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 p-modulus 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 \(\mathfrak {n}\) on the considered family of measures \(\Sigma \) on X. This measure, called optimal plan, satisfies the following remarkable identity

$$\begin{aligned} \mathrm{mod}_p(\Sigma )^{-1} \int \limits _{X} f_{\Sigma }^{p-1}(x)f(x)\,d\mathfrak {m}=\int \limits _{\Sigma }\int \limits _{X} f(x)\,d\mu (x)\,d\mathfrak {n}(\mu ),\quad f\ge 0, \end{aligned}$$
(1)

where \(\mathrm{mod}_p(\Sigma )\) is the p-modulus of \(\Sigma \) (with respect to a fixed Borel measure \(\mathfrak {m}\)) and \(f_{\Sigma }\) 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, \(f_{\Sigma }\) 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 \(f_{\Sigma }\) and under two natural conditions we prove that \(f_{\Sigma }\) is in fact extremal for the p-modulus of \(\Sigma \) (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 \(\Sigma \). We heavily rely on (1) and Proposition 3. We rewrite (1) assuming there is a Borel bijection \(\Sigma \mapsto Y\) for some Polish space Y and we push forward the optimal plan \(\mathfrak {n}\) to a probability measure on Y. In a sense, Y counts elements of the family \(\Sigma \) or, in other words, using geometrical language, is a transversal to \(\Sigma \). We consider the following families:

  1. 1.

    measures which are absolutely continuous with respect to the reference measure \(\mathfrak {m}\),

  2. 2.

    ‘regular’ family of measures with disjoint supports, in particular,

  3. 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.

2 Preliminary facts

Let (X, d) be a Polish space and let \(\mathfrak {m}\) be a (reference) Borel measure on X. Consider a family of Borel measures \(\Sigma \). Let \({\mathcal {L}}(X,\mathfrak {m})\) (resp. \({\mathcal {L}}^p_+(X,\mathfrak {m})\)) be the set of all (resp. nonnegative) p–integrable functions on X. We say that \(f\in {\mathcal {L}}^p_+(X,\mathfrak {m})\) is admissible (for \(\Sigma \) or, more precisely, for the p-modulus of \(\Sigma \)) if

$$\begin{aligned} \int \limits _{X} f(x)\,d\mu (x)\ge 1 \end{aligned}$$

for all \(\mu \in \Sigma \). Denote the family of such f by \(\mathrm{adm}_p(\Sigma )\). Then the p-modulus of \(\Sigma \) is the following number

$$\begin{aligned} \mathrm{mod}_p(\Sigma )={\mathop {\inf \limits _{f\in \mathrm{adm}(\Sigma )}}}\int \limits _{X} f^p(x)\,d\mathfrak {m}(x) \end{aligned}$$

if \(\mathrm{adm}_p(\Sigma )\) is nonempty, otherwise we put \(\mathrm{mod}_p(\Sigma )=\infty \). 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 \(f_{\Sigma }\).

We say that a certain condition (P) holds for p–a.e. \(\mu \in \Sigma \) if there is a subfamily \(\Pi \subset \Sigma \) such that (P) holds for every \(\mu \in \Pi \) and \(\mathrm{mod}_p(\Sigma {\setminus }\Pi )=0\). The p-modulus has the following important properties [5]:

  1. (M1)

    \(\mathrm{mod}_p(\Sigma )=0\) if and only if there is an admissible function f such that \(\int _X f(x)\,d\mu (x)=\infty \) for every \(\mu \in \Sigma \),

  2. (M2)

    if \(f\in {\mathcal {L}}^p(X,\mathfrak {m})\), then \(f\in {\mathcal {L}}^1(X,\mu )\) for p–a.e. \(\mu \in \Sigma \),

  3. (M3)

    if \(\Pi \subset \Sigma \), then \(\mathrm{mod}_p(\Pi )\le \mathrm{mod}_p(\Sigma )\),

  4. (M4)

    if \(\Pi \subset \bigcup _i\Sigma _i\), then \(\mathrm{mod}_p(\Pi )\le \sum _i \mathrm{mod}_p(\Sigma _i)\).

