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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| Published in | Advances in mathematics (New York. 1965) Vol. 453; p. 109852 |
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| Format | Journal Article |
| Language | English |
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01.09.2024
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| ISSN | 0001-8708 1090-2082 |
| DOI | 10.1016/j.aim.2024.109852 |
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| Abstract | 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. |
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| AbstractList | 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. |
| ArticleNumber | 109852 |
| Author | Jiang, Renjin Lin, Fanghua |
| Author_xml | – sequence: 1 givenname: Renjin surname: Jiang fullname: Jiang, Renjin email: rejiang@cnu.edu.cn organization: Academy for Multidisciplinary Studies, Capital Normal University, Beijing, 100048, China – sequence: 2 givenname: Fanghua surname: Lin fullname: Lin, Fanghua email: linf@cims.nyu.edu organization: Courant Institute of Mathematical Sciences, 251 Mercer Street, New York, NY 10012, USA |
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| Keywords | Dirichlet operators Riesz transform Exterior Lipschitz domain 35J05 42B37 35J25 Neumann operators Harmonic function |
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| Snippet | 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... |
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| SubjectTerms | Dirichlet operators Exterior Lipschitz domain Harmonic function Neumann operators Riesz transform |
| Title | Riesz transform on exterior Lipschitz domains and applications |
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