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

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 Calderón--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 $[b,T]:L^p(\mu )\to L^q(\lambda )$ 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 $$M_{\nu }^{\#}b:=\underset{Q}{\sup }\frac{1_Q}{\nu (Q)}{\int }_Q|b-⟨b{⟩}_Q|\mathrm{d}x$$ and $\nu$ is the Bloom weight defined by ${\nu }^{1/p+1/q^{\prime }}:={\mu }^{1/p}{\lambda }^{-1/q}$. In the unweighted case $\mu =\lambda =1$, by a result of Hytönen 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 $p\ne q$. 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 $p,q\in (1,\infty )$.

arXiv comment37 pages, typos corrected. To appear in Journal de Mathématiques Pures et Appliquées

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