Resistance Conditions and Applications

Juha Kinnunen; Pilar Silvestre

Analysis and Geometry in Metric Spaces (2013)

  • Volume: 1, page 276-294
  • ISSN: 2299-3274

Abstract

top
This paper studies analytic aspects of so-called resistance conditions on metric measure spaces with a doubling measure. These conditions are weaker than the usually assumed Poincaré inequality, but however, they are sufficiently strong to imply several useful results in analysis on metric measure spaces. We show that under a perimeter resistance condition, the capacity of order one and the Hausdorff content of codimension one are comparable. Moreover, we have connections to the Sobolev inequality for compactly supported Lipschitz functions on balls as well as capacitary strong type estimates for the Hardy-Littlewood maximal function. We also consider extensions to Sobolev type inequalities with two different measures and Lorentz type estimates.

How to cite

top

Juha Kinnunen, and Pilar Silvestre. "Resistance Conditions and Applications." Analysis and Geometry in Metric Spaces 1 (2013): 276-294. <http://eudml.org/doc/266944>.

@article{JuhaKinnunen2013,
abstract = {This paper studies analytic aspects of so-called resistance conditions on metric measure spaces with a doubling measure. These conditions are weaker than the usually assumed Poincaré inequality, but however, they are sufficiently strong to imply several useful results in analysis on metric measure spaces. We show that under a perimeter resistance condition, the capacity of order one and the Hausdorff content of codimension one are comparable. Moreover, we have connections to the Sobolev inequality for compactly supported Lipschitz functions on balls as well as capacitary strong type estimates for the Hardy-Littlewood maximal function. We also consider extensions to Sobolev type inequalities with two different measures and Lorentz type estimates.},
author = {Juha Kinnunen, Pilar Silvestre},
journal = {Analysis and Geometry in Metric Spaces},
keywords = {Metric measure space; resistance condition; Poincaré inequality; Hausdorff content of codimension one; Hardy-Littlewood maximal function; Sobolev type inequalities; metric measure space},
language = {eng},
pages = {276-294},
title = {Resistance Conditions and Applications},
url = {http://eudml.org/doc/266944},
volume = {1},
year = {2013},
}

TY - JOUR
AU - Juha Kinnunen
AU - Pilar Silvestre
TI - Resistance Conditions and Applications
JO - Analysis and Geometry in Metric Spaces
PY - 2013
VL - 1
SP - 276
EP - 294
AB - This paper studies analytic aspects of so-called resistance conditions on metric measure spaces with a doubling measure. These conditions are weaker than the usually assumed Poincaré inequality, but however, they are sufficiently strong to imply several useful results in analysis on metric measure spaces. We show that under a perimeter resistance condition, the capacity of order one and the Hausdorff content of codimension one are comparable. Moreover, we have connections to the Sobolev inequality for compactly supported Lipschitz functions on balls as well as capacitary strong type estimates for the Hardy-Littlewood maximal function. We also consider extensions to Sobolev type inequalities with two different measures and Lorentz type estimates.
LA - eng
KW - Metric measure space; resistance condition; Poincaré inequality; Hausdorff content of codimension one; Hardy-Littlewood maximal function; Sobolev type inequalities; metric measure space
UR - http://eudml.org/doc/266944
ER -

References

top
  1. [1] M. T. Barlow, R. F. Bass and T. Kumagai, Stability of parabolic Harnack inequalities on metric measure spaces, J. Math. Soc. Japan 58 (2006), no. 2, 485–519. Zbl1102.60064
  2. [2] C. Bennett and S. Sharpley, Interpolation of Operators, Pure and Applied Mathematics, 129. Academic Press, Inc., Boston, MA, 1988. Zbl0647.46057
  3. [3] A. Björn and J. Björn, Nonlinear Potential Theory on Metric Spaces, Tracts in Mathematics 17, European Mathematical Society, 2011. Zbl1231.31001
  4. [4] Yu. A. Brudnyi and N. Ya. Krugljak, Interpolation functors and interpolation spaces, Vol. I. North-Holland Mathematical Library, 47. North-Holland Publishing Co., Amsterdam, 1991. 
  5. [5] J. Cerdà, Lorentz capacity spaces, Interpolation theory and applications, Contemp. Math. 445, 45–59, Amer. Math. Soc., Providence, RI, 2007. Zbl1141.46313
  6. [6] J. Cerdà, J. Martín and P. Silvestre, Capacitary function spaces, Collectanea Math. 62 (2011), no. 1, 95–118. Zbl1225.46021
  7. [7] J. Cerdà, J. Martín and P. Silvestre, Conductor Sobolev type estimates and isocapacitary inequalities, to appear in Indiana Univ. Math. J. Zbl1281.46033
  8. [8] S. Costea and V. G. Maz’ya, Conductor inequalities and criteria for Sobolev-Lorentz two-weight inequalities, Sobolev spaces in mathematics. II, 103–121, Int. Math. Ser. (N. Y.) 9 (2009), Springer, New York. Zbl1165.26009
  9. [9] A. Grigor’yan and A. Telcs, Harnack inequalities and sub-Gaussian estimates for random walks, Math. Ann. 324 (2002), no. 3, 521–556. Zbl1011.60021
  10. [10] H. Hakkarainen and J. Kinnunen, The BV-capacity in metric spaces, Manuscripta Math. 132 (2010), no. 1-2, 51–73. Zbl1194.28001
  11. [11] H. Hakkarainen and N. Shanmugalingam, Comparisons of relative BV-capacities and Sobolev capacity in metric spaces, Nonlinear Anal. 74 (2011), no. 16, 5525–5543. Zbl1248.28002
  12. [12] J. Kinnunen, R. Korte, N. Shanmugalingam and H. Tuominen, Lebesgue points and capacities via boxing inequality in metric spaces, Indiana Univ. Math. J. 57 (2008), no. 1, 401–430. [WoS] Zbl1146.46018
  13. [13] J. Kinnunen, R. Korte, N. Shanmugalingam and H. Tuominen, The DeGiorgi measure and an obstacle problem related to minimal surfaces in metric spaces, J. Math. Pures Appl. (9) 93 (2010), no. 6, 599–622. [WoS] Zbl1211.49055
  14. [14] V. G. Maz’ya, Conductor and capacitary inequalities for functions on topological spaces and their applications to Sobolev type imbeddings, J. Funct. Anal. 224 (2005), no. 2, 408–430. 
  15. [15] V. G. Maz’ya, Conductor inequalities and criteria for Sobolev type two-weight imbeddings. J. Comput. Appl. Math. 194 (2006), no. 11, 94–114. Zbl1104.46020
  16. [16] M. Miranda, Functions of bounded variation on "good" metric spaces, J. Math. Pures Appl. (9) 82 (2003), no. 8, 975–1004. Zbl1109.46030
  17. [17] J. Orobitg and J. Verdera, Choquet integrals, Hausdorff content and the Hardy-Littlewood maximal operator, Bull. London Math. Soc. 30 (1998), no. 2, 145–150. Zbl0921.42016
  18. [18] P. Silvestre, Capacitary function spaces and applications, PhD-thesis (2012), TDR, B. 8121-2012. www.tesisenred.net/handle/10803/77717 

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

                
Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.