Off-diagonal estimates for bi-commutators

Airta, Emil; Hytönen, Tuomas; Li, Kangwei; Martikainen, Henri; Oikari, Tuomas

doi:10.48550/arxiv.2005.03548

Cited by 2 publications

(7 citation statements)

References 34 publications

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“…The following oscillatory characterization is recorded e.g. as Proposition 3.2. in [1]. Let r ∈ (1, ∞), then there holds that…”

Section: The Casementioning

confidence: 99%

“…Then, let F Q stand for the collection of all 1/2-sparse collections of cubes, and let F R stand for the collection of all 1/2-sparse collections of rectangles. Then, a straightforward adaptation of Proposition 3.2. in [1] would allows us to actually show that…”

Section: The Casementioning

confidence: 99%

“…Hence, we need not trouble ourselves with recording their coefficient size, nonetheless, we mention that the coefficient size is measured by the norm of the product BMO space of Chang and Fefferman, see e.g. Section 7 in [1].…”

Section: Proposition Let Pmentioning

confidence: 99%

“…has a built-in order to it. These limitations, however, are not new and were expected, as we already saw them in our earlier paper with Airta, Hytönen, Li and Martikainen in [1] that dealt with the mixed norm bounds for the other natural bi-parameter commutator [T 1 , [b, T 2 ]], where each T i is a one-parameter Calderón-Zygmund operator.…”

Section: Introductionmentioning

confidence: 99%

“…On the off-diagonal in [1] we achieved a fully satisfactory characterization of the boundedness of the commutator in 5 out of the 9 cases, on the diagonal, this is when p 1 = q 1 and p 2 = q 2 , we provided new and simplified proofs of existing results on lower bounds of the commutator [T 1 , [b, T 2 ]], we also proved the natural expected upper bounds in the unfinished cases, and in addition we developed some vector valued theory of commutators.…”

Section: Introductionmentioning

confidence: 99%

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Off-diagonal estimates for commutators of bi-parameter singular integrals

Oikari¹

2020

Preprint

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We study the commutators [b, T ] of pointwise multiplications and bi-parameter Calderón-Zygmund operators and characterize their off-diagonal L p 1 L p 2 → L q 1 L q 2 boundedness in the range (1, ∞) for several of the mixed norm integrability exponents. CONTENTS 1. Introduction 1 2. Definitions and preliminaries 3 3. Approximate weak factorization in the bi-parameter setting 6 4. Upper and lower bounds 12 5. Upper bounds for the cases p 1 = q 1 , p 2 < q 2 and p 1 < q 1 , p 2 = q 2 23 References 352010 Mathematics Subject Classification. 42B20.

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“…The following oscillatory characterization is recorded e.g. as Proposition 3.2. in [1]. Let r ∈ (1, ∞), then there holds that…”

Section: The Casementioning

confidence: 99%

Section: The Casementioning

confidence: 99%

Section: Proposition Let Pmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Off-diagonal estimates for commutators of bi-parameter singular integrals

Oikari¹

2020

Preprint

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Commutators of Cauchy-Szego type integrals for domains in C^n with minimal smoothness

Duong¹,

Lacey

Li³

et al. 2021

Indiana Univ. Math. J.

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Let D ⊂ C n be a bounded, strongly pseudoconvex domain whose boundary bD satisfies the minimal regularity condition of class C 2 . We characterize boundedness and compactness in L p (bD, ω), for 1 < p < ∞, of the commutator [b, Sω] where Sω is the Cauchy-Szegő (orthogonal) projection of L 2 (bD, ω) onto the holomorphic Hardy space H 2 (bD, ω) and the measure ω belongs to a family (the "Leray Levi-like" measures) that includes induced Lebesgue measure σ. We next consider a much larger family of measures {Ω} modeled after the Muckenhoupt weights for σ: we define the holomorphic Hardy spaces H p (bD, Ω) for any Ap-like measure Ω and we characterize boundedness and compactness of [b, SΩ] in L 2 (bD, Ω) for any A2-like measure Ω. Earlier closely related results rely upon an asymptotic expansion, and subsequent pointwise estimates, of the Cauchy-Szegő kernel that are not available in the settings of minimal regularity of bD and/or Ap-like measures.Among the new main tools are • operator identities for [b, Sω] and for [b, SΩ] inspired by the classical Kerzman-Stein equation for Sσ in L 2 (bD, σ) [22] and its 2017 variant in L p (bD, ω) [32], where p = 2; • sharp weighted estimates in L p (bD, Ω) for a family of Cauchy type integral operators {Cǫ}ǫ (related to SΩ via the Kerzman-Stein equations) and for their commutators {[b, Cǫ]}ǫ; • cancellation estimates in L p (bD, Ω) for the symmetrized truncation of the aforementioned {Cǫ}ǫ with quantitative norm bounds displaying explicit dependence on ǫ.

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Off-diagonal estimates for bi-commutators

Cited by 2 publications

References 34 publications

Off-diagonal estimates for commutators of bi-parameter singular integrals

Off-diagonal estimates for commutators of bi-parameter singular integrals

Commutators of Cauchy-Szego type integrals for domains in C^n with minimal smoothness

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