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 \(\Sigma \). Let us be precise. Firstly, consider a space \({\mathcal {M}}(X)\) 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 \({\mathcal {M}}(X)\). We say that such a measure \(\mathfrak {n}\) is a plan with barycenter in \(L^q(X,\mathfrak {m})\), where q is a coefficient conjugate to p, if there is a constant \(c(\mathfrak {n})\) such that

$$\begin{aligned} \int \limits _{{\mathcal {M}}(X)}\int \limits _{X} f(x)\,d\mu (x)\,d\mathfrak {n}(\mu )\le c(\mathfrak {n})\Vert f\Vert _p,\quad f\in {\mathcal {L}}_+(X,\mathfrak {m}). \end{aligned}$$
(2)

The explanation of the use of \(L^q(X,\mathfrak {m})\) is the following. Define a Borel measure \({\underline{\mu }}\) on X as follows

$$\begin{aligned} {\underline{\mu }}(A)=\int \limits _{\Sigma } \mu (A)\,d\mathfrak {n}(\mu ). \end{aligned}$$

Then \({\underline{\mu }}\) is absolutely continuous with respect to \(\mathfrak {m}\) with a density \(\rho \) belonging to \(L^q(X,\mathfrak {m})\). The authors in [1] prove the existence and remarkable properties of the optimal plan.

Theorem 1

([1]). Assume \(\Sigma \subset {\mathcal {M}}(X)\) is a Suslin set such that \(\mathrm{mod}_p(\Sigma )>0\) and \(\mathrm{sup}_{\mu \in \Sigma }\mu (X)<\infty \). Then there is a plan \(\mathfrak {n}\) with barycenter in \(L^q(X,\mathfrak {m})\) such that \(\mathfrak {n}\) is concentrated on \(\Sigma \), \(c(\mathfrak {n})=\mathrm{mod}_p(\Sigma )^{-\frac{1}{p}}\) is the best possible constant, and the following formulas hold

$$\begin{aligned} \int \limits _{X} f_{\Sigma }(x)\,d\mu (x)=1\quad \text {for }\mathfrak {n}\text {--a.e. } \mu \quad \text {and}\quad \rho =\frac{f_{\Sigma }^{p-1}}{\Vert f_{\Sigma }\Vert _p^p}, \end{aligned}$$

where \(f_{\Sigma }\) is the extremal function for \(\Sigma \).

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 \(\rho \), we immediately see that a formula for \(\rho \) in the above Theorem implies

$$\begin{aligned} \int \limits _{\Sigma }\int \limits _{X} f(x)\,d\mu (x)\,d\mathfrak {n}(\mu )=\mathrm{mod}_p(\Sigma )^{-1}\int \limits _{X} f_{\Sigma }^{p-1}(x)f(x)\,d\mathfrak {m}(x) \end{aligned}$$
(3)

for any nonnegative Borel function f.

2.1 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 \(f_{\Sigma }\). 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 \(\eta \) on \({\mathcal {M}}(X)\) such that for any nonnegative Borel f,

$$\begin{aligned} \int \limits _{{\mathcal {M}}(X)}\int \limits _X f(x)\,d\mu (x)\,d\eta (\mu )=\int \limits _X f_{\eta }^{p-1}(x)f(x)\,d\mathfrak {m}(x) \end{aligned}$$
(4)

for some function \(f_{\eta }\in {\mathcal {L}}_+^p(X,\mathfrak {m})\). Then, by the Hölder inequality,

$$\begin{aligned} \int \limits _{{\mathcal {M}}(X)}\int \limits _X f(x)\,d\mu (x)\,d\eta (\mu )\le \Vert f_{\eta }^{p-1}\Vert _q\cdot \Vert f\Vert _p, \end{aligned}$$

hence \(\eta \) is a plan with barycenter in \(L^q(X,\mathfrak {m})\) with \(c(\eta )=\Vert f_{\eta }^{p-1}\Vert _q\). Moreover, if \(f\in {\mathcal {L}}_+^p(X,\mathfrak {m})\) is admissible, then

$$\begin{aligned} \eta (\Sigma )\le \Vert f_{\eta }^{p-1}\Vert _q\cdot \Vert f\Vert _p. \end{aligned}$$
(5)

Hence, taking the infimum with respect to all admissible functions f, we get

$$\begin{aligned} \frac{\eta (\Sigma )^p}{\Vert f_{\eta }^{p-1}\Vert ^p_q}\le \mathrm{mod}_p(\Sigma )=\frac{1}{c(\mathfrak {n})}. \end{aligned}$$

Consider the following two assumptions:

  1. (E1)

    the measure \(\eta \) is concentrated on \(\Sigma \),

  2. (E2)

    there is a constant \(c>0\) such that for every \(\mu \in \Sigma \),

    $$\begin{aligned} \int \limits _X f_{\eta }(x)\,d\mu (x)=c. \end{aligned}$$
    (6)

