Commutators with fractional integral operators

The Unified Theory for the Necessity of Bounded Commutators and Applications

“…Invoking Theorem 2.1 and Proposition 2.3 again, by (4.5) and the similar arguments to in the proof of Corollary 4.8, we can get the following result, which essentially improve and extend the result of [21] and present the converse result of (4.5).…”

On Bloom type estimates for iterated commutators of fractional integrals

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In this paper we provide quantitative Bloom type estimates for iterated commutators of fractional integrals improving and extending results from [15]. We give new proofs for those inequalities relying upon a new sparse domination that we provide as well in this paper and also in techniques developed in the recent paper [22].

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“…We give new proofs for those inequalities relying upon a new sparse domination that we provide as well in this paper and also in techniques developed in the recent paper [22]. We extend as well the necessity established in [15] to iterated commutators providing a new proof. As a consequence of the preceding results we recover the one weight estimates in [7, 1] and establish the sharpness in the iterated case.…”

A revisit on the compactness of commutators

“…A p and A p,q weights. In this subsection, we recall some useful properties of A p and A p, q weights; see [20,8,10,9]. Define the A ∞ class of weights by A ∞ := ∪ p>1 A p , and recall the Fujii…”