Computing the p -modulus of systems of measures via optimal plans
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Archiv der Mathematik
Computing the p-modulus of systems of measures via optimal plans Malgorzata Ciska-Niedzialomska
Abstract. We use the notion of the optimal plan associated with the Fuglede p-modulus of a family of Borel measures to derive formulas for the p-modulus and the extremal function in many special cases. Among others, we deduce Rodin’s formula for a family of Hausdorff measures associated with leaves of a foliation defined by a single chart. Mathematics Subject Classification. 28A25, 28A33. Keywords. Fuglede p-modulus, Extremal function, Optimal plan, Rodin’s formula.
1. Introduction. The notion of p-modulus was introduced by Fuglede [5] in order to study functional completion. Its reciprocal—extremal length, for p = 2—had been considered before and is closely related with the study of conformality and quasiconformality of maps on a plane. Since the work of Fuglede, the p-modulus became a powerful tool in geometric measure theory. For example, Shanmugalingam [7] uses the p-modulus to consider p–harmonic maps on metric measure spaces. In such case, the pmodulus is considered for a family of curves, more precisely, for the family of 1–dimensional Hausdorff measures on continuous curves. Computation of the p-modulus for a concrete family of measures is not easy and there are not many explicit formulas. The majority of them is related to a family of Hausdorff measures on curves or hypersurfaces and is related to the notion of q–capacity (here p and q are conjugate coefficients). Recently, Ambrosio, di Marino, and Savare [1] characterised the p-modulus in terms of existence of a certain measure n on the considered family of measures Σ on X. This measure, called optimal plan, satisfies the following remarkable identity
M. Ciska-Niedzialomska −1
modp (Σ)
fΣp−1 (x)f (x) dm
X
Arch. Math.
=
f (x) dμ(x) dn(μ),
f ≥ 0,
(1)
Σ X
where modp (Σ) is the p-modulus of Σ (with respect to a fixed Borel measure m) and fΣ is an extremal function, i.e., a function which realizes the p-modulus. It can be shown that, up to a subfamily of modulus zero, fΣ is unique. Equality (1) allows us to derive formulas for the p-modulus and the extremal function in many special cases. We begin with a general fact. We consider (1) without the assumption on extremality of fΣ and under two natural conditions we prove that fΣ is in fact extremal for the p-modulus of Σ (Proposition 3). The main part of this note is devoted to formulas for the p-modulus and the extremal function for different families of measures Σ. We heavily rely on (1) and Proposition 3. We rewrite (1) assuming there is a Borel bijection Σ → Y for some Polish space Y and we push forward the optimal plan n to a probability measure on Y . In a sense, Y counts elements of the family Σ or, in other words, using geometrical language, is a transversal to Σ. We consider the following families: 1. measures which are absolutely continuous with respect to the reference measure m, 2. ‘regular’ family of measures with disjoint supports, in particular, 3. Lebesgue measures
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