Some properties of the extremal function for the Fuglede p-modulus

Abstract

We deal with the Fuglede p-modulus of a system of measures, focusing on three aspects. First, we combine results concerning Badger’s criterion for the extremal function, i.e., the function which realizes the p-modulus, and plans with barycenter in $L^q$, which give an alternative—in a sense, probabilistic—approach to p-modulus. It seems that the correlation of these results has not yet been established. Second, we deal with families of measures for which the integral of the extremal function is one. On such a family, the p-modulus as well as the optimal plans are concentrated. We consider closures of these families and relate them with generic families of measures for which the extremal function exists. Finally, we compute the p-modulus and extremal function for finite families of measures.

1 Introduction

Consider a measurable space $(X,{{\mathcal {M}}})$ and fix a reference measure $\mathfrak {m}$. For a family of measures $\Sigma$ defined on a $\sigma$-algebra ${{\mathcal {M}}}$ and coefficient $p>1$, we may consider a notion of p-modulus [3]. This is a number which plays a remarkable role in many areas, such as: quasiconformal mappings, geometric measure theory, analysis on manifolds, etc. Let us introduce this notion. Denote by ${{\mathcal {L}}}^p(X,\mathfrak {m})$ a set of all functions on X with finite p-norm. We do not identify functions equal on a set of full measure. We say that $f\in {{\mathcal {L}}}(X,\mathfrak {m})$ is p-admissible if $f\ge 0$ and $\int _X f(x)\,\text {d}\mu (x)\ge 1$ for all $\mu \in \Sigma$. The p-modulus ${\textrm{mod}}_p(\Sigma ,\mathfrak {m})$ it defined as

$$\begin{aligned} {\textrm{mod}}_p(\Sigma ,\mathfrak {m})={\textrm{inf}}\left\{ \int \limits _X f^p(x)\,\text {d}\mathfrak {m}\mid \,f \,\text {is}\, p\text {-admissible}\right\} . \end{aligned}$$

A function $f_{\Sigma }$ which realizes the infimum is called extremal.

The important consequence is the generalization on the notion: almost everywhere (with respect to some measure). In this case, we deal with a family of measures. Namely, we say that some property holds p-almost everywhere (p-a.e.) with respect to $\Sigma$, if there is a subfamily $T\subset \Sigma$, such that ${\textrm{mod}}_p(T)=0$ and this property holds for any measure in $\Sigma \setminus T$.

The basic example [3] is the following: Let $\Sigma _A$ be a family of Dirac measures $\delta _x$, where $x\in A$, and $A\in {{\mathcal {M}}}$. Then, it is easy to see that the “best” p-admissible function is $f\equiv 1$ on A, which implies ${\textrm{mod}}_p(\Sigma _A)=\mathfrak {m}(A)$. Notice that here p-modulus does not depend on p. In this case, p-a.e. is equivalent to a.e. in a classical sense.

Although, the importance of p-modulus was first discovered in the context of capacity and quasiconformality, now it plays central role in many other aspects. It was a starting point for considerations of p-harmonic functions and Newton–Sobolev spaces on measure metric spaces [4]. In another remarkable article [1], the authors introduce the notion of a plan with barycenter in ${{\mathcal {L}}}^q$, where q is a coefficient conjugate to p, to study the p-modulus from the probabilistic perspective and its consequences for week gradients on metric measure spaces (which are fundamental for Newton–Sobolev spaces and p-harmonicity). Another important observation has been made by Badger [2], where he generalizes Beurling’s criteria for the existence of the extermal function. Let us enlarge on these two results.

The main result in [2] is the following.

Theorem 1.1

Let $\Sigma$ be a family of measures on X and fix a reference measure $\mathfrak {m}$. Then, a p-admissible function $f_{\Sigma }$ is extremal if and only if the following holds: there exists a family of measures F, such that

• (i) ${\textrm{mod}}_p(\Sigma )={\textrm{mod}}_p(\Sigma \cup F)$,

• (ii) for every $\nu \in F$, $\int _X f(x)\,d\nu (x)=1$,

• (iii) for $f\in {{\mathcal {L}}}^p(X,\mathfrak {m})$ taking values in $[-\infty ,\infty ]$ if $\int _X f(x)\,d\nu (x)\ge 0$ for all $\nu \in F$, then $\int _X f_{\Sigma }^{p-1}(x)f(x)\,d\mathfrak {m}\ge 0$.