Proposition 3

Assume \(\Sigma \subset {\mathcal {M}}(X)\) is a Suslin set such that \(\mathrm{mod}_p(\Sigma )>0\) and \(\mathrm{sup}_{\mu \in \Sigma }\mu (X)<\infty \). Assume, moreover, that there is a probability measure \(\eta \) on \({\mathcal {M}}(X)\) such that (4) holds for some \(f_{\eta }\in {\mathcal {L}}_+^p(X)\). If (E1), (E2) hold, then

  1. (i)

    \(c=\mathrm{mod}_p(\Sigma )^{1-q}\),

  2. (ii)

    \(f_{\Sigma }=\mathrm{mod}_p(\Sigma )^{q-1}f_{\eta }\) is extremal for the p-modulus of \(\Sigma \),

  3. (iii)

    \(\eta =\mathfrak {n}\), i.e., \(\eta \) is the optimal plan with barycenter in \(L^q(X,\mathfrak {m})\),

  4. (iv)

    (4) is equivalent to (3).

Proof

Substituting \(f=f_{\eta }\) in (4), we have

$$\begin{aligned} c=\int \limits _X f_{\eta }^p(x)\,d\mathfrak {m}(x). \end{aligned}$$

Thus, by (3) with \(f=f_{\eta }\), we have

$$\begin{aligned} c=\mathrm{mod}_p(\Sigma )^{-1}\int \limits _X f_{\Sigma }^{p-1}(x)f_{\eta }(x)\,d\mathfrak {m}(x)\le \mathrm{mod}_p(\Sigma )^{-\frac{1}{p}}c^{\frac{1}{p}}. \end{aligned}$$

Hence

$$\begin{aligned} c^{\frac{1}{q}}\mathrm{mod}_p(\Sigma )^{\frac{1}{p}}\le 1. \end{aligned}$$

On the other hand, \(f_{\Sigma }\) is admissible for the p-modulus, so by (5),

$$\begin{aligned} 1\le c^{\frac{1}{q}}\mathrm{mod}_p(\Sigma )^{\frac{1}{p}}. \end{aligned}$$

Together with the previous inequality, we conclude (i). For the proof of (ii), notice that \(c^{-1}f_{\eta }\) is admissible for the p-modulus of \(\Sigma \) and

$$\begin{aligned} \mathrm{mod}_p(\Sigma )=c^{1-p}=c^{-p}\int \limits _X f_{\eta }^p(x)\,d\mathfrak {m}(x)=\int \limits _X (c^{-1}f_{\eta })^p(x)\,d\mathfrak {m}(x), \end{aligned}$$

which proves the extremality of \(c^{-1}f_{\eta }\). By the uniqueness (up to a subset of zero measure), we get (ii). The remaining conditions follow immediately by (i) and (ii). \(\square \)

3 Results

Adopt notation from the previous section. Fix \(p>1\) and let \(\Sigma \) be a family of Borel measures such that \(\mathrm{mod}_p(\Sigma )>0\) and \(\mathrm{sup}_{\mu \in \Sigma }\mu (X)<\infty \). Assume that the family \(\Sigma \subset {\mathcal {M}}(X)\) can be realized as follows: there is a Borel bijective map \(\mu :Y\rightarrow \Sigma \) with a Borel inverse map \(\mu ^{-1}:\Sigma \rightarrow Y\), where Y is a Polish space. Hence, we may index the family \(\Sigma \) with elements from Y, i.e., \(\Sigma =\{\mu _y\}_{y\in Y}\). We may push-forward the optimal plan \(\mathfrak {n}\) with respect to \(\mu ^{-1}\) to obtain a Borel probability measure \(\lambda =\lambda _{\mathfrak {n}}\) on Y. Then formula (3) may be rewritten as follows

$$\begin{aligned} \int \limits _Y \int \limits _X f(x)\,d\mu _y(x)\,d\lambda (y)=\mathrm{mod}_p(\Sigma )^{-1}\int \limits _X f_{\Sigma }^{p-1}(x)f(x)\,d\mathfrak {m} \end{aligned}$$
(7)

for any nonnegative Borel function f.

3.1 Measures absolutely continuous with respect to \(\mathfrak {m}\)

In this section, we provide a formula for the p-modulus in the case when each \(\mu _y\in \Sigma \) is absolutely continuous with respect to \(\mathfrak {m}\).

