Computing the p-modulus of systems of measures via optimal plans

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.

where mod p (Σ) is the p-modulus of Σ (with respect to a fixed Borel measure m) and f Σ 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, f Σ 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 f Σ and under two natural conditions we prove that f Σ 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 associated with leaves of a foliation defined by a single chart (Rodin's formula) obtaining new, generalizing, or reproving well known relations.

Preliminary facts.
Let (X, d) be a Polish space and let m be a (reference) Borel measure on X. Consider a family of Borel measures Σ. Let L(X, m) (resp. L p + (X, m)) be the set of all (resp. nonnegative) p-integrable functions on X. We say that f ∈ L p + (X, m) is admissible (for Σ or, more precisely, for for all μ ∈ Σ. Denote the family of such f by adm p (Σ). Then the p-modulus of Σ is the following number if adm p (Σ) is nonempty, otherwise we put mod p (Σ) = ∞. An admissible function which realizes the p-modulus is called extremal. It can be shown, that up to a subfamily of full p-modulus, there is a unique extremal function [5].
Denote it by f Σ . We say that a certain condition (P ) holds for p-a.e. μ ∈ Σ if there is a subfamily Π ⊂ Σ such that (P ) holds for every μ ∈ Π and mod p (Σ\Π) = 0. The p-modulus has the following important properties [5]: (M1) mod p (Σ) = 0 if and only if there is an admissible function f such that The p-modulus is a powerful tool, especially in the context of families of Hausdorff measures on curves, in geometric measure theory [7].
In the article [1], the authors prove the existence of an optimal plan associated with a family Σ. Let us be precise. Firstly, consider a space M(X) of positive and finite Borel measures on a Polish space X. We can equip this space with the weak- * topology and consider Borel probability measures on M(X). We say that such a measure n is a plan with barycenter in L q (X, m), where q is a coefficient conjugate to p, if there is a constant c(n) such that The explanation of the use of L q (X, m) is the following. Define a Borel measure μ on X as follows Then μ is absolutely continuous with respect to m with a density ρ belonging to L q (X, m). The authors in [1] prove the existence and remarkable properties of the optimal plan.
Then there is a plan n with barycenter in L q (X, m) such that n is concentrated on Σ, c(n) = mod p (Σ) − 1 p is the best possible constant, and the following formulas hold where f Σ is the extremal function for Σ.

Remark 2.
Let us recall (see [3]) that a Polish space is Suslin and the Borel image of a Suslin set is again Suslin.
By the definition of ρ, we immediately see that a formula for ρ in the above Theorem implies for any nonnegative Borel function f .

An important consequence.
Let us prove a general consequence of the formula (3). Now, we give one implication of the formula (3) but without the assumption on extremality of f Σ . This approach has been partially noticed in the paper [1] but in the context with relation (2) and with different conclusions.
Assume there is a probability measure η on M(X) such that for any nonnegative Borel f , for some function f η ∈ L p + (X, m). Then, by the Hölder inequality, Hence, taking the infimum with respect to all admissible functions f , we get Consider the following two assumptions: (E1) the measure η is concentrated on Σ, (E2) there is a constant c > 0 such that for every μ ∈ Σ, Thus, by (3) with f = f η , we have On the other hand, f Σ is admissible for the p-modulus, so by (5), Together with the previous inequality, we conclude (i). For the proof of (ii), notice that c −1 f η is admissible for the p-modulus of Σ and which proves the extremality of c −1 f η . By the uniqueness (up to a subset of zero measure), we get (ii). The remaining conditions follow immediately by (i) and (ii).

Results.
Adopt notation from the previous section. Fix p > 1 and let Σ be a family of Borel measures such that mod p (Σ) > 0 and sup μ∈Σ μ(X) < ∞.
Assume that the family Σ ⊂ M(X) can be realized as follows: there is a Borel bijective map μ : Y → Σ with a Borel inverse map μ −1 : Σ → Y , where Y is a Polish space. Hence, we may index the family Σ with elements from Y , i.e., Σ = {μ y } y∈Y . We may push-forward the optimal plan n with respect to μ −1 to obtain a Borel probability measure λ = λ n on Y . Then formula (3) may be rewritten as follows for any nonnegative Borel function f .

Measures absolutely continuous with respect to m.
In this section, we provide a formula for the p-modulus in the case when each μ y ∈ Σ is absolutely continuous with respect to m.

Proposition 4. Assume there is a Borel function
Proof. In this case, formula (7) simplifies to Since f ≥ 0 is arbitrary, by the Fubini-Tonelli theorem, we get Hence f Σ (x) = (mod p (Σ)α(x)) q−1 . Thus which implies the formula for the p-modulus and for the extremal function.
If α has separate variables, the formulas are much simpler and, what is probably the most important, we get some additional information about the measure λ.

Corollary 5. Assume α is of the form α(x, y) = β(x)γ(y). Then
Proof. It suffices to notice that and use Proposition 4.
Denoting the denominator on the right hand side of the above formula by c, we have γ(y 0 ) = c for λ-a.e. y 0 ∈ Y .

'Regular' family of measures with disjoint supports.
Here, we consider a special case of a separate system of measures Σ. Then, for λ-a.e. y 0 , and Proof. By the assumptions, formula (7) simplifies to (for any nonnegative Borel By Theorem 1, for λ-a.e. y, we have This also implies the formula for f Σ . Remark 8. Notice that the function of the variable y on the right hand side of (9) is constant on a set of full λ-measure in Y .
Let us prove, in a sense, the converse statement by using Proposition 3.
If, additionally, β γ • α −1 ∈ L q (X, m) and the function is constant, then the extremal function f Σ for the p-modulus of Σ and the p-modulus mod p (Σ) are given by (8) and (9) (for any y 0 ).
Proof. By assumptions and the Fubini-Tonelli theorem, we have for any Borel function f ≥ 0. Thus (4) holds with Hence, we may apply Proposition 3. It was generalized to arbitrary dimension [2] (still for a family of curves) and recently, by the author and K. Niedzia lomski, to any family of k-dimensional surfaces defined by a single chart [4]. Let us recall this result and show how it can be obtained from Proposition 9.
Theorem 11 ([4]). Let L and Y be two domains in R m and R n−m , respectively. Let α : L × Y → Ω be a C 1 -smooth diffeomorphism onto Ω ⊂ R n . Denote by Σ the family of m-dimensional surfaces σ y , y ∈ Y , being the images of L with respect to f , σ y = α(L, y). Then the extremal function for the p-modulus of Σ is the following where Moreover, the p-modulus of Σ equals Recall that J z α is the Jacobian of a map z → α(z, y) with fixed z, whereas J α is the 'full' Jacobian of α.
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