Regularity of a $$\overline{\partial }$$-Solution Operator for Strongly $$\mathbf{C}$$-Linearly Convex Domains with Minimal Smoothness

Gong, Xianghong; Lanzani, Loredana

doi:10.1007/s12220-020-00443-w

Cited by 5 publications

(3 citation statements)

References 20 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…The C k+1/2 solutions for f ∈ C k (k ∈ N) was obtained by Siu [54] for (0, 1) forms and by Lieb-Range [40] for forms with q ≥ 1 when ∂D is sufficiently smooth boundary. For ∂D ∈ C 2 , Theorem 1.1 and analogous result for a homotopy formula were proved in [17] through the construction of a homotopy formula. These results were extended by Shi [51] to a weighted Sobolev spaces with a gain less than 1/2 derivative and by Shi-Yao [52,53] to H s+1/2,p space gaining 1/2 derivative for s > 1/p when ∂D ∈ C 2 and for s ∈ R when ∂D is sufficiently smooth.…”

Section: Introductionmentioning

confidence: 81%

“…These results were extended by Shi [51] to a weighted Sobolev spaces with a gain less than 1/2 derivative and by Shi-Yao [52,53] to H s+1/2,p space gaining 1/2 derivative for s > 1/p when ∂D ∈ C 2 and for s ∈ R when ∂D is sufficiently smooth. It is worthy to point out that Shi-Yao achieved the first regularity result for negative order s. Furthermore, Gong-Lanzani [17] obtained Λ r+1/2 (with r > 1) regularity gaining 1/2 derivative on strongly C-linear convex C 1,1 domains D.…”

Section: Introductionmentioning

confidence: 89%

“…When ∂D i ∈ C 2 , one can still verify the integral formula on the domain D 1 ∩ D 2 by smoothing g j . For instance, see [15,17] for details.…”

Section: A Local Approximate Homotopy Formulamentioning

confidence: 99%

See 2 more Smart Citations

On regularity of $\overline\partial$-solutions on $a_q$ domains with $C^2$ boundary in complex manifolds

Gong¹

2022

Preprint

Self Cite

View full text Add to dashboard Cite

We study regularity of solutions u to ∂u = f on a relatively compact C 2 domain D in a complex manifold of dimension n, where f is a (0, q) form. Assume that there are either (q + 1) negative or (n − q) positive Levi eigenvalues at each point of boundary ∂D. Under the necessary condition that a locally L 2 solution exists on the domain, we show the existence of the solutions on the closure of the domain that gain 1/2 derivative when q = 1 and f is in the Hölder-Zygmund space Λ r (D) with r > 1. For q > 1, the same regularity for the solutions is achieved when ∂D is either sufficiently smooth or of (n − q) positive Levi eigenvalues everywhere on ∂D.

show abstract

Section: Introductionmentioning

confidence: 81%

Section: Introductionmentioning

confidence: 89%

See 1 more Smart Citation

On regularity of $\overline\partial$-solutions on $a_q$ domains with $C^2$ boundary in complex manifolds

Gong¹

2022

Preprint

Self Cite

View full text Add to dashboard Cite

show abstract

Regular multi-types and the Bloom conjecture

Huang

Yin

2021

Journal de Mathématiques Pures et Appliquées

View full text Add to dashboard Cite

A solution operator for the \overline∂ equation in Sobolev spaces of negative index

Shi,

Yao

2023

Trans. Amer. Math. Soc.

View full text Add to dashboard Cite

Let Ω \Omega be a strictly pseudoconvex domain in C n \mathbb {C}^n with C k + 2 C^{k+2} boundary, k ≥ 1 k \geq 1 . We construct a ∂ ¯ \overline \partial solution operator (depending on k k ) that gains 1 2 \frac 12 derivative in the Sobolev space H s , p ( Ω ) H^{s,p} (\Omega ) for any 1 > p > ∞ 1>p>\infty and s > 1 p − k s>\frac {1}{p} -k . If the domain is C ∞ C^{\infty } , then there exists a ∂ ¯ \overline \partial solution operator that gains 1 2 \frac 12 derivative in H s , p ( Ω ) H^{s,p}(\Omega ) for all s ∈ R s \in \mathbb {R} . We obtain our solution operators via the method of homotopy formula. A novel technique is the construction of “anti-derivative operators” for distributions defined on bounded Lipschitz domains.

show abstract

Regularity of a $$\overline{\partial }$$-Solution Operator for Strongly $$\mathbf{C}$$-Linearly Convex Domains with Minimal Smoothness

Cited by 5 publications

References 20 publications

On regularity of $\overline\partial$-solutions on $a_q$ domains with $C^2$ boundary in complex manifolds

On regularity of $\overline\partial$-solutions on $a_q$ domains with $C^2$ boundary in complex manifolds

Regular multi-types and the Bloom conjecture

A solution operator for the \overline∂ equation in Sobolev spaces of negative index

Contact Info

Product

Resources

About