Proposition 4

Assume there is a Borel function \(\alpha =\alpha (x,y):X\times Y\rightarrow {\mathbb {R}}_+\) such that \(\mu _y=\alpha (\cdot ,y)\mathfrak {m}\). Put

$$\begin{aligned} {\hat{\alpha }}(x)=\int \limits _Y \alpha (x,y)\,d\lambda (y). \end{aligned}$$

Then

$$\begin{aligned} f_{\Sigma }(x)=\frac{{\hat{\alpha }}^{q-1}(x)}{\int _X {\hat{\alpha }}^q(x)\,d\mathfrak {m}(x)}\quad \text {and}\quad \mathrm{mod}_p(\Sigma )=\left( \int \limits _X {\hat{\alpha }}^q(x)\,d\mathfrak {m}(x)\right) ^{1-p}. \end{aligned}$$

Proof

In this case, formula (7) simplifies to

$$\begin{aligned} \int \limits _Y\int \limits _X \alpha (x,y)f(x)\,d\mathfrak {m}\,d\lambda (y)=\mathrm{mod}_p(\Sigma )^{-1}\int \limits _X f_{\Sigma }^{p-1}(x)f(x)\,d\mathfrak {m}. \end{aligned}$$

Since \(f\ge 0\) is arbitrary, by the Fubini–Tonelli theorem, we get

$$\begin{aligned} \int \limits _Y \alpha (x,y)\,d\lambda (y)=\mathrm{mod}_p(\Sigma )^{-1}f_{\Sigma }^{p-1}(x),\quad x\in X. \end{aligned}$$

Hence \(f_{\Sigma }(x)=(\mathrm{mod}_p(\Sigma ){\hat{\alpha }}(x))^{q-1}\). Thus

$$\begin{aligned} \mathrm{mod}_p(\Sigma )=\mathrm{mod}_p(\Sigma )^q\int \limits _X{\hat{\alpha }}^q(x)\,d\mathfrak {m}, \end{aligned}$$

which implies the formula for the p-modulus and for the extremal function. \(\square \)

If \(\alpha \) has separate variables, the formulas are much simpler and, what is probably the most important, we get some additional information about the measure \(\lambda \).

Corollary 5

Assume \(\alpha \) is of the form \(\alpha (x,y)=\beta (x)\gamma (y)\). Then

$$\begin{aligned} f_{\Sigma }(x)=\frac{\beta ^{q-1}(x)}{\left( \int _Y\gamma (y)\,d\lambda (y)\right) ^{q-1}\int _X \beta ^q(x)\,d\mathfrak {m}(x)} \end{aligned}$$

and

$$\begin{aligned} \mathrm{mod}_p(\Sigma )=\left( \int \limits _Y\gamma (y)\,d\lambda (y)\right) ^{-p}\left( \int \limits _X \beta ^q(x)\,d\mathfrak {m}\right) ^{1-p}. \end{aligned}$$

Proof

It suffices to notice that

$$\begin{aligned} {\hat{\alpha }}(x)=\left( \int \limits _Y \gamma (y)\,d\lambda (y)\right) \beta (x), \end{aligned}$$

and use Proposition 4. \(\square \)

Corollary 6

If \(\alpha (x,y)=\beta (x)\gamma (y)\), then \(\gamma \) is constant on a subset of Y of full \(\lambda \)-measure.

Proof

By Theorem 1, we know that \(\int _X f_{\Sigma }(x)\,d\mu _{y_0}(x)=1\) for \(\lambda \)–a.e. \(y_0\in Y\). Thus, for \(\lambda \)–a.e. \(y_0\in Y\),

$$\begin{aligned} 1=\frac{\int _X \beta ^{q-1}(x)\beta (x)\gamma (y_0)\,d\mathfrak {m}}{\left( \int _Y\gamma (y)\,d\lambda (y)\right) ^{q-1}\int _X \beta ^q(x)\,d\mathfrak {m}(x)}=\frac{\gamma (y_0)}{\int _Y \gamma (y)\,d\lambda (y)}. \end{aligned}$$

Denoting the denominator on the right hand side of the above formula by c, we have \(\gamma (y_0)=c\) for \(\lambda \)–a.e. \(y_0\in Y\). \(\square \)

3.2 ‘Regular’ family of measures with disjoint supports

Here, we consider a special case of a separate system of measures \(\Sigma \).

