New Estimates of Rychkov's Universal Extension Operators for Lipschitz Domains and Some Applications

Shi, Ziming; Yao, Liding

doi:10.48550/arxiv.2110.14477

Cited by 5 publications

(5 citation statements)

References 15 publications

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“…We refer the reader to [SY21b] for some more detailed properties of the Rychkov extension operator. The following anti-derivative operator, which we introduced in [SY21a], is used in the construction of our ∂ solution operator.…”

Section: Preliminariesmentioning

confidence: 99%

On $1/2$ estimate for global Newlander-Nirenberg theorem

Shi¹

2023

Preprint

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Given a formally integrable almost complex structure X defined on the closure of a bounded domain D ⊂ C n , and provided that X is sufficiently close to the standard complex structure, the global Newlander-Nirenberg problem asks whether there exists a global diffeomorphism defined on D that transform X into the standard complex structure, under certain geometric and regularity assumptions on D. In this paper we prove a regularity result of this problem. Assuming D is a strictly pseudoconvex domain in C n with C 2 boundary, and that the almost structure X is of the Hölder-Zygmund class Λ r (D) for r > 1, we prove the existence of a global diffeomorphism (independent of r) in the class Λ r+ 1 2 −ε (D), for any ε > 0. Contents 1. Introduction 2. Preliminaries 2.1. Function space and stability of constant 2.2. Smoothing operator on bounded Lipschitz domains 2.3. Paraproduct estimate 2.4. ∂ equation in Hölder space of negative smooth index 3. Norm estimates for the error term 4. Iteration Scheme and convergence of maps References

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Section: Preliminariesmentioning

confidence: 99%

On $1/2$ estimate for global Newlander-Nirenberg theorem

Shi¹

2023

Preprint

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“…By assumption (4.7) we get supp φ j ⊂ −K ∩ {x 1 < −c 1 2 −j } for all j ≥ 0. A simple calculation shows that (see [SY21b,Lemma 5…”

Section: Tangential Commutator Estimate and Strong Hardy-littlewood L...mentioning

confidence: 99%

“…In order to ensure the integral operators make sense we express the given forms as the derivatives of functions with positive index. For k ≥ 1 we constructed the anti-derivative operators {S k,α } |α|≤k in [SY21b] such that if a function g is supported outside Ω then g = |α|≤k D α S k,α g with all summand supported outside Ω as well. See Proposition 4.13.…”

Section: Introductionmentioning

confidence: 99%

Sobolev and Hölder Estimates for Homotopy Operators of $\overline\partial$-Equations on Convex Domains of Finite Multitype

Yao¹

2022

Preprint

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We construct homotopy formulas for ∂-equations on convex domains of finite type that have optimal Sobolev and Hölder estimates. For a bounded smooth finite type convex domain Ω ⊂ C n that has q-type mq for 1 ≤ q ≤ n, our ∂ solution operator Tq on (0, q)-forms has (fractional) Sobolev boundedness Tq : H s,p → H s+1/mq ,p and Hölder-Zygmund boundedness Tq : C s → C s+1/mq for all s ∈ R and 1 < p < ∞. We also show the L p -boundedness Tq : H s,p → H s,prq /(rq −p) for all s ∈ R and 1 < p < rq, where rq := (n − q + 1)mq + 2q.

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“…The technique of anti-derivatives could be useful in studying integral operators. In a recent preprint [SY21a], the authors develop this technique in full generality and demonstrate some applications.…”

Section: Introductionmentioning

confidence: 99%

Existence of Solutions for $\overline\partial$ Equation in Sobolev Spaces of Negative Index

Shi¹,

Yao²

2021

Preprint

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Let Ω be a strictly pseudoconvex domain in C n with C k+2 boundary, k ≥ 1. We construct a ∂ solution operator (depending on k) that gains 1 2 derivative in the Sobolev space H s,p (Ω) for any 1 < p < ∞ and s > 1 p − k. If the domain is C ∞ , then there exists a ∂ solution operator that gains 1 2 derivative in H s,p (Ω) for all s ∈ R. We obtain our solution operators through the method of homotopy formula; a new feature is the construction of "anti-derivative operators" on distributions defined on bounded Lipschitz domains.

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New Estimates of Rychkov's Universal Extension Operators for Lipschitz Domains and Some Applications

Cited by 5 publications

References 15 publications

On $1/2$ estimate for global Newlander-Nirenberg theorem

On $1/2$ estimate for global Newlander-Nirenberg theorem

Sobolev and Hölder Estimates for Homotopy Operators of $\overline\partial$-Equations on Convex Domains of Finite Multitype

Existence of Solutions for $\overline\partial$ Equation in Sobolev Spaces of Negative Index

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