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On the Coincidence between Campanato Functions and Lipschitz Functions: A New Approach via Elliptic PDES.
- Source :
-
Quarterly Journal of Mathematics . Jun2024, Vol. 75 Issue 2, p663-693. 31p. - Publication Year :
- 2024
-
Abstract
- Let |$({\mathcal{M}},d,\mu)$| be the metric measure space with a Dirichlet form |$\mathscr{E}$|. In this paper, we obtain that the Campanato function and the Lipschitz function do always coincide. Our approach is based on the harmonic extension technology, which extends a function u on |${\mathcal{M}}$| to its Poisson integral P t u on |${\mathcal{M}}\times\mathbb{R}_+$|. With this tool in hand, we can utilize the same Carleson measure condition of the Poisson integral to characterize its Campanato/Lipschitz trace, and hence, they are equivalent to each other. This equivalence was previously obtained by Macías–Segovia [Adv. Math. 1979]. However, we provide a new proof, via the boundary value problem for the elliptic equation. This result indicates the famous saying of Stein–Weiss at the beginning of Chapter II in their book [Princeton Mathematical Series, No. 32, 1971]. [ABSTRACT FROM AUTHOR]
Details
- Language :
- English
- ISSN :
- 00335606
- Volume :
- 75
- Issue :
- 2
- Database :
- Academic Search Index
- Journal :
- Quarterly Journal of Mathematics
- Publication Type :
- Academic Journal
- Accession number :
- 178134855
- Full Text :
- https://doi.org/10.1093/qmath/haae019