The important observation by Badger is that F does not need to be inside $\Sigma$. In fact, from the proof of above theorem it follows that F may be chosen as a family of measures $c\mu$, where $\mu \in \Sigma$ and $c=c(\mu )$ is such that $\int _X f_{\Sigma }(x)\,d(c\mu )(x)=1$. This immediately implies the following important corollary.

Corollary 1.2

Let $\Sigma$ be a family of measures on X and fix a reference measure $\mathfrak {m}$. If $f_{\Sigma }$ is is extremal for the p-modulus of $\Sigma$, then the following condition holds:

• (iii’) for $f\in {{\mathcal {L}}}^p(X,\mathfrak {m})$ taking values in $[-\infty ,\infty ]$ if $\int _X f(x)\,d\mu (x)\ge 0$ for all $\mu \in \Sigma$, then $\int _X f_{\Sigma }^{p-1}(x)f(x)\,d\mathfrak {m}\ge 0$.

Moreover, the following example from [2] shows that F may be disjoint with $\Sigma$: Let $\Sigma$ be a family of Lebesgue measures on curves joining two horizontal edges of a rectangle, where a is a length of vertical edge and b is a length of horizontal edge. It can be shown that the p-modulus of F equals the p-modulus of a family of vertical lines (Fig. 1), which is $a^{1-p}b$. Denote a family of vertical lines by F. The extremal function $f_{\Sigma }$ in this case is constant end equals $f=\frac{1}{a}$. Thus, since any curve in $\Sigma '=\Sigma {\setminus } F$ has length greater that curves in F, it follows that $\int _X f_{\Sigma }(x)\,\text {d}\mu (x)>1$ for any $\mu \in \Sigma '$. However, since any curve in F can be approximated by curves in $\Sigma '$ it follows (see details in [2]) that

$$\begin{aligned} {\textrm{mod}}_p(\Sigma ')={\textrm{mod}}_p(\Sigma )={\textrm{mod}}_p(F). \end{aligned}$$

Fig. 1

Family of lines joining horizontal edges in a rectangle

On the other hand, we have results by Ambrosio et al. [1] treating existence of a plan associated with family of measures $\Sigma$. The idea is to find a measure $\eta$ on $\Sigma$, such that the continuity condition holds

$$\begin{aligned} \int \limits _{\Sigma } \int \limits _X f(x)\,\text {d}\mu (x)\,\text {d}\eta (\mu )\le c\Vert f\Vert _p, \end{aligned}$$

for any $f\in {{\mathcal {L}}}^p(X,\mathfrak {m})$ with non-negative values and for some constant c depending only on $\eta$ itself and the coefficient q conjugate to p (We will denote the constant c by $c_q(\eta )$ if it is optimal in a sense of Theorem 1.3 and we will call it the optimal constant). The implications of this condition are huge. Firstly, considering a barycenter ${{\underline{\eta }}}$ associated with $\Sigma$. It is a measure on X defined by

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

From the definition of a plan (applied with characteristic function $f=\chi _A$), it follows that barycenter ${{\underline{\eta }}}$ is absolutely continuous with respect to $\mathfrak {m}$ and the density $\rho =\text {d}{{\underline{\eta }}}/\text {d}\mathfrak {m}$ is q-integrable. Authors in [1] prove the following remarkable result.

Theorem 1.3

Assume $\Sigma$ is a Suslin set,

$$\begin{aligned} {\textrm{mod}}_p(\Sigma )>0,\quad C_{p,\mathfrak {m}}(\Sigma )={\textrm{sup}}_{c(\eta )>0}\frac{\eta (\Sigma )}{c(\eta )}>0,\quad {\textrm{inf}}_{\mu \in \Sigma }\mu (X)<\infty . \end{aligned}$$

Then, there exists optimal plan, i.e., a plan for which $C_{p,\mathfrak {m}}(\Sigma )$ is attained among all plans with $c(\eta )>0$. This plan satisfies, among other properties

• (a) $\rho ={\textrm{mod}}_p(\Sigma )^{-1}f_{\Sigma }^{p-1}$,

• (b) $\int _X f_{\Sigma }(x)\,d\mu (x)=1$ for $\eta$-a.e. $\mu \in \Sigma$,

• (c) $\eta$ is concentrated on $\Sigma$.

