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.
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