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Register a new userAspects of area formulas by way of Luzin, Rado, and Reichelderfer on metric measure spaces
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We consider some measure-theoretic properties of functions belonging to a Sobolev-type class on metric measure spaces that admit a Poincare inequality and are equipped with a doubling measure. The properties we have selected to study are those that are related to area formulas.
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We study mean value properties of harmonic functions in metric measure spaces. The metric measure spaces we consider have a doubling measure and support a (1,1)- Poincare inequality. The notion of harmonicity is based on the Dirichlet form defined in terms of a Cheeger differentiable structure. By studying fine properties of the Green function on balls, we characterize harmonic functions in terms of a mean value property. As a consequence, we obtain a detailed description of Poisson kernels. We shall also obtain a Gauss-Green type formula for sets of finite perimeter which posses a Minkowski content characterization of the perimeter. For the Gauss-Green formula we introduce a suitable notion of the interior normal trace of a regular ball.
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