Notice, that from condition (b) above, it implies that $\Sigma$ contains measure family $\Sigma _{{\textrm{ext}}}$ with measures for which the integral of the extremal function equals 1. This condition may not be satisfied with Badger’s approach. The explanation may be the following: in some cases family $\Sigma$ is such that $C_{p,\mathfrak {m}}(\Sigma )=0$. On the other hand, condition (a) above reminds of condition (iii) in Badger’s theorem. In fact, for Suslin sets $\Sigma$, (a) implies (iii). Let us explain this. From the definition of $\rho$, we have the following integral formula:

$$\begin{aligned} \int \limits _{\Sigma } \int \limits _X f(x)\,\text {d}\mu (x)\,\text {d}\eta (\mu )={\textrm{mod}}_p(\Sigma )^{-1}\int \limits _X f_{\Sigma }^{p-1}(x)f(x)\,\text {d}\mathfrak {m}, \end{aligned}$$

for any non-negative function $f\in {{\mathcal {L}}}^p(X,\mathfrak {m})$. Clearly, taking non-negative and nonpositive part of a function using linearity of both sides with respect to f, it follows that above formula holds for any $f\in {{\mathcal {L}}}^p(X,\mathfrak {m})$. Therefore, taking $F=\{\mu \in \Sigma \mid \int _X f_{\Sigma }(x)\,\text {d}\mu (x)=1\}$ and $f\in {{\mathcal {L}}}^p(X,\mathfrak {m})$ taking values in $[-\infty ,\infty ]$, if $\int _X f(x)\,\text {d}\mu (x)\ge 0$ for all $\mu \in F$, then

$$\begin{aligned} {\textrm{mod}}_p(\Sigma )^{-1}\int \limits _X f_{\Sigma }^{p-1}(x)f(x)\,\text {d}\mathfrak {m}=\int \limits _F \int \limits _X f(x)\,\text {d}\mu (x)\,\text {d}\eta (\mu )\ge 0, \end{aligned}$$

(1.1)

since F is of full measure in $\Sigma$. Notice moreover, that such chosen F satisfies also conditions (i) and (ii) of Badger’s criterion. It seems that the correlation between results in [2] and [1] has not been established anywhere in the literature. Moreover, it is interesting to notice that condition (iii) in Badger’s Theorem 1.1 means that a functional $L:{{\mathcal {W}}}\mapsto {{\mathbb {R}}}$:

$$\begin{aligned} L(\Phi _f)={\textrm{mod}}_p(\Sigma )^{-1}\int \limits _X f_{\Sigma }^{p-1}f\,\text {d}\mathfrak {m},\quad \Phi _f(\mu )=\int \limits _X f\,\text {d}\mu , \end{aligned}$$

where ${{\mathcal {W}}}=\{\Phi _f\mid f\in C_b(X)\}$, is positive. The existence of optimal plan is proved in [1] using geometric Hahn–Banach theorem and Riesz representation theorem for positive functional. We will use above observation to present slightly different proof of Theorem 1.3.

In this note, we study the extremal function from three perspective. First, we use Badger’s result to give alternative proof (in a compact case) of the existence of optimal plan (Theorem 1.3). Second, we focus on the set of Borel measures for which the integral of the extremal function equals one. Finally, we deal with the finite families $\Sigma$ of measures. We give explicit formulas for the extremal function and the p-modulus.

2 Basic definitions and facts

Let X be a Polish space, $\mathfrak {m}$ a Borel measure on X. Denote by ${{\mathcal {L}}}^p(X,\mathfrak {m})$ (and ${{\mathcal {L}}}_+^p(X,\mathfrak {m}$) spaces of p-integrable (positive) functions on X. Let ${{\mathcal {M}}}_+(X)$ be a space of Borel non-negative measures on X. For a function $f\in {{\mathcal {L}}}_+^p(X,\mathfrak {m})$ denote by $\Phi _f:{{\mathcal {M}}}_+(X)\rightarrow {{\mathbb {R}}}$ the operator given by

$$\begin{aligned} \Phi _f(\mu )=\int \limits _X f(x)\,\text {d}\mu (x). \end{aligned}$$

It can be proven that $\Phi _f$ is a Borel map.

Fix a subset $\Sigma \subset {{\mathcal {M}}}_+(X)$. We say that a function $f\in {{\mathcal {L}}}^p_+(X,\mathfrak {m})$ is p-admissible if $\Phi _f\ge 1$ on $\Sigma$. Following [3], we define p-modulus of $\Sigma$ by

$$\begin{aligned} {\textrm{mod}}_p(\Sigma )={\textrm{inf}}\left\{ \int \limits _X f^p(x)\,\text {d}\mathfrak {m}(x)\mid \,f \,\text {is\,admissible}\right\} . \end{aligned}$$

