Asymptotically Mean Value Harmonic Functions in Doubling Metric Measure Spaces
We consider functions with an asymptotic mean value property, known to characterize harmonicity in Riemannian manifolds and in doubling metric measure spaces. We show that the strongly amv-harmonic functions are Hölder continuous for any exponent below one. More generally, we define the class of fun...
| Main Authors: | , , |
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| Format: | Article |
| Language: | English |
| Published: |
De Gruyter
2022-11-01
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| Series: | Analysis and Geometry in Metric Spaces |
| Subjects: | |
| Online Access: | https://doi.org/10.1515/agms-2022-0143 |
| _version_ | 1811320092703588352 |
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| author | Adamowicz Tomasz Kijowski Antoni Soultanis Elefterios |
| author_facet | Adamowicz Tomasz Kijowski Antoni Soultanis Elefterios |
| author_sort | Adamowicz Tomasz |
| collection | DOAJ |
| description | We consider functions with an asymptotic mean value property, known to characterize harmonicity in Riemannian manifolds and in doubling metric measure spaces. We show that the strongly amv-harmonic functions are Hölder continuous for any exponent below one. More generally, we define the class of functions with finite amv-norm and show that functions in this class belong to a fractional Hajłasz–Sobolev space and their blow-ups satisfy the mean-value property. Furthermore, in the weighted Euclidean setting we find an elliptic PDE satisfied by amv-harmonic functions. |
| first_indexed | 2024-04-13T12:54:14Z |
| format | Article |
| id | doaj.art-d643be447bab4064a3e6f04619557749 |
| institution | Directory Open Access Journal |
| issn | 2299-3274 |
| language | English |
| last_indexed | 2024-04-13T12:54:14Z |
| publishDate | 2022-11-01 |
| publisher | De Gruyter |
| record_format | Article |
| series | Analysis and Geometry in Metric Spaces |
| spelling | doaj.art-d643be447bab4064a3e6f046195577492022-12-22T02:46:07ZengDe GruyterAnalysis and Geometry in Metric Spaces2299-32742022-11-0110134437210.1515/agms-2022-0143Asymptotically Mean Value Harmonic Functions in Doubling Metric Measure SpacesAdamowicz Tomasz0Kijowski Antoni1Soultanis Elefterios2Institute of Mathematics, Polish Academy of Sciences, Warsaw, PolandAnalysis on Metric Spaces Unit, OIST, Okinawa, JapanRadboud University, IMAPP, Nijmegen, The NetherlandsWe consider functions with an asymptotic mean value property, known to characterize harmonicity in Riemannian manifolds and in doubling metric measure spaces. We show that the strongly amv-harmonic functions are Hölder continuous for any exponent below one. More generally, we define the class of functions with finite amv-norm and show that functions in this class belong to a fractional Hajłasz–Sobolev space and their blow-ups satisfy the mean-value property. Furthermore, in the weighted Euclidean setting we find an elliptic PDE satisfied by amv-harmonic functions.https://doi.org/10.1515/agms-2022-0143asymptotic mean value propertyelliptic pdesharmonic functionsgromov–hausdorff convergencehölder continuitymean value propertysobolev spacesweighted euclidean spacesprimary: 31e05secondary: 53c2335r03 |
| spellingShingle | Adamowicz Tomasz Kijowski Antoni Soultanis Elefterios Asymptotically Mean Value Harmonic Functions in Doubling Metric Measure Spaces Analysis and Geometry in Metric Spaces asymptotic mean value property elliptic pdes harmonic functions gromov–hausdorff convergence hölder continuity mean value property sobolev spaces weighted euclidean spaces primary: 31e05 secondary: 53c23 35r03 |
| title | Asymptotically Mean Value Harmonic Functions in Doubling Metric Measure Spaces |
| title_full | Asymptotically Mean Value Harmonic Functions in Doubling Metric Measure Spaces |
| title_fullStr | Asymptotically Mean Value Harmonic Functions in Doubling Metric Measure Spaces |
| title_full_unstemmed | Asymptotically Mean Value Harmonic Functions in Doubling Metric Measure Spaces |
| title_short | Asymptotically Mean Value Harmonic Functions in Doubling Metric Measure Spaces |
| title_sort | asymptotically mean value harmonic functions in doubling metric measure spaces |
| topic | asymptotic mean value property elliptic pdes harmonic functions gromov–hausdorff convergence hölder continuity mean value property sobolev spaces weighted euclidean spaces primary: 31e05 secondary: 53c23 35r03 |
| url | https://doi.org/10.1515/agms-2022-0143 |
| work_keys_str_mv | AT adamowicztomasz asymptoticallymeanvalueharmonicfunctionsindoublingmetricmeasurespaces AT kijowskiantoni asymptoticallymeanvalueharmonicfunctionsindoublingmetricmeasurespaces AT soultaniselefterios asymptoticallymeanvalueharmonicfunctionsindoublingmetricmeasurespaces |