Geodesic
Sobolev spaces and Poincaré inequalities on the Vicsek fractal
Fabrice Baudoin1 ; Li Chen2
1 University of Connecticut, Department of Mathematics
2 Louisiana State University, Department of Mathematics
Annales Fennici Mathematici, Tome 48 (2023) no. 1, pp. 3-26.

Voir la notice de l'article provenant de la source Journal.fi

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 $L^p$-Poincaré inequalities for all values of $p \ge 1$.
DOI : 10.54330/afm.122168
Keywords: Vicsek set, Sobolev spaces, Poincaré inequalities, p-energies, real interpolation

Fabrice Baudoin 1 ; Li Chen 2

1 University of Connecticut, Department of Mathematics
2 Louisiana State University, Department of Mathematics 
@article{AFM_2023_48_1_a0,
     author = {Fabrice Baudoin and Li Chen},
     title = {Sobolev spaces and {Poincar\'e} inequalities  on the {Vicsek} fractal},
     journal = {Annales Fennici Mathematici},
     pages = {3--26},
     publisher = {mathdoc},
     volume = {48},
     number = {1},
     year = {2023},
     doi = {10.54330/afm.122168},
     language = {en},
     url = {http://geodesic.mathdoc.fr/articles/10.54330/afm.122168/}
}
TY  - JOUR
AU  - Fabrice Baudoin
AU  - Li Chen
TI  - Sobolev spaces and Poincaré inequalities  on the Vicsek fractal
JO  - Annales Fennici Mathematici
PY  - 2023
SP  - 3
EP  - 26
VL  - 48
IS  - 1
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/articles/10.54330/afm.122168/
DO  - 10.54330/afm.122168
LA  - en
ID  - AFM_2023_48_1_a0
ER  - 
%0 Journal Article
%A Fabrice Baudoin
%A Li Chen
%T Sobolev spaces and Poincaré inequalities  on the Vicsek fractal
%J Annales Fennici Mathematici
%D 2023
%P 3-26
%V 48
%N 1
%I mathdoc
%U http://geodesic.mathdoc.fr/articles/10.54330/afm.122168/
%R 10.54330/afm.122168
%G en
%F AFM_2023_48_1_a0
Fabrice Baudoin; Li Chen. Sobolev spaces and Poincaré inequalities  on the Vicsek fractal. Annales Fennici Mathematici, Tome 48 (2023) no. 1, pp. 3-26. doi : 10.54330/afm.122168. http://geodesic.mathdoc.fr/articles/10.54330/afm.122168/

Cité par Sources :