Sobolev spaces and Poincaré inequalities on the Vicsek fractal

Fabrice Baudoin; Li Chen

doi:10.54330/AFM.122168

Sobolev spaces and Poincaré inequalities on the Vicsek fractal

^*Corresponding author af dette arbejde

Publikation: Bidrag til tidsskrift/Konferencebidrag i tidsskrift /Bidrag til avis › Tidsskriftartikel › Forskning › peer review

Abstract

In this paper we prove that several natural approaches to Sobolev spaces coincide on the Vicsek fractal. More precisely, we show that the metric approach of Korevaar–Schoen, the approach by limit of discrete p-energies and the approach by limit of Sobolev spaces on cable systems all yield the same functional space with equivalent norms for p > 1. As a consequence we prove that the Sobolev spaces form a real interpolation scale. We also obtain Lp-Poincaré inequalities for all values of p ≥ 1.

Originalsprog	Engelsk
Tidsskrift	Annales Fennici Mathematici
Vol/bind	48
Nummer	1
Sider (fra-til)	3-26
Antal sider	24
ISSN	2737-0690
DOI	https://doi.org/10.54330/AFM.122168
Status	Udgivet - 2022
Udgivet eksternt	Ja

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AB - In this paper we prove that several natural approaches to Sobolev spaces coincide on the Vicsek fractal. More precisely, we show that the metric approach of Korevaar–Schoen, the approach by limit of discrete p-energies and the approach by limit of Sobolev spaces on cable systems all yield the same functional space with equivalent norms for p > 1. As a consequence we prove that the Sobolev spaces form a real interpolation scale. We also obtain Lp-Poincaré inequalities for all values of p ≥ 1.

KW - P-energies

KW - Poincaré inequalities

KW - Real interpolation

KW - Sobolev spaces

KW - Vicsek set

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Sobolev spaces and Poincaré inequalities on the Vicsek fractal

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