This paper provides a characterization of functions of bounded variation (BV) in a compact Riemannian manifold in terms of the short time behavior of the heat semigroup. In particular, the main result proves that the total variation of a function equals the limit characterizing the space BV. The proof is carried out following two fully independent approaches, a probabilistic and an analytic one; each method presents different advantages.
Ce papier caractérise sur les variétés riemanniennes compactes les fonctions a variation bornée á l’aide d’asymptotiques en temps petit du semigroupe de la chaleur. En particulier, on montre comment la variation totale d’une fonction peut être calculée à partir du noyau de la chaleur. Nous utilisons deux approches disjointes, une approche probabiliste et une approche analytique ; ces deux approches ont des avantages différents et potentiellement peuvent être généralisées dans des contextes distincts.
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Keywords: BV functions, heat semigroup, short-time heat flow, Riemannian manifolds, Stochastic analysis on manifolds.
Patricia Alonso Ruiz 1; Fabrice Baudoin 2
CC-BY 4.0
@article{AFST_2023_6_32_3_577_0,
author = {Patricia Alonso Ruiz and Fabrice Baudoin},
title = {Yet another heat semigroup characterization of {BV} functions on {Riemannian} manifolds},
journal = {Annales de la Facult\'e des sciences de Toulouse : Math\'ematiques},
pages = {577--606},
publisher = {Universit\'e Paul Sabatier, Toulouse},
volume = {Ser. 6, 32},
number = {3},
year = {2023},
doi = {10.5802/afst.1745},
language = {en},
url = {https://afst.centre-mersenne.org/articles/10.5802/afst.1745/}
}
TY - JOUR AU - Patricia Alonso Ruiz AU - Fabrice Baudoin TI - Yet another heat semigroup characterization of BV functions on Riemannian manifolds JO - Annales de la Faculté des sciences de Toulouse : Mathématiques PY - 2023 SP - 577 EP - 606 VL - 32 IS - 3 PB - Université Paul Sabatier, Toulouse UR - https://afst.centre-mersenne.org/articles/10.5802/afst.1745/ DO - 10.5802/afst.1745 LA - en ID - AFST_2023_6_32_3_577_0 ER -
%0 Journal Article %A Patricia Alonso Ruiz %A Fabrice Baudoin %T Yet another heat semigroup characterization of BV functions on Riemannian manifolds %J Annales de la Faculté des sciences de Toulouse : Mathématiques %D 2023 %P 577-606 %V 32 %N 3 %I Université Paul Sabatier, Toulouse %U https://afst.centre-mersenne.org/articles/10.5802/afst.1745/ %R 10.5802/afst.1745 %G en %F AFST_2023_6_32_3_577_0
Patricia Alonso Ruiz; Fabrice Baudoin. Yet another heat semigroup characterization of BV functions on Riemannian manifolds. Annales de la Faculté des sciences de Toulouse : Mathématiques, Serie 6, Volume 32 (2023) no. 3, pp. 577-606. doi : 10.5802/afst.1745. https://afst.centre-mersenne.org/articles/10.5802/afst.1745/
[1] Gagliardo–Nirenberg, Trudinger–Moser and Morrey inequalities on Dirichlet spaces, J. Math. Anal. Appl., Volume 497 (2021) no. 2, 124899, 27 pages | Zbl
[2] Besov class via heat semigroup on Dirichlet spaces I: Sobolev type inequalities, J. Funct. Anal., Volume 278 (2020) no. 11, 108459, 48 pages
[3] Besov class via heat semigroup on Dirichlet spaces II: BV functions and Gaussian heat kernel estimates, Calc. Var. Partial Differ. Equ., Volume 59 (2020) no. 3, 103, 32 pages
[4] Besov class via heat semigroup on Dirichlet spaces III: BV functions and sub-Gaussian heat kernel estimates, Calc. Var. Partial Differ. Equ., Volume 60 (2021) no. 5, 170, 38 pages
[5] Equivalent definitions of space and of total variation on metric measure spaces, J. Funct. Anal., Volume 266 (2014) no. 7, pp. 4150-4188
[6] Geometric inequalities on Riemannian and sub-Riemannian manifolds by heat semigroups techniques (2018) (https://arxiv.org/abs/1801.05702, to appear in Levico Summer School Lecture notes)
[7] Another look at Sobolev spaces, Optimal control and partial differential equations (IOS Press), Ohmsha, 2001, pp. 439-455
[8] Comparison theorems for exit times, Geom. Funct. Anal., Volume 11 (2001) no. 4, pp. 651-692
[9] A note on bounded variation and heat semigroup on Riemannian manifolds, Bull. Aust. Math. Soc., Volume 76 (2007) no. 1, pp. 155-160
[10] Eigenvalues in Riemannian geometry, Pure and Applied Mathematics, 115, Academic Press Inc., 1984 (including a chapter by Burton Randol, with an appendix by Jozef Dodziuk)
[11] On an open question about functions of bounded variation, Calc. Var. Partial Differ. Equ., Volume 15 (2002) no. 4, pp. 519-527
[12] On the space of BV functions and a related stochastic calculus in infinite dimensions, J. Funct. Anal., Volume 183 (2001) no. 1, pp. 245-268
[13] Riemannian geometry, Universitext, Springer, 1990
[14] Heat kernel and analysis on manifolds, AMS/IP Studies in Advanced Mathematics, 47, American Mathematical Society, 2009
[15] Functions with bounded variation on a class of Riemannian manifolds with Ricci curvature unbounded from below, Math. Ann., Volume 363 (2015) no. 3, pp. 1307-1331
[16] Nonlinear analysis on manifolds: Sobolev spaces and inequalities, Courant Lecture Notes in Mathematics, 5, American Mathematical Society, 1999
[17] Stochastic analysis on manifolds, Graduate Studies in Mathematics, 38, American Mathematical Society, 2002
[18] Sobolev spaces and harmonic maps for metric space targets, Commun. Anal. Geom., Volume 1 (1993) no. 4, pp. 561-659
[19] Fractional Sobolev norms and BV functions on manifolds, Nonlinear Anal., Theory Methods Appl., Volume 187 (2019), pp. 450-466
[20] Semigroup proofs of the isoperimetric inequality in Euclidean and Gauss space, Bull. Sci. Math., Volume 118 (1994) no. 6, pp. 485-510
[21] Characterization of Sobolev and spaces, J. Funct. Anal., Volume 261 (2011) no. 10, pp. 2926-2958
[22] Strong short-time asymptotics and convolution approximation of the heat kernel, Ann. Global Anal. Geom., Volume 55 (2019) no. 2, pp. 371-394
[23] Characterizations of sets of finite perimeter using heat kernels in metric spaces, Potential Anal., Volume 45 (2016) no. 4, pp. 609-633
[24] Functions of bounded variation on “good” metric spaces, J. Math. Pures Appl., Volume 82 (2003) no. 8, pp. 975-1004
[25] Heat semigroup and functions of bounded variation on Riemannian manifolds, J. Reine Angew. Math., Volume 613 (2007), pp. 99-119
[26] Short-time heat flow and functions of bounded variation in , Ann. Fac. Sci. Toulouse, Math., Volume 16 (2007) no. 1, pp. 125-145
[27] A semigroup version of the isoperimetric inequality, Semigroup Forum, Volume 68 (2004) no. 2, pp. 233-245
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