If the set of p-admissible functions is empty, we put ${\textrm{mod}}_p(\Sigma )=\infty$. We will need also a version with respect to continuous and bounded functions

$$\begin{aligned} {\textrm{mod}}_{p,c}(\Sigma )={\textrm{inf}}\left\{ \int \limits _X f^p(x)\,\text {d}\mathfrak {m}(x)\mid f\in C_b(X),\, \Phi _f\ge 1 \,\text {on}\, \Sigma \right\} . \end{aligned}$$

p-Moduli ${\textrm{mod}}_p$ and ${\textrm{mod}}_{p,c}$ have the following properties (see Fuglede [3]):

• 1. if ${{\mathcal {T}}}\subset \Sigma$, then ${\textrm{mod}}_p({{\mathcal {T}}})\le {\textrm{mod}}_p(\Sigma )$,

• 2. for families $\Sigma _i$, $i\in {{\mathbb {N}}}$, ${\textrm{mod}}_p(\bigcup _i\Sigma _i)^{\frac{1}{p}}\le \sum _i{\textrm{mod}}_p(\Sigma _i)^{\frac{1}{p}}$,

• 3. if $p>1$ and ${\textrm{mod}}_p(\Sigma )>0$, then there is a function $f_{\Sigma }\in {{\mathcal {L}}}^p_+(X,\mathfrak {m})$, such that $\Phi _{f_{\Sigma }}\ge 1$ up to a subfamily of p-modulus zero and ${\textrm{mod}}_p(\Sigma )=\Vert f_{\Sigma }\Vert ^p$,

• 4. ${\textrm{mod}}_p(\Sigma )=0$ if and only if there is a function $f\in {{\mathcal {L}}}^p_+(X,\mathfrak {m})$, such that $\Phi _f=\infty$ on $\Sigma$.

Let us move to description of plans with barycenter in $L^q(X,\mathfrak {m})$ studied in [1]. Firstly, recall that we equip ${{\mathcal {M}}}_+(X)$ with Polish topology of weak convergence (in duality with $C_b(X)$). We say that a Borel probability measure $\eta$ on ${{\mathcal {M}}}_+(X)$ is a plan with barycenter in $L^q(X,\mathfrak {m})$ if there is non-negative constant $c_q(\eta )$, such that

$$\begin{aligned} \int \limits _{{{\mathcal {M}}}_+(X)}\Phi _f(\mu )\,\text {d}\eta (\mu )\le c_q(\eta )\Vert f\Vert _p,\quad f\in {{\mathcal {L}}}^p_+(X,\mathfrak {m}). \end{aligned}$$

(2.1)

The use of such terminology in motivated by the following observation [1]: define a measure ${{\underline{\eta }}}$ on X as a barycenter:

$$\begin{aligned} {{\underline{\eta }}}(A)=\int \limits _{{{\mathcal {M}}}_+(X)}\mu (A)\,\text {d}\eta (\mu ). \end{aligned}$$

Then, ${{\underline{\eta }}}(A)\le c_q(\eta )(\mathfrak {m}(A))^{\frac{1}{p}}$, hence there exists an integrable function $\rho$, such that ${{\underline{\eta }}}=\rho \mathfrak {m}$. By condition (2.1) it follows that $\rho \in L^q(X,\mathfrak {m})$. Moreover, by the definition of ${{\underline{\eta }}}$ from the “layer cake” representation we have

$$\begin{aligned} \int \limits _{{{\mathcal {M}}}_+(X)}\Phi _f(\mu )\,\text {d}\eta (\mu )=\int \limits _X \rho (x)f(x)\,\text {d}\mathfrak {m}(x). \end{aligned}$$

(2.2)

Choose a subset $\Sigma$ in ${{\mathcal {M}}}_+(X)$ and consider a number

$$\begin{aligned} C_{p,\mathfrak {m}}(\Sigma )={\textrm{sup}}_{c(\eta )>0}\frac{\eta (\Sigma )}{c(\eta )}, \end{aligned}$$

(2.3)

where the supremum is taken with respect to all plans $\eta$ with barycenter in $L^q(X,\mathfrak {m})$, such that $c(\eta )>0$. If such plans do not exist, we put $C_{p,\mathfrak {m}}(\Sigma )=-\infty$. It follows by the definition of p-modulus taking p-admissible functions in (2.1) that

$$\begin{aligned} C_{p,\mathfrak {m}}(\Sigma )\le {\textrm{mod}}_p(\Sigma )^{\frac{1}{p}}. \end{aligned}$$

(2.4)

The main results in [1] concerning p-modulus in general are Theorem 1.3 and the following lemma.

