Geometric Hardy inequalities for the sub-elliptic Laplacian on convex domains in the Heisenberg group
Abstract We prove geometric $$L^p$$ L p versions of Hardy’s inequality for the sub-elliptic Laplacian on convex domains $$\Omega $$ Ω in the Heisenberg group $$\mathbb {H}^n$$ H n , where convex is meant in the Euclidean sense. When $$p=2$$ p = 2 and $$\Omega $$ Ω is the half-space given by $$\langl...
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| Format: | Article |
| Language: | English |
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World Scientific Publishing
2016-04-01
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| Series: | Bulletin of Mathematical Sciences |
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| Online Access: | http://link.springer.com/article/10.1007/s13373-016-0083-4 |
| _version_ | 1818517196150669312 |
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| author | Simon Larson |
| author_facet | Simon Larson |
| author_sort | Simon Larson |
| collection | DOAJ |
| description | Abstract We prove geometric $$L^p$$ L p versions of Hardy’s inequality for the sub-elliptic Laplacian on convex domains $$\Omega $$ Ω in the Heisenberg group $$\mathbb {H}^n$$ H n , where convex is meant in the Euclidean sense. When $$p=2$$ p = 2 and $$\Omega $$ Ω is the half-space given by $$\langle \xi , \nu \rangle > d$$ ⟨ ξ , ν ⟩ > d this generalizes an inequality previously obtained by Luan and Yang. For such p and $$\Omega $$ Ω the inequality is sharp and takes the form $$\begin{aligned} \int _\Omega |\nabla _{\mathbb {H}^n}u|^2 \, d\xi \ge \frac{1}{4}\int _{\Omega } \sum _{i=1}^n\frac{\langle X_i(\xi ), \nu \rangle ^2+\langle Y_i(\xi ), \nu \rangle ^2}{{{\mathrm{\text {dist}}}}(\xi , \partial \Omega )^2}|u|^2\, d\xi , \end{aligned}$$ ∫ Ω | ∇ H n u | 2 d ξ ≥ 1 4 ∫ Ω ∑ i = 1 n ⟨ X i ( ξ ) , ν ⟩ 2 + ⟨ Y i ( ξ ) , ν ⟩ 2 dist ( ξ , ∂ Ω ) 2 | u | 2 d ξ , where $${{\mathrm{\text {dist}}}}(\, \cdot \,, \partial \Omega )$$ dist ( · , ∂ Ω ) denotes the Euclidean distance from $$\partial \Omega $$ ∂ Ω . |
| first_indexed | 2024-12-11T00:52:55Z |
| format | Article |
| id | doaj.art-79e11e4f3be945b39eb2088ced218b8f |
| institution | Directory Open Access Journal |
| issn | 1664-3607 1664-3615 |
| language | English |
| last_indexed | 2024-12-11T00:52:55Z |
| publishDate | 2016-04-01 |
| publisher | World Scientific Publishing |
| record_format | Article |
| series | Bulletin of Mathematical Sciences |
| spelling | doaj.art-79e11e4f3be945b39eb2088ced218b8f2022-12-22T01:26:35ZengWorld Scientific PublishingBulletin of Mathematical Sciences1664-36071664-36152016-04-016333535210.1007/s13373-016-0083-4Geometric Hardy inequalities for the sub-elliptic Laplacian on convex domains in the Heisenberg groupSimon Larson0Department of Mathematics, Royal Institute of TechnologyAbstract We prove geometric $$L^p$$ L p versions of Hardy’s inequality for the sub-elliptic Laplacian on convex domains $$\Omega $$ Ω in the Heisenberg group $$\mathbb {H}^n$$ H n , where convex is meant in the Euclidean sense. When $$p=2$$ p = 2 and $$\Omega $$ Ω is the half-space given by $$\langle \xi , \nu \rangle > d$$ ⟨ ξ , ν ⟩ > d this generalizes an inequality previously obtained by Luan and Yang. For such p and $$\Omega $$ Ω the inequality is sharp and takes the form $$\begin{aligned} \int _\Omega |\nabla _{\mathbb {H}^n}u|^2 \, d\xi \ge \frac{1}{4}\int _{\Omega } \sum _{i=1}^n\frac{\langle X_i(\xi ), \nu \rangle ^2+\langle Y_i(\xi ), \nu \rangle ^2}{{{\mathrm{\text {dist}}}}(\xi , \partial \Omega )^2}|u|^2\, d\xi , \end{aligned}$$ ∫ Ω | ∇ H n u | 2 d ξ ≥ 1 4 ∫ Ω ∑ i = 1 n ⟨ X i ( ξ ) , ν ⟩ 2 + ⟨ Y i ( ξ ) , ν ⟩ 2 dist ( ξ , ∂ Ω ) 2 | u | 2 d ξ , where $${{\mathrm{\text {dist}}}}(\, \cdot \,, \partial \Omega )$$ dist ( · , ∂ Ω ) denotes the Euclidean distance from $$\partial \Omega $$ ∂ Ω .http://link.springer.com/article/10.1007/s13373-016-0083-435A2335H20 |
| spellingShingle | Simon Larson Geometric Hardy inequalities for the sub-elliptic Laplacian on convex domains in the Heisenberg group Bulletin of Mathematical Sciences 35A23 35H20 |
| title | Geometric Hardy inequalities for the sub-elliptic Laplacian on convex domains in the Heisenberg group |
| title_full | Geometric Hardy inequalities for the sub-elliptic Laplacian on convex domains in the Heisenberg group |
| title_fullStr | Geometric Hardy inequalities for the sub-elliptic Laplacian on convex domains in the Heisenberg group |
| title_full_unstemmed | Geometric Hardy inequalities for the sub-elliptic Laplacian on convex domains in the Heisenberg group |
| title_short | Geometric Hardy inequalities for the sub-elliptic Laplacian on convex domains in the Heisenberg group |
| title_sort | geometric hardy inequalities for the sub elliptic laplacian on convex domains in the heisenberg group |
| topic | 35A23 35H20 |
| url | http://link.springer.com/article/10.1007/s13373-016-0083-4 |
| work_keys_str_mv | AT simonlarson geometrichardyinequalitiesforthesubellipticlaplacianonconvexdomainsintheheisenberggroup |