The weak Cartan property for the p-fine topology on metric spaces
Jana BjornAnders BjornVisa Latvala
Primary: 31E05Secondary: 30L9931C4031C4535J9249Q20.capacitycoarsest topologydoublingfine topologyfinely continuousmetric spacep-harmonicPoincare inequalityquasicontinuoussuperharmonicthickthinweak Cartan propertyWiener criterion
We study the $p\mspace{1mu}$-fine topology on complete metric spaces e\-quipped with a doubling measure supporting a $p\mspace{1mu}$-Poincar\'e inequality, $1<p<\infty$. We establish a weak Cartan property, which yields characterizations of the $p\mspace{1mu}$-thinness and the $p\mspace{1mu}$-fine continuity, and allows us to show that the $p\mspace{1mu}$-fine topology is the coarsest topology making all $p\mspace{1mu}$-superharmonic functions continuous. Our $p\mspace{1mu}$-harmonic and superharmonic functions are defined by means of scalar-valued upper gradients, and do not rely on a vector-valued differentiable structure.
Indiana University Mathematics Journal
2015
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10.1512/iumj.2015.64.5527
10.1512/iumj.2015.64.5527
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Indiana Univ. Math. J. 64 (2015) 915 - 941
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