Riesz transform on exterior Lipschitz domains and applications

Let L=−divA∇ be a uniformly elliptic operator on Rn, n≥2. Let Ω be an exterior Lipschitz domain, and let LD and LN be the operator L on Ω subject to the Dirichlet and Neumann boundary values, respectively. We establish the boundedness of the Riesz transforms ∇LD−1/2, ∇LN−1/2 in Lp spaces. As a bypro...

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Bibliographic Details
Published inAdvances in mathematics (New York. 1965) Vol. 453; p. 109852
Main Authors Jiang, Renjin, Lin, Fanghua
Format Journal Article
LanguageEnglish
Published Elsevier Inc 01.09.2024
Subjects
Dirichlet operators
Exterior Lipschitz domain
Harmonic function
Neumann operators
Riesz transform
Dirichlet operators
Riesz transform
Exterior Lipschitz domain
35J05
42B37
35J25
Neumann operators
Harmonic function
Online AccessGet full text
ISSN0001-8708
1090-2082
DOI10.1016/j.aim.2024.109852

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Summary:Let L=−divA∇ be a uniformly elliptic operator on Rn, n≥2. Let Ω be an exterior Lipschitz domain, and let LD and LN be the operator L on Ω subject to the Dirichlet and Neumann boundary values, respectively. We establish the boundedness of the Riesz transforms ∇LD−1/2, ∇LN−1/2 in Lp spaces. As a byproduct, we show the reverse inequality ‖LD1/2f‖Lp(Ω)≤C‖∇f‖Lp(Ω) holds for any 1<p<∞. The proof can be generalized to show the boundedness of the Riesz transforms, for operators with VMO coefficients on exterior Lipschitz or C1 domains. The estimates can be also applied to the inhomogeneous Dirichlet and Neumann problems. These results are new even for the Dirichlet and Neumann of the Laplacian operator on the exterior Lipschitz and C1 domains.
ISSN:0001-8708
1090-2082
DOI:10.1016/j.aim.2024.109852