Commutators in the two-weight setting

By I. Holmes, Michael T. Lacey, Brett D. Wick. 2015

124 citations recorded by the source corpus.

Let R be the vector of Riesz transforms on $$ \mathbb R ^{n}$$Rn, and let $$\mu ,\lambda \in A_p$$μ,λ∈Ap be two weights on $$ \mathbb R ^{n}$$Rn, $$ 1< p < \infty $$1<p<∞. The two-weight norm inequality for the commutator $$ ||[b, R] \;:\; L ^{p} (\mu ) \rightarrow L ^{p} (\lambda )||$$||[b,R]:Lp(μ)→Lp(λ)|| is shown to be equivalent to the function b being in a BMO space adapted to $$ \mu $$μ and $$ \lambda $$λ. This is a common extension of a result of Coifman–Rochberg–Weiss in the case of both $$ \lambda $$λ and $$ \mu $$μ being Lebesgue measure, and Bloom in the case of dimension one.

Subfields: Applied Mathematics

View Paper: https://www.semanticscholar.org/paper/bb4f1cbefee42144d9808d4a10e063c86cf2fe00

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In Applied Mathematics

Citations in the corpus

Commutators of singular integrals revisited (2017, methodology): “In [10] this part was obtained, similar to [4], assuming that [b,Rj ] is bounded from L(μ) to L(λ) for every Riesz transform Rj .”
On pointwise and weighted estimates for commutators of Calder\'on-Zygmund operators (2016, background): “In the particular case when μ = λ = w ∈ A2 the approach in [13] recovers (1.”
Weighted little bmo and two-weight inequalities for Journé commutators (2017, methodology): “In [11], it was shown that, when b ∈ BMO(ν), the operators Πb, Πb , and Γb are bounded L(μ) → L(λ).”
Two Weight Commutators on Spaces of Homogeneous Type and Applications (2018, methodology): “4 in [21] and its proof), as a consequence of the duality of weighted Hardy space H1 ν (X) and BMOν(X) (Theorem 6.”