Applications of the Cauchy integral on Lipschitz curves

Calderón, A. P.; Calderón, Calixto P.; Fabes, Eugene B.; Jodeit, Max; Rivière, N. M.

doi:10.1090/s0002-9904-1978-14478-4

Cited by 38 publications

(31 citation statements)

References 7 publications

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“…-u = g on bD (1) for boundary data# in L 2 (do). A corollary is that Vu attains its boundary values nontangentially pointwise almost everywhere and through dominated convergence in L 2 on level sets that tend to bD.…”

mentioning

confidence: 99%

The Neumann problem on Lipschitz domains

Jerison¹,

Kenig²

1981

Bull. Amer. Math. Soc.

291

248

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Let D be a Lipschitz domain in R", n > 2. Let o denote surface measure on 3A and let b/bn denote the normal derivative on bD. In this note we use an a priori estimate due to Payne and Weinberger [6], to bound the nontangential maximal function of the gradient sju of a (generalized) solution to the Neumann problem we found an elementary integral formula (7) and used it to prove a theorem of Dahlberg (Theorem 1) on Lipschitz domains. Unknown to us, this formula had already been discovered long ago by Payne and Weinberger and applied to the Dirichlet problem in smooth domains. Moreover, they used a second formula (2), which is a variant of a formula due to F. Rellich [7], to study the Neumann problem in smooth domains. We show here that the same strategy as in [4] applied to the second formula (2) coupled with Dahlberg's theorem yields our main result. Thus integral formulas give appropriate estimates for the solution of not only the Dirichlet problem, but also the Neumann problem on Lipschitz domains. We will present a more general version that applies to variable coefficient operators, systems, and other elliptic problems in a later

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mentioning

confidence: 99%

The Neumann problem on Lipschitz domains

Jerison¹,

Kenig²

1981

Bull. Amer. Math. Soc.

291

248

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“…It is well known that the study of this commutator is closely connected to the Cauchy integral on Lipschitz curves and the elliptic boundary value problem on non-smooth domain (see [4], [3], [5] and [16]). In [5], by using the method of rotation, (iii) F (t) = F (−t) for t ∈ R and F (t) is real analytic in {|t| ≤ ∇A ∞ }.…”

Section: General Calderón Commutatormentioning

confidence: 99%

Weak type (1,1) bound criterion for singular integrals with rough kernel and its applications

Ding

Lai

2018

Trans. Amer. Math. Soc.

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In this paper, a weak type (1,1) bound criterion is established for singular integral operator with rough kernel. As some applications of this criterion, we show some important operators with rough kernel in harmonic analysis, such as Calderón commutator, higher order Calderón commutator, general Calderón commutator, Calderón commutator of Bajsanski-Coifman type and general singular integral of Muckenhoupt type, are all of weak type (1,1).

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“…Proof. Case n = 2 can be established using Theorem 4 of [3] following the approach taken in the paper of Coifman, David and Meyer [4]. Namely with x, y ∈ Γ ⊂ R 2 we get that…”

Section: γ2mentioning

confidence: 99%

Hardy spaces and potential theory on 𝐶¹ domains in Riemannian manifolds

Dindoš¹

2008

Memoirs of the AMS

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We study Hardy spaces on C 1 and Lipschitz domains in Riemannian manifolds. Hardy spaces, originally introduced in 1920 in complex analysis setting, are invaluable tool in harmonic analysis. For this reason these spaces have been studied extensively by many authors.Our main result is an equivalence theorem proving that the definition of Hardy spaces by conjugate harmonic functions is equivalent to the atomic definition of these spaces. We establish this theorem in any dimension if the domain is C 1 , in case of a Lipschitz domain the result holds if dim M ≤ 3. The remaining cases for Lipschitz domains remain open. This result is a nontrivial generalization of flat (R n ) equivalence theorems due to Fefferman, Stein, Dahlberg and others.The material presented here required to develop potential theory approach for C 1 domains on Riemannian manifolds in the spirit of earlier works by Fabes, Jodeit and Rivière and recent results by Mitrea and Taylor. In particular, the first part of this work is of interest in itself, since we consider the boundary value problems for the Laplace-Beltrami operator. We prove that both Dirichlet and Neumann problem for Laplace-Beltrami equation are solvable for any given boundary data in L p (∂Ω), where 1 < p < ∞. Same remains true in Hardy spaces p (∂Ω) for (n − 1)/n < p ≤ 1.In the whole work we work with Riemannian metric g with smallest possible regularity. In particular, mentioned results for the Laplace-Beltrami equation require Hölder class regularity of the metric tensor; the equivalence theorem requires g in C 1,1 .

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Applications of the Cauchy integral on Lipschitz curves

Cited by 38 publications

References 7 publications

The Neumann problem on Lipschitz domains

The Neumann problem on Lipschitz domains

Weak type (1,1) bound criterion for singular integrals with rough kernel and its applications

Hardy spaces and potential theory on 𝐶¹ domains in Riemannian manifolds

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