Resistance conditions, Poincaré inequalities, the Lip-lip condition and hardy’s inequalities

• Autorzy: Kinnunen J. , Silvestre P.
• Języki publikacji: EN

Abstrakty

EN

This note investigates weaker conditions than a Poincaré inequality in analysis on metric measure spaces. We discuss two resistance conditions which are stated in terms of capacities. We show that these conditions can be characterized by versions of Sobolev–Poincaré inequalities. As a consequence, we obtain so-called Lip-lip condition related to pointwise Lipschitz constants. Moreover, we show that the pointwise Hardy inequalities and uniform fatness conditions are equivalent under an appropriate resistance condition.

Słowa kluczowe

EN

capacity; Sobolev–Poincaré inequality; Hardy inequality; analysis on metric measure spaces

PL

przepustowość; nierówności Poincarégo; nierówności Soboleva; nierówność Hardy'ego

• Wydawca: De Gruyter
• Czasopismo: Demonstratio Mathematica
• Rocznik: 2016
• Tom: Vol. 49, nr 1
• Strony: 50--63
• Opis fizyczny: Bibliogr. 11 poz.

Twórcy

autor

Kinnunen J.

• Department of Mathematics Aalto University Fi-00076 Aalto, Finland

autor

Silvestre P.

• Department of Mathematics Aalto University Fi-00076 Aalto, Finland

Bibliografia

• [1] M. T. Barlow, R. F. Bass, T. Kumagai, Stability of parabolic Harnack inequalities on metric measure spaces, J. Math. Soc. Japan 58(2) (2006), 485–519.
• [2] J. Björn, Boundary continuity for quasiminimizers on metric spaces, Illinois J. Math. 46(2) (2002), 383–403.
• [3] A. Björn, J. Björn, Nonlinear potential theory on metric spaces, Tracts in Mathematics 17, European Mathematical Society, 2011.
• [4] A. Grigor’yan, A. Telcs, Harnack inequalities and sub-Gaussian estimates for random walks, Math. Ann. 324(3) (2002), 521–556.
• [5] J. Heinonen, Lectures on analysis on metric spaces, Universitext, Springer-Verlag, New York, 2001.
• [6] J. Heinonen, P. Koskela, A note on Lipschitz functions, upper gradients, and the Poincaré inequality, New Zealand J. Math. 28 (1999), 37–42.
• [7] S. Keith, A differentiable structure for metric measure spaces, Adv. Math. 183(2) (2004), 27–315.
• [8] J. Kinnunen, R. Korte, Characterizations of Sobolev inequalities on metric spaces, J. Math. Anal. Appl. 344 (2008), 1093–1104.
• [9] J. Kinnunen, P. Silvestre, Resistance conditions and applications, Anal. Geom. Metric Spaces 1 (2013), 276–294.
• [10] R. Korte, J. Lehrbäck, H. Tuominen, The equivalence between pointwise Hardy inequalities and uniform fatness, Math. Ann. 351 (2011), 711–731.
• [11] V. Maz’ya, Sobolev spaces with applications to elliptic partial differential equations, Grundlehren der Mathematischen Wissenschaften 342, Springer, 2011.

Uwagi

Opracowanie ze środków MNiSW w ramach umowy 812/P-DUN/2016 na działalność upowszechniającą naukę.