Lemma 2.1

[1] If $\Sigma$ is universally measurable, $C_{p,\mathfrak {m}}(\Sigma )>0$, ${\textrm{sup}}_{\mu \in \Sigma }\mu (X)<\infty$, then there is a plan $\eta$ with barycenter in $L^q(X,\mathfrak {m})$ optimal for (2.3). Moreover, this plan is concentrated on $\Sigma$, hence $C_{p,\mathfrak {m}}(\Sigma )=c(\eta )^{-1}$.

Recall that a set $\Sigma \subset {{\mathcal {M}}}_+(X)$ is universally measurable if it is measurable for any measure $\sigma \in {{\mathcal {M}}}_+({{\mathcal {M}}}_+(X))$.

3 Main results

Adopt notation from the previous section.

3.1 Alternative proof of Theorem 1.3 in a compact case

We will give a proof in the case of $\Sigma$ compact. Assume $C_{p,\mathfrak {m}}(\Sigma )>0$ and ${\textrm{sup}}_{\mu \in \Sigma }\mu (X)<\infty$. Denote by $f_{\Sigma }$ the extremal function for the p-modulus of $\Sigma$. Put

$$\begin{aligned} {{\mathcal {W}}}=\{\Phi _f\mid f\in C(\Sigma )\},\quad K=\{F\in C(\Sigma )\mid F\ge 0\}. \end{aligned}$$

Then, ${{\mathcal {W}}}$ is a subspace in $C(\Sigma )$ and K is a cone in $C(\Sigma )$. Consider a linear functional $L:{{\mathcal {W}}}\rightarrow {{\mathbb {R}}}$ given by

$$\begin{aligned} L(\Phi _f)={\textrm{mod}}_p(\Sigma )^{-1}\int \limits _X f_{\Sigma }^{p-1}f\,\text {d}\mathfrak {m}. \end{aligned}$$

For $\Phi _f\in {{\mathcal {W}}}\cap K$, by Corollary 1.2 of Badger’s Theorem, we have $L(\Phi _f)\ge 0$. Thus, L is positive on ${{\mathcal {W}}}\cap K$. We will show that for any $F\in C(\Sigma )$ there exists $f\in C(X)$, such that $F-\Phi _f\in K$, i.e., $F\ge \Phi _f$. Indeed, by assumption $d={\textrm{sup}}_{\mu \in \Sigma }\mu (X)<\infty$. Since F is continuous on a compact set $\Sigma$, it attains its minimum $m={\textrm{inf}}_{\Sigma }F$. It suffices to put $f=\frac{m}{d}$. Then, for any $\mu \in \Sigma$:

$$\begin{aligned} \Phi _f(\mu )=\int \limits _X f\,\text {d}\mu =\frac{m}{d}\mu (X)\le m\le F(\mu ). \end{aligned}$$

Hence, $\Phi _f\le F$. Thus we may apply Riesz extension theorem. It implies existence of positive functional ${\tilde{L}}$ on $C(\Sigma )$ extending L. Now, by Riesz representation theorem, there is a Borel measure $\eta$ on $\Sigma$, such that

$$\begin{aligned} {\tilde{L}}(F)=\int \limits _{\Sigma } F(\mu )\,\text {d}\eta (\mu ). \end{aligned}$$

In particular, for $F=\Phi _f$ we have

$$\begin{aligned} \int \limits _{\Sigma } \Phi _f(\mu )\,\text {d}\eta (\mu )={\textrm{mod}}_p(\Sigma )^{-1}\int \limits _X f_{\Sigma }^{p-1}f\,\text {d}\mathfrak {m}. \end{aligned}$$

Hence, $\eta$ is a plan with $c(\eta )={\textrm{mod}}_p(\Sigma )^{-\frac{1}{p}}$. Putting any p-admissible f we get $\eta (\Sigma )\le 1$. Moreover, for the extremal function $f_{\Sigma }$ we have

$$\begin{aligned} \int \limits _{\Sigma }\Phi _{f_{\Sigma }}\,\text {d}\eta =1. \end{aligned}$$

Since $\Phi _{f_{\Sigma }}\ge 1$ it follows that $\eta (\Sigma )>0$. Now we proceed analogously as in [1]. If $\eta (\Sigma )<1$, define $\eta '=\eta (\Sigma )^{-1}\eta$. Then, $\eta '$ is a Borel probability measure concentrated on $\Sigma$ and

$$\begin{aligned} \int \limits _{\Sigma } \Phi _f(\mu )\,\text {d}\eta '(\mu )=(\eta (\Sigma ){\textrm{mod}}_p(\Sigma ))^{-1}\int \limits _X f_{\Sigma }(x)f(x)\,\text {d}\mathfrak {m}(x). \end{aligned}$$