Proposition 7

Assume there is a Borel map \(\alpha :L\times Y\rightarrow X\) and Borel functions \(\beta ,\gamma :L\times Y\rightarrow {\mathbb {R}}_+\), where \((L,\nu )\) is a Polish space with a Borel measure \(\nu \), such that

$$\begin{aligned} \alpha (\cdot ,y)_{\sharp }(\beta (\cdot ,y)\nu )=\mu _y,\quad \alpha _{\sharp }(\gamma (\nu \otimes \lambda ))=\mathfrak {m}. \end{aligned}$$

Then, for \(\lambda \)–a.e. \(y_0\),

$$\begin{aligned} f_{\Sigma }(\alpha (z,y))=\left( \int \limits _L \frac{\beta ^q(z,y_0)}{\gamma ^{q-1}(z,y_0)}\,d\nu (z)\right) ^{-1} \frac{\beta ^{q-1}(z,y)}{\gamma ^{q-1}(z,y)} \end{aligned}$$
(8)

and

$$\begin{aligned} \mathrm{mod}_p(\Sigma )=\left( \int \limits _L\frac{\beta ^q(z,y_0)}{\gamma ^{q-1}(z,y_0) }\,d\nu (z)\right) ^{1-p}. \end{aligned}$$
(9)

Proof

By the assumptions, formula (7) simplifies to (for any nonnegative Borel map f)

$$\begin{aligned}&\int \limits _{L\times Y} f(\alpha (z,y))\beta (z,y)\,d\nu (z)\,d\lambda (y)\\&\quad =\mathrm{mod}_p(\Sigma )^{-1}\int \limits _{L\times Y} f_{\Sigma }^{p-1}(\alpha (z,y))f(\alpha (z,y))\gamma (z,y)\,d\nu (z)\,d\lambda (y). \end{aligned}$$

Hence

$$\begin{aligned} \beta (z,y)=\mathrm{mod}_p(\Sigma )^{-1}f_{\Sigma }^{p-1}(\alpha (z,y))\gamma (z,y). \end{aligned}$$

Therefore

$$\begin{aligned} f_{\Sigma }(\alpha (z,y))=\mathrm{mod}_p(\Sigma )^{q-1}\left( \frac{\beta (z,y)}{\gamma (z,y)}\right) ^{q-1}. \end{aligned}$$

By Theorem 1, for \(\lambda \)–a.e. y, we have

$$\begin{aligned} \mathrm{mod}_p(\Sigma )=\left( \int \limits _L\frac{\beta ^q(z,y)}{\gamma ^{q-1}(z,y)}\,d\nu (z)\right) ^{1-p}. \end{aligned}$$

This also implies the formula for \(f_{\Sigma }\). \(\square \)

Remark 8

Notice that the function of the variable y on the right hand side of (9) is constant on a set of full \(\lambda \)-measure in Y.

Let us prove, in a sense, the converse statement by using Proposition 3.

Proposition 9

Assume there is a Borel bijective map \(\alpha :L\times Y\rightarrow X\) and Borel positive functions \(\beta ,\gamma :L\times Y\rightarrow {\mathbb {R}}\), where \((L,\nu )\) and \((Y,\eta )\) are Polish spaces with a Borel measure \(\nu \) and a Borel probability measure \(\eta \), such that

$$\begin{aligned} \alpha (\cdot ,y)_{\sharp }(\beta (\cdot ,y)\nu )=\mu _y,\quad \alpha _{\sharp }(\gamma (\nu \otimes \eta ))=\mathfrak {m}. \end{aligned}$$

If, additionally, \(\frac{\beta }{\gamma }\circ \alpha ^{-1}\in {\mathcal {L}}^q(X,\mathfrak {m})\) and the function

$$\begin{aligned} y\mapsto \int \limits _L \frac{\beta ^q(z,y)}{\gamma ^{q-1}(z,y)}\,d\nu (z) \end{aligned}$$

is constant, then the extremal function \(f_{\Sigma }\) for the p-modulus of \(\Sigma \) and the p-modulus \(\mathrm{mod}_p(\Sigma )\) are given by (8) and (9) (for any \(y_0\)).

