Distortion of Hausdorff measures under Orlicz-Sobolev maps
A comprehensive theory of the effect of Orlicz-Sobolev maps, between Euclidean spaces, on subsets with zero or finite Hausdorff measure is offered. Arbitrary Orlicz-Sobolev spaces embedded into the space of continuous function and Hausdorff measures built upon general gauge functions are included in...
| Main Authors: | , , |
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| Format: | Journal article |
| Language: | English |
| Published: |
European Mathematical Society
2024
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| _version_ | 1811139256672845824 |
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| author | Cianchi, A Korobkov, M Kristensen, J |
| author_facet | Cianchi, A Korobkov, M Kristensen, J |
| author_sort | Cianchi, A |
| collection | OXFORD |
| description | A comprehensive theory of the effect of Orlicz-Sobolev maps, between Euclidean spaces, on
subsets with zero or finite Hausdorff measure is offered. Arbitrary Orlicz-Sobolev spaces embedded
into the space of continuous function and Hausdorff measures built upon general gauge functions
are included in our discussion. An explicit formula for the distortion of the relevant gauge function
under the action of these maps is offered in terms of the Young function defining the OrliczSobolev space. Classical results, dealing with standard Sobolev spaces and Hausdorff measures,
are recovered, and their optimality is shown to hold in a refined stronger sense. Special instances
available in the literature, concerning Young functions and gauge functions of non-power type,
are also reproduced and, when not sharp, improved. New phenomena and features, related to the
flexibility in the definition of the degree of integrability of weak derivatives of maps and in the
notion of measure of sets, are detected. For instance, perhaps surprisingly, Hausdorff measures,
other than the Lebesgue measure, which are not distorted by classical Sobolev maps are exhibited.
The relevant measures are associated with gauge functions with a logarithmic decay near zero,
which cannot be contemplated by the standard theory. |
| first_indexed | 2024-03-07T08:16:49Z |
| format | Journal article |
| id | oxford-uuid:2f264135-dafb-4c12-a61b-9be555aa338a |
| institution | University of Oxford |
| language | English |
| last_indexed | 2024-09-25T04:03:12Z |
| publishDate | 2024 |
| publisher | European Mathematical Society |
| record_format | dspace |
| spelling | oxford-uuid:2f264135-dafb-4c12-a61b-9be555aa338a2024-05-08T09:44:37ZDistortion of Hausdorff measures under Orlicz-Sobolev mapsJournal articlehttp://purl.org/coar/resource_type/c_dcae04bcuuid:2f264135-dafb-4c12-a61b-9be555aa338aEnglishSymplectic ElementsEuropean Mathematical Society2024Cianchi, AKorobkov, MKristensen, JA comprehensive theory of the effect of Orlicz-Sobolev maps, between Euclidean spaces, on subsets with zero or finite Hausdorff measure is offered. Arbitrary Orlicz-Sobolev spaces embedded into the space of continuous function and Hausdorff measures built upon general gauge functions are included in our discussion. An explicit formula for the distortion of the relevant gauge function under the action of these maps is offered in terms of the Young function defining the OrliczSobolev space. Classical results, dealing with standard Sobolev spaces and Hausdorff measures, are recovered, and their optimality is shown to hold in a refined stronger sense. Special instances available in the literature, concerning Young functions and gauge functions of non-power type, are also reproduced and, when not sharp, improved. New phenomena and features, related to the flexibility in the definition of the degree of integrability of weak derivatives of maps and in the notion of measure of sets, are detected. For instance, perhaps surprisingly, Hausdorff measures, other than the Lebesgue measure, which are not distorted by classical Sobolev maps are exhibited. The relevant measures are associated with gauge functions with a logarithmic decay near zero, which cannot be contemplated by the standard theory. |
| spellingShingle | Cianchi, A Korobkov, M Kristensen, J Distortion of Hausdorff measures under Orlicz-Sobolev maps |
| title | Distortion of Hausdorff measures under Orlicz-Sobolev maps |
| title_full | Distortion of Hausdorff measures under Orlicz-Sobolev maps |
| title_fullStr | Distortion of Hausdorff measures under Orlicz-Sobolev maps |
| title_full_unstemmed | Distortion of Hausdorff measures under Orlicz-Sobolev maps |
| title_short | Distortion of Hausdorff measures under Orlicz-Sobolev maps |
| title_sort | distortion of hausdorff measures under orlicz sobolev maps |
| work_keys_str_mv | AT cianchia distortionofhausdorffmeasuresunderorliczsobolevmaps AT korobkovm distortionofhausdorffmeasuresunderorliczsobolevmaps AT kristensenj distortionofhausdorffmeasuresunderorliczsobolevmaps |