Thus

$$\begin{aligned} \frac{\eta '(\Sigma )}{c(\eta ')}=\frac{\eta (\Sigma )}{c(\eta )}=\eta (\Sigma ){\textrm{mod}}_p(\Sigma )^{\frac{1}{p}}. \end{aligned}$$

We have shown that there exists plan with barycenter in $L^q(X,\mathfrak {m})$ concentrated on $\Sigma$ with $c>0$. Now, classical argument with considering maximizing sequence shows that there is an optimal plan (see proof of Lemma 4.4 in [1])

3.2 Sets of extremal measures

Define

$$\begin{aligned} \Sigma _{{\textrm{ext}}}=\{\mu \in {{\mathcal {M}}}_+(X)\mid \Phi _{f_{\Sigma }}(\mu )=1\},\quad \Sigma ^0_{{\textrm{ext}}}=\Sigma _{{\textrm{ext}}}\cap \Sigma . \end{aligned}$$

If $C_{p,\mathfrak {m}}=0$, then for any plan with positive $c(\eta )$ we have $\eta (\Sigma )=0$. Hence, $\eta (\Sigma ^0_{{\textrm{ext}}})=0$. Let us move to more detailed description of the sets $\Sigma _{{\textrm{ext}}}$ and $\Sigma ^0_{{\textrm{ext}}}$. We are motivated by the following two examples.

Example 3.1

(See [1]) Let $X=[0,1]$ be the unit interval, $\mathfrak {m}={{\mathcal {L}}}$ the Lebesgue measure. Put

$$\begin{aligned} \Sigma =\left\{ {{\mathcal {L}}},{{\mathcal {L}}}_{[0,\frac{1}{2}]},{{\mathcal {L}}}_{[\frac{1}{2},1]}\right\} . \end{aligned}$$

Then, any p-admissible function f satisfies

$$\begin{aligned} \int \limits _0^1 f(x)\,\text {d}{{\mathcal {L}}}(x)=\int \limits _0^{\frac{1}{2}} f(x)\,\text {d}{{\mathcal {L}}}(x)+\int \limits _{\frac{1}{2}}^1 f(x)\,\text {d}{{\mathcal {L}}}(x)\ge 2. \end{aligned}$$

Thus, the extremal function equals $f_{\Sigma }=2$ and ${\textrm{mod}}_p(\Sigma )=2^p$. Additionally, $\Sigma ^0_{{\textrm{ext}}}=\left\{ {{\mathcal {L}}}_{\left[ 0,\frac{1}{2}\right] },{{\mathcal {L}}}_{\left[ \frac{1}{2},1\right] }\right\} \ne \Sigma$ and ${\textrm{mod}}_p(\Sigma )={\textrm{mod}}_p(\Sigma ^0_{{\textrm{ext}}})$. Hence, the optimal plan $\eta$ is concentrated on $\Sigma ^0_{{\textrm{ext}}}$. It is easy to see that

$$\begin{aligned} \eta =\frac{1}{2}\delta _{{{\mathcal {L}}}_{\left[ 0,\frac{1}{2}\right] }}+\frac{1}{2}\delta _{}{{\mathcal {L}}}_{\left[ \frac{1}{2},1\right] }. \end{aligned}$$

Example 3.2

Let $\Sigma$ be a family of Lebesgue measures on curves joining two concentric circles on a plane (as in the Introduction) except for the rays. It is known that the extremal function $f_{\Sigma }$ is constant and that the p-modulus of $\Sigma$ equals q-capacity of considered condenser, q and p are conjugate coefficients. However, for the family ${{\mathcal {T}}}$ of Lebesgue measures on rays we have ${{\mathcal {T}}}\subset \Sigma _{{\textrm{ext}}}$, hence ${{\mathcal {T}}}$ and $\Sigma _{{\textrm{ext}}}$ are disjoint with $\Sigma$. Moreover, ${\textrm{mod}}_p(\Sigma )={\textrm{mod}}_p(\Sigma \cup {{\mathcal {T}}})$.

The first example suggests that it is convenient to consider the closure ${\textrm{cl}}\Sigma$ of $\Sigma$. We will consider in this case the p-modulus ${\textrm{mod}}_{p,c}$ defined with respect to continuous bounded functions.

