Uniqueness theorems for weighted harmonic functions in the upper half-plane
We consider a class of weighted harmonic functions in the open upper half-plane known as α -harmonic functions. Of particular interest is the uniqueness problem for such functions subject to a vanishing Dirichlet boundary value on the real line and an appropriate vanishing condition at infinity. We...
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| Published in | Journal d'analyse mathématique (Jerusalem) Vol. 152; no. 1; pp. 317 - 359 |
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| Main Authors | , |
| Format | Journal Article |
| Language | English |
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Jerusalem
The Hebrew University Magnes Press
01.04.2024
Springer Nature B.V |
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| Abstract | We consider a class of weighted harmonic functions in the open upper half-plane known as
α
-harmonic functions. Of particular interest is the uniqueness problem for such functions subject to a vanishing Dirichlet boundary value on the real line and an appropriate vanishing condition at infinity. We find that the non-classical case (
α
≠ 0) allows for a considerably more relaxed vanishing condition at infinity compared to the classical case (
α
= 0) of usual harmonic functions in the upper half-plane. The reason behind this dichotomy is different geometry of zero sets of certain polynomials naturally derived from the classical binomial series. These findings shed new light on the theory of harmonic functions, for which we provide sharp uniqueness results under vanishing conditions at infinity along geodesics or along rays emanating from the origin. |
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| AbstractList | We consider a class of weighted harmonic functions in the open upper half-plane known as α-harmonic functions. Of particular interest is the uniqueness problem for such functions subject to a vanishing Dirichlet boundary value on the real line and an appropriate vanishing condition at infinity. We find that the non-classical case (α ≠ 0) allows for a considerably more relaxed vanishing condition at infinity compared to the classical case (α = 0) of usual harmonic functions in the upper half-plane. The reason behind this dichotomy is different geometry of zero sets of certain polynomials naturally derived from the classical binomial series. These findings shed new light on the theory of harmonic functions, for which we provide sharp uniqueness results under vanishing conditions at infinity along geodesics or along rays emanating from the origin. We consider a class of weighted harmonic functions in the open upper half-plane known as α -harmonic functions. Of particular interest is the uniqueness problem for such functions subject to a vanishing Dirichlet boundary value on the real line and an appropriate vanishing condition at infinity. We find that the non-classical case ( α ≠ 0) allows for a considerably more relaxed vanishing condition at infinity compared to the classical case ( α = 0) of usual harmonic functions in the upper half-plane. The reason behind this dichotomy is different geometry of zero sets of certain polynomials naturally derived from the classical binomial series. These findings shed new light on the theory of harmonic functions, for which we provide sharp uniqueness results under vanishing conditions at infinity along geodesics or along rays emanating from the origin. |
| Author | Wittsten, Jens Olofsson, Anders |
| Author_xml | – sequence: 1 givenname: Anders surname: Olofsson fullname: Olofsson, Anders organization: Centre for Mathematical Sciences, Lund University – sequence: 2 givenname: Jens surname: Wittsten fullname: Wittsten, Jens email: jens.wittsten@math.lu.se organization: Centre for Mathematical Sciences, Lund University, Department of Engineering, University of Borås |
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| Cites_doi | 10.1007/978-81-322-2113-5_8 10.1007/s11854-014-0019-4 10.1112/plms/pdm025 10.1016/j.jmaa.2015.12.026 10.2307/1969638 10.1007/b97238 10.2140/pjm.1951.1.485 10.7146/math.scand.a-12218 10.4171/rmi/668 10.2140/pjm.1996.175.571 10.1512/iumj.1996.45.1961 10.7146/math.scand.a-11697 10.1112/plms/s3-13.1.639 10.1016/j.aim.2014.07.020 10.1007/BF02392249 10.2969/jmsj/06520447 10.1080/17476933.2019.1669572 10.1016/j.bulsci.2015.04.004 10.1017/CBO9781107325937 10.1090/S0002-9904-1953-09651-3 10.1215/S0012-7094-80-04722-5 10.2140/pjm.1992.153.1 10.1016/j.bulsci.2019.102809 10.1007/s13324-021-00561-w |
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| Snippet | We consider a class of weighted harmonic functions in the open upper half-plane known as
α
-harmonic functions. Of particular interest is the uniqueness... We consider a class of weighted harmonic functions in the open upper half-plane known as α-harmonic functions. Of particular interest is the uniqueness problem... |
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| SubjectTerms | Abstract Harmonic Analysis Analysis Dirichlet problem Dynamical Systems and Ergodic Theory Functional Analysis Geodesy Half planes Harmonic functions Infinity Matematik Matematisk analys Mathematical Analysis Mathematical Sciences Mathematics Mathematics and Statistics Natural Sciences Naturvetenskap Partial Differential Equations Polynomials Uniqueness theorems |
| Title | Uniqueness theorems for weighted harmonic functions in the upper half-plane |
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