Carleson Perturbations for the Regularity Problem

Dai, Zanbing; Feneuil, Joseph; Mayboroda, Svitlana

doi:10.48550/arxiv.2203.07992

Cited by 2 publications

(2 citation statements)

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“…Corollary 1.22 fully answers the question of the solvability of (R L p ) for some p > 1 when L is a DKP operator on a bounded rough domain, but we mention again that this problem had been open even when Ω is the unit ball. If in addition to the hypothesis of Corollary 1.22 we have that Ω satisfies the Harnack chain condition, then Ω is a chord-arc domain [AHMNT17] (see Section 2.1.2 for definitions), and in this case we can use the theory of Carleson perturbations for the regularity problem [KP95,DFM] to deduce the following improvement to Corollary 1.22.…”

Section: Hu Dmmentioning

confidence: 99%

Solvability of the Poisson-Dirichlet problem with interior data in $L^{p'}$-Carleson spaces and its applications to the $L^{p}$-regularity problem

Mourgoglou¹,

Poggi²,

Tolsa³

2022

Preprint

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We prove that the L p ′ -solvability of the homogeneous Dirichlet problem for an elliptic operator L = − div A∇ with real and merely bounded coefficients is equivalent to the L p ′ -solvability of the Poisson Dirichlet problem Lw = H − div F, assuming that H and F lie in certain Carleson-type spaces, and that the domain Ω ⊂ R n+1 , n ≥ 2, satisfies the corkscrew condition and has n-Ahlfors regular boundary. The L p ′solvability of the Poisson problem (with an L p ′ estimate on the non-tangential maximal function) is new even when L = −∆ and Ω is the unit ball. In turn, we use this result to show that, in a bounded domain with uniformly n-rectifiable boundary that satisfies the corkscrew condition, L p ′ -solvability of the homogeneous Dirichlet problem for an operator L = − div A∇ satisfying the Dahlberg-Kenig-Pipher condition (of arbitrarily large constant) implies solvability of the L p -regularity problem for the adjoint operator L * = − div A T ∇, where 1/p + 1/p ′ = 1 and A T is the transpose matrix of A. Contents 4.2. The starlike Lipschitz subdomains Ω ± R 31 4.3. The corona decomposition of Ω 36 4.4. The properties of Ω R 38 5. The almost L-elliptic extension 41 6. The Regularity Problem for DKP operators 42 6.1. A n.t. maximal function estimate for the almost L-elliptic extension 44 6.2. Proof of Proposition 6.2 49 Appendix A. Proofs of auxiliary results 54 References 63

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Section: Hu Dmmentioning

confidence: 99%

Solvability of the Poisson-Dirichlet problem with interior data in $L^{p'}$-Carleson spaces and its applications to the $L^{p}$-regularity problem

Mourgoglou¹,

Poggi²,

Tolsa³

2022

Preprint

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“…Finally, we give one reference for the perturbation results for the Regularity theory that are needed. There were several advances in perturbation in rougher domains than Lipschitz, but the latest one can be found in [9]. There, the authors show that the solvability of the Dirichlet problem for an operator L 1 which is a Carleson perturbation of L 0 , leads to a comparison of the nontangential maximal function of gradients of solutions with the same boundary values.…”

Section: Weaker Geometric Conditions On the Boundary Of The Domainmentioning

confidence: 99%