Proposition 3.3

We have ${\textrm{mod}}_{p,c}(\Sigma )={\textrm{mod}}_{p,c}({\textrm{cl}}\Sigma )$.

Proof

Since $\Sigma \subset {\textrm{cl}}\Sigma$ it follow that ${\textrm{mod}}_{p,c}(\Sigma )\le {\textrm{mod}}_{p,c}({\textrm{cl}}\Sigma )$. To prove the converse inequality it suffices to show that any p-admissible function for the p-modulus of $\Sigma$ is also p-admissible for the p-modulus of ${\textrm{cl}}\Sigma$. Let $f\in C_b(X)$ be such that $\Phi _f(\mu )\ge 1$ for $\mu \in \Sigma$. Let $\mu _0\in {\textrm{cl}}\Sigma$ and let $(\mu _n)\subset \Sigma$ be a sequence convergent to $\mu _0$. Then

$$\begin{aligned} \int \limits _X f\,\text {d}\mu _n\rightarrow \int \limits _X f\,\text {d}\mu _0. \end{aligned}$$

Since $\int _X f\,\text {d}\mu \ge$ it follows that $\int _X f\,\text {d}\mu _0\ge 1$. Thus f is p-admissible for ${\textrm{cl}}\Sigma$. $\square$

Corollary 3.4

The extremal function $f_{\Sigma }$ for the p-modulus of $\Sigma$ is also the extremal function for the p-modulus of ${\textrm{cl}}\Sigma$, i.e., $f_{\Sigma }=f_{{\textrm{cl}}\Sigma }$.

Proposition 3.5

Assume $f_{\Sigma }\in C_b(X)$ and $\Sigma$ is a bounded set in ${{\mathcal {M}}}_+(X)$. Then, $({\textrm{cl}}\Sigma )_{{\textrm{ext}}}^0$ is nonempty.

Proof

Suppose $({\textrm{cl}}\Sigma )_{{\textrm{ext}}}^0$ is empty. Let

$$\begin{aligned} c={\textrm{inf}}\left\{ \Phi _{f_{\Sigma }}(\mu )\mid \mu \in \Sigma \right\} . \end{aligned}$$

Then $c=1$, since otherwise $\frac{1}{c}f_{\Sigma }$ would be p-admissible for the p-modulus of $\Sigma$ with the p-norm smaller that the p-norm of $f_{\Sigma }$. Take a minimizing sequence $(\mu _n)$, i.e. $\Phi _{f_{\Sigma }}(\mu _n)\rightarrow c=1$. Thus $\int _X f_{\Sigma }\,\text {d}\mu _n\rightarrow 1$. Since ${\textrm{cl}}\Sigma$ is compact, by boundedness of $\Sigma$, it follows that $(\mu _n)$ has convergent subsequence $(\mu _{n_k})$. Denote a limit measure by $\mu _0\in {\textrm{cl}}\Sigma$. Then

$$\begin{aligned} \int \limits _X f_{\Sigma }\,\text {d}\mu _{n_k}\rightarrow \int \limits _X f_{\Sigma }\,\text {d}\mu . \end{aligned}$$

It implies $\Phi _{f_{\Sigma }}(\mu _0)=1$. Hence, $\mu _0\in ({\textrm{cl}}\Sigma )_{{\textrm{ext}}}^0$. $\square$

3.3 Finite sets of measures

We now move to a description of families $\Sigma$ consisting of finite number of measures. Let us begin with the simplest case of one measure.

Proposition 3.6

Assume $\Sigma =\{\mu \}$, where $\mu =\rho \mathfrak {m}$ for some $\rho \in L^q(X,\mathfrak {m})$. Then

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

Proof

We apply Theorem 1.1. Let $f_{\Sigma }=m^{q-1}\rho ^{q-1}$, where q is a coefficient conjugate to p and $m=\left( \int _X \rho ^q\,\text {d}\mathfrak {m}\right) ^{1-p}$. Let $F=\{\mu \}=\Sigma$. Then

$$\begin{aligned} \int \limits _X f_{\Sigma }\,\text {d}\mu =m^{q-1}\int \limits _X \rho ^{q-1}\rho \,\text {d}\mathfrak {m}=1. \end{aligned}$$

Moreover, for any $f\in {{\mathcal {L}}}^p(X,\mathfrak {m})$ possibly taking values $\pm \infty$, and such that $\int _X f\,\text {d}\mu \ge 0$, we have

$$\begin{aligned} \int \limits _X f_{\Sigma }^{p-1}f\,\text {d}\mathfrak {m}=m\int \limits _X \rho f\,\text {d}\mathfrak {m}=m\int \limits _X f\,\text {d}\mu \ge 0. \end{aligned}$$

Applying Badger’s Theorem 1.1 we see that $f_{\Sigma }$ is extremal for the p-modulus of $\Sigma$ and that ${\textrm{mod}}_p(\Sigma )=m$. This completes the proof. $\square$

Notice that in this case, i.e. $\Sigma =\{\mu \}$, the optimal plan is a Dirac measure at $\mu$, $\eta =\delta _{\mu }$.

