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.
 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.
 C. Bennett and S. Sharpley, Interpolation of Operators, Pure and Applied Mathematics, 129. Academic Press, Inc.,
Boston, MA, 1988.
 A. Björn and J. Björn, Nonlinear Potential Theory on Metric Spaces, Tracts in Mathematics 17, European Mathematical
 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.
 J. Cerdà, Lorentz capacity spaces, Interpolation theory and applications, Contemp. Math. 445, 45–59, Amer. Math.
Soc., Providence, RI, 2007.
 J. Cerdà, J. Martín and P. Silvestre, Capacitary function spaces, Collectanea Math. 62 (2011), no. 1, 95–118.
 J. Cerdà, J. Martín and P. Silvestre, Conductor Sobolev type estimates and isocapacitary inequalities, to appear in
Indiana Univ. Math. J.
 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.
 A. Grigor’yan and A. Telcs, Harnack inequalities and sub-Gaussian estimates for random walks, Math. Ann. 324
(2002), no. 3, 521–556.
 H. Hakkarainen and J. Kinnunen, The BV-capacity in metric spaces, Manuscripta Math. 132 (2010), no. 1-2, 51–73.
 H. Hakkarainen and N. Shanmugalingam, Comparisons of relative BV-capacities and Sobolev capacity in metric
spaces, Nonlinear Anal. 74 (2011), no. 16, 5525–5543.
 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.
 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.
 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.
 V. G. Maz’ya, Conductor inequalities and criteria for Sobolev type two-weight imbeddings. J. Comput. Appl. Math.
194 (2006), no. 11, 94–114.
 M. Miranda, Functions of bounded variation on "good" metric spaces, J. Math. Pures Appl. (9) 82 (2003), no. 8,
 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.
 P. Silvestre, Capacitary function spaces and applications, PhD-thesis (2012), TDR, B. 8121-2012.