Proof

By assumptions and the Fubini–Tonelli theorem, we have

$$\begin{aligned} \int \limits _Y\int \limits _X f(x)\,d\mu _y(x)\,d\eta (y)&=\int \limits _L\int \limits _Y f(\alpha (z,y))\beta (z,y)\,d\eta (y)\,d\nu (z)\\&=\int \limits _X \frac{\beta }{\gamma }\circ \alpha ^{-1}(x)f(x)\,d\mathfrak {m}(x) \end{aligned}$$

for any Borel function \(f\ge 0\). Thus (4) holds with \(f_{\eta }=\left( \frac{\beta }{\gamma }\circ \alpha ^{-1}\right) ^{q-1}\). Moreover, for any \(y\in Y\),

$$\begin{aligned} \int \limits _X f_{\eta }(x)\,d\mu _y(x)=\int \limits _L \frac{\beta ^q(z,y)}{\gamma ^{q-1}(z,y)}\,d\nu (z). \end{aligned}$$

Hence, we may apply Proposition 3. \(\square \)

Example 10

(Rodin’s formula). The original Rodin formula [6] is the formula for the extremal function for the p-modulus of a family of curves on the Euclidean plane, which are images of horizontal lines in the rectangle \([0,1]\times [0,b]\). It was generalized to arbitrary dimension [2] (still for a family of curves) and recently, by the author and K. Niedziałomski, 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 \({\mathbb {R}}^m\) and \({\mathbb {R}}^{n-m}\), respectively. Let \(\alpha :L\times Y\rightarrow \Omega \) be a \(C^1\)–smooth diffeomorphism onto \(\Omega \subset {\mathbb {R}}^n\). Denote by \(\Sigma \) the family of m–dimensional surfaces \(\sigma _y\), \(y\in Y\), being the images of L with respect to f, \(\sigma _y=\alpha (L,y)\). Then the extremal function for the p-modulus of \(\Sigma \) is the following

$$\begin{aligned} f_{\Sigma }(x)=\frac{1}{l(z)}\left( \frac{|J^z_{\alpha }|}{|J_{\alpha }|}\right) ^{q-1}\circ \alpha ^{-1}(x),\quad x=\alpha (z,y), \end{aligned}$$
(10)

where

$$\begin{aligned} l(y)=\int \limits _L \left( \frac{|J^z_{\alpha }|}{|J_{\alpha }|}\right) ^q J_{\alpha }\,dz. \end{aligned}$$
(11)

Moreover, the p-modulus of \(\Sigma \) equals

$$\begin{aligned} \mathrm{mod}_p(\Sigma )=\int \limits _Y l(y)^{1-p}\,dy. \end{aligned}$$
(12)

Recall that \(J^z_{\alpha }\) is the Jacobian of a map \(z\mapsto {\alpha }(z,y)\) with fixed z, whereas \(J_{\alpha }\) is the ’full’ Jacobian of \(\alpha \).

Let \(\nu =dz\) be the Lebesgue measure on L and \(\mathfrak {m}=dx\) be the Lebesgue measure on \(\Omega \) and let \(\mu _y\) be the Lebesgue measure on \(\sigma _y\). Let us define \(\beta \) and \(\gamma \) by

$$\begin{aligned} \beta (z,y)=|J^z\alpha (z,y)|,\quad \gamma (z,y)=\frac{|J_{\alpha }(z,y)|}{\rho (y)}, \end{aligned}$$

where \(\rho (y)=c\,l(y)^{1-p}\) and c is a constant such that the measure

$$\begin{aligned} \eta =\rho \,dy \end{aligned}$$

is a probability measure on Y, i.e., \(c^{-1}=\int _Y l(y)^{1-p}\,dy\). Since, by the Fubini–Tonelli theorem, for any nonnegative Borel f,

$$\begin{aligned} \int \limits _Y\int \limits _L f(\alpha (z,y))\gamma (z,y)\,d\nu (z)\,d\eta (y)&=\int \limits _Y\int \limits _L f(\alpha (z,y))|J_{\alpha }|\,dz\,dy\\&=\int \limits _X f(x)\,d\mathfrak {m}(x), \end{aligned}$$

we have \(\alpha _{\sharp }(\gamma \,\nu \otimes \eta )=\mathfrak {m}\). Clearly, \(\alpha (\cdot ,y)_{\sharp }(\beta (\cdot ,y)\nu )=\mu _y\). Moreover,

$$\begin{aligned} \int \limits _L \frac{\beta ^q(z,y)}{\gamma ^{q-1}(z,y)}\,d\nu (z)=\int \limits _L c^{q-1}\frac{|J^z_{\alpha }(z,y)|^q}{|J_{\alpha }(z,y)|^{q-1}}l(y)^{-1}\,dz=c^{q-1} \end{aligned}$$

for any \(y\in Y\). Thus the assumptions of Proposition 9 are satisfied. Hence (12) and (11) hold.