Let us remark, that if $\mu$ is not absolutely continuous with respect to $\mathfrak {m}$, then ${\textrm{mod}}_p(\Sigma )=0$. Indeed, there exists Borel set A, such that $\mathfrak {m}(A)=0$ and $\mu (A)>0$. In particular taking

$$\begin{aligned} f=\left\{ \begin{array}{ll} \frac{1}{\mu (A)} &{}\text {on}\, A\\ 0 &{} \text {otherwise} \end{array}\right. \end{aligned}$$

we have $\int _X f\,\text {d}\mu =1$ and $\int _X f^p\,\text {d}\mathfrak {m}=0$.

Assume now $\Sigma =\{\mu _1,\ldots ,\mu _n\}$, where $\mu _i=\rho _i\mathfrak {m}$ and $\rho _i\in L^q(X,\mathfrak {m})$.

Proposition 3.7

For above finite family $\Sigma$ we have

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

where $\rho =\frac{1}{n}\sum _i \rho _i$. Moreover, the optimal plan equals

$$\begin{aligned} \eta =\frac{1}{n}\sum _i \delta _{\mu _i}. \end{aligned}$$

Proof

Consider a one element family $F=\{\rho \,\mathfrak {m}\}$. Let

$$\begin{aligned} f_{\Sigma }=m^{q-1}\rho ^{q-1},\quad \text {where}\quad m=\left( \int \limits _X\rho ^q\,\text {d}\mathfrak {m}\right) ^{1-p}. \end{aligned}$$

Then

• 1. ${\textrm{mod}}_p(\Sigma )={\textrm{mod}}_p(\Sigma \cup F)$. Indeed, the inequality ${\textrm{mod}}_p(\Sigma )\le {\textrm{mod}}_p(\Sigma \cup F)$ is obvious by the monotonicity of p-modulus. Choose an admissible function f for the p-modulus of $\Sigma$. Then

  $$\begin{aligned} \int \limits _X f\,\text {d}(\rho \mathfrak {m})=\int \limits _X \rho f\,\text {d}\mathfrak {m}=\frac{1}{n}\sum _i \int \limits _X \rho _if\,\text {d}\mathfrak {m}\ge 1. \end{aligned}$$

  Thus f is p-admissible for $\Sigma \cup F$. This implies that ${\textrm{mod}}_p(\Sigma )\ge {\textrm{mod}}_p(\Sigma \cup F)$.

• 2. By the definition of a constant m, we have

  $$\begin{aligned} \int \limits _X f_{\Sigma }\,\text {d}(\rho \mathfrak {m})=\frac{m}{n}\sum _i\int \limits _X \rho _i^q\,\text {d}\mathfrak {m}=1. \end{aligned}$$

• 3. For any $f\in {{\mathcal {L}}}^p(X,\mathfrak {m})$ taking values in $[-\infty ,\infty ]$ if $\int _X f\,\text {d}(\rho \mathfrak {m})\ge 0$, then

  $$\begin{aligned} \int \limits _X f_{\Sigma }^{p-1}f\,\text {d}\mathfrak {m}=\frac{m}{n}\sum _i \int \limits _X \rho _if\,\text {d}\mathfrak {m}=m\int \limits _X f\,\text {d}(\rho \mathfrak {m})\ge 0. \end{aligned}$$

Therefore, all assumptions of Badger’s Theorem 1.1 are satisfied. It suffices to prove the formula for the optimal plan $\eta$. We have

$$\begin{aligned} \int \limits _{\Sigma }\Phi _f\,\text {d}\eta =\frac{1}{n}\sum _i\int \limits _X \rho _if\,\text {d}\mathfrak {m}={\textrm{mod}}_p(\Sigma )^{-1}\int \limits _X f_{\Sigma }^{p-1}f\,\text {d}\mathfrak {m}. \end{aligned}$$

Hence $\eta$ is a (probability) plan with barycenter in $L^q(X,\mathfrak {m})$ with the optimal constant $C_{p,\mathfrak {m}}(\Sigma )$. $\square$