Local and semilocal Poincaré inequalities on metric spaces

Abstract

We consider several local versions of the doubling condition and Poincaré inequalities on metric measure spaces. Our first result is that in proper connected spaces, the weakest local assumptions self-improve to semilocal ones, i.e. holding within every ball.We then study various geometrical and analytical consequences of such local assumptions, such as local quasiconvexity, self-improvement of Poincaré inequalities, existence of Lebesgue points, density of Lipschitz functions and quasicontinuity of Sobolev functions. It turns out that local versions of these properties hold under local assumptions, even though they are not always straightforward.We also conclude that many qualitative, as well as quantitative, properties of pharmonic functions on metric spaces can be proved in various forms under such local assumptions, with the main exception being the Liouville theorem, which fails without global assumptions.Résumé. Nous considérons plusieurs versions locales des conditions de doublement et des inégalités de Poincaré dans des espaces métriques mesurés. Notre premier résultat stipule que dans un espace propre connexe, les hypothèses locales les plus faibles s'améliorent en semi-locales, c.à.d. elles sont valables dans chaque boule.Nousétudions ensuite certaines conséquences géométriques et analytiques de telles hypothèses locales tel que la quasi-convexité locale, l'auto amélioration des inégalités de Poincaré, l'existence des points Lebesgue, la densité des fonctions Lipschitz et la quasicontinuité des fonctions Sobolev. Il s'avère que les versions locales de ces propriétés restent valables sous les hypothèses locales même si elles ne sont pas toujours immédiates.Nous concluonségalement que sous telles hypothéses locales, plusieurs propriétés qualitatives, ainsi que quantitatives, des fonctions p-harmoniques sur des espaces métriques peuventêtre prouvées sous diverses formes, l'exception principaleétant le théorème de Liouville quiéchoue sans hypothéses globales.

Poincaré inequalities and Newtonian Sobolev functions on noncomplete metric spaces

“…by Cheeger [13], Danielli-Garofalo-Marola [14], Garofalo-Marola [17] and Holopainen-Shanmugalingam [23]. In the following definition we follow the recent terminology from Björn-Björn [6], where a more extensive discussion of these assumptions can be found. x 0 ∈ X there is r 0 > 0 (depending on x 0 ) such that the doubling property or the p-Poincaré inequality holds within B(x 0 , r 0 ).…”

“…When X is complete and µ is globally doubling, a deep result due to Keith-Zhong [24, Theorem 1.0.1] shows that the Poincaré inequality is an open-ended property, in the sense that if µ supports a global p-Poincaré inequality then it also supports a global q-Poincaré inequality for some q < p. Counterexamples due to Koskela [29] show that this is false for locally compact X. Nevertheless, by localizing the arguments in [24] local versions of this self-improvement result were obtained in [6] for locally compact spaces. In Section 5, we further generalize these results to non-locally compact spaces, using our extension theorem as the key tool.…”

Self-improvement of pointwise Hardy inequality

“…We impose the following localized doubling condition on the weight w. We remark that there are also other uses for the term semilocally doubling in the literature, see e.g. [2]. In our definition "local" refers to the fact that the condition is required only for points x ∈ Ω, but "semi" is added since the balls need not be contained in Ω.…”

Locally p-admissible measures on R

“…Intervals centred at x ∈ X \ [− 1 2 M, 1 2 M ], and of length at most 1 2 M , can be treated similarly by reflecting at M . Since X is compact, the global doubling and p-Poincaré inequality then follow for all I ⊂ X, by Proposition 1.2 and Theorem 1.3 in Björn-Björn[3].By symmetry, we can assume that 0 ≤ x ≤ 2r. (If 2r < x ≤ 1 2 M then 2I ⊂ [0, M ] and the doubling property forμ and I is immediate.)…”

“…This complements some results in Kilpeläinen-Koskela-Masaoka [21], where such questions were studied for global A p weights and globally p-admissible measures on R n . As a byproduct, we provide an elementary proof of [21,Proposition 4.3], see Lemma 5.3. This note is a continuation of the systematic development of local and semilocal doubling measures and Poincaré inequalities on metric spaces from Björn-Björn [3] and [4]. Local assumptions are also natural for studying p-harmonic and quasiharmonic functions, and Theorem 1.2 plays a role in Liouville type theorems on the real line, see Björn-Björn-Shanmugalingam [5].…”