Hostname: page-component-cd9895bd7-gvvz8 Total loading time: 0 Render date: 2024-12-27T10:52:07.406Z Has data issue: false hasContentIssue false
  • English
  • Français

Weighted Poincaré inequalities and Minkowski content

Published online by Cambridge University Press:  14 November 2011

D. E. Edmunds  and
R. Hurri-Syrjänen
D. E. Edmunds
Affiliation:
Mathematics Division, University of Sussex, Falmer, Brighton, East Sussex BN1 9QH, U.K.
R. Hurri-Syrjänen
Affiliation:
Department of Mathematics, University of Helsinki, Hallituskatu 15, SF-00100 Helsinki, Finland
Get access Rights & Permissions [Opens in a new window]

Abstract

Weighted Poincaré inequalities are established in any bounded domain D in ℝn (n ≧ 2), and their connection with the Minkowski content of ∂D is explored.

Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 125 , Issue 4 , 1995 , pp. 817 - 825
Copyright
Copyright © Royal Society of Edinburgh 1995

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

1Chanillo, S. and Wheeden, R.. Weighted Poincaré and Sobolev inequalities and estimates for weighted Peano maximal functions. Amer. J. Math. 107 (1985), 1191–226.CrossRefGoogle Scholar
2Chanillo, S. and Wheeden, R.. Poincaré inequalities for a class of non-Ap weights. Indiana Math. J. 41 (1992), 605–24.CrossRefGoogle Scholar
3Edmunds, D. E. and Opic, B.. Weighted Poincaré and Friedrichs inequalities. J. London Math. Soc. 47 (1993), 79–91.Google Scholar
4Fabes, E., Kenig, C. and Serapioni, R.. The local regularity of solutions of degenerate elliptic equations. Comm. Partial Differential Equations 7 (1982), 77–116.CrossRefGoogle Scholar
5Hurri, R.. Poincaré domains in ℝn. Ann. Acad. Sci. Venn. Ser. A I Math. Dissertationes 71 (1988), 1–42.Google Scholar
6Hurri, R.. The weighted Poincaré inequalities. Math. Scand. 67 (1990), 145–60.CrossRefGoogle Scholar
7Hurri-Syrjänen, R.. An improved Poincaré inequality. Proc. Amer. Math. Soc. 120 (1994), 213–22.Google Scholar
8Koskela, P.. Natural weights for the Poincaré inequality. Proc. Finnish-Polish Summer School 1992.Google Scholar
9Martio, O. and Vuorinen, M.. Whitney cubes, p-capacity and Minkowski content. Exposition. Math. 5 (1987), 17–40.Google Scholar
10Sawyer, E. and Wheeden, R.. Weighted inequalities for fractional integrals on Euclidean and homogeneous spaces. Amer. J. Math. 114 (1992), 813–74.CrossRefGoogle Scholar
11Smith, W. and Stegenga, D.. Exponential integrability of the quasi-hyperbolic metric on Hölder domains. Ann. Acad. Sci. Fenn. Ser. A I Math. 16 (1991), 345–60.CrossRefGoogle Scholar
12Stein, E. M.. Singular Integrals and Differentiability Properties of Functions (Princeton: Princeton University Press, 1970).Google Scholar