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3 Magnani V.

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An area formula in metric spaces

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Magnani V.

Colloquium Mathematicum

2011

tom 124

nr 2

275-283

We present an area formula for continuous mappings between metric spaces, under minimal regularity assumptions. In particular, we do not require any notion of differentiability. This is a consequence of a measure-theoretic notion of Jacobian, defined as the density of a suitable "pull-back measure". Finally, we give some applications and examples.

Blow-up of regular submanifolds in Heisenberg groups and applications

100%

Magnani V.

Open Mathematics

2006

tom 4

nr 1

82-109

We obtain a blow-up theorem for regular submanifolds in the Heisenberg group, where intrinsic dilations are used. Main consequence of this result is an explicit formula for the density of (p+1)-dimensional spherical Hausdorff measure restricted to a p-dimensional submanifold with respect to the Riemannian surface measure. We explicitly compute this formula in some simple examples and we present a lower semicontinuity result for the spherical Hausdorff measure with respect to the weak convergence of currents. Another application is the proof of an intrinsic coarea formula for vector-valued mappings on the Heisenberg group.

Differentiability from the representation formula and the Sobolev-Poincaré inequality

100%

Magnani V.

Studia Mathematica

2005

tom 168

nr 3

251-272

In the geometries of stratified groups, we provide differentiability theorems for both functions of bounded variation and Sobolev functions. Proofs are based on a systematic application of the Sobolev-Poincaré inequality and the so-called representation formula.