commutator $[b,T]\colon L^p(\mu)\to L^q(\lambda)$ is bounded if and only if $b$ satisfies the following new, cancellative condition: $$M^\#_\nu b\in L^{pq/(p-q)}(\nu),$$ where $M^\#_\nu b$ is the weighted sharp maximal function defined by $$ M^\#_\nu b:=\sup_{Q} \frac{\mathbf{1}_Q}{\nu(Q)} \int_{Q} |b-\langle b\rangle_Q |\,\mathrm{d}x$$ and $\nu$ is the Bloom weight defined by $\nu^{1/p+1/q'}:= \mu^{1/p} \lambda^{-1/q}$. In the unweighted case $\mu=\lambda=1$, by a result of Hyt\"onen the boundedness of the commutator $[b,T]$ is, after factoring out constants, characterized by the boundedness of pointwise multiplication by $b$, which amounts to the non-cancellative condition $b\in L^{pq/(p-q)}$. We provide a counterexample showing that this characterization breaks down in the weighted case $\mu\in A_p$ and $\lambda\in A_q$. Therefore, the introduction of our new, cancellative condition is necessary. In parallel to commutators, we also characterize the weighted boundedness of dyadic paraproducts $\Pi_b$ in the missing exponent range $p\neq q$. Combined with previous results in the complementary exponent ranges, our results complete the characterisation of the weighted boundedness of both commutators and of paraproducts for all exponents $p,q \in (1,\infty)$.