How to think about dual space of a certain space of Lipschitz functions

Question

Consider the following Banach space (for concreteness):

$$X=Lip(\bar{\mathbb{B}}^n)=\{f\in C^0(\bar{\mathbb{B}}^n): \Vert f \Vert_L<\infty \}$$ where $$ \bar{\mathbb{B}}^n=\{\mathbf{x}\in \mathbb{R}^n: |\mathbf{x}|\leq 1\} $$ is the closed ball and $$ \Vert f \Vert_L=\sup_{\bar{\mathbb{B}}^n}|f| +\sup_{\mathbf{x}\neq \mathbf{y}\in \bar{\mathbb{B}}^n} \frac{|f(\mathbf{x})-f(\mathbf{y})|}{|\mathbf{x}-\mathbf{y}|} $$ is one of the usual versions of a lipschitz norm.

I'm curious what is the best way to think about the (topological) dual space $X^*$ of $X$ as this space is a bit mysterious to me.

For instance, it's clear that any finite (signed) measure on $\bar{\mathbb{B}}^n$ can be thought of as an element of $X^*$, but one should also have elements that look like differences of infinite measures whose supports are sufficiently close and whose ``relative mass" is finite. Are there other natural elements?

Any references (in particular those that are more concrete and less abstract) would be appreciated.

Edit: I guess (please correct me if I am wrong) a fairly pathological element in the dual would be something like $$ \mu=\sum_{i=1}^\infty \left( 2^i \delta_{2^{-2i}}-2^{i} \delta_{-2^{-2i}}\right) $$ on $\mathbb{B}=[-1,1]$.

Are there any natural subsets on which one can restrict and have a less crazy situation? For instance, fix a Radon measures $\mu$ on $\mathbb{B}^n$ (the open ball) with $\mu(\mathbb{B}^n)=\infty$. Is the subspace

$$Z=\{T\in X^*: T=\nu-\mu: \nu \mbox{ a Radon measure}\}$$ any nicer?

Answer 1

This would be a really long comment, so I've decided to post it as an answer. Hope it helps!

Disclaimer. I'm still learning FA, and my answer is based on my blurred understanding of the subject. I hope experts here will (in)validate this.

On a metric space $X=(X,d)$, let $\mathcal M(X)$ be the set of all measures and $\mathcal M_0(X)$ be the subset of measures $\mu$ for which $\mu(X)=0$. Define $\|\cdot\|_{KR}:\mathcal M(X) \rightarrow \mathbb R$ by

$$ \|\mu\|_{KR} := \inf_{\nu \in \mathcal M_0(X)}\|\nu\|_0 + \|\mu-\nu\|_{TV}, $$

where

• $\|\mu-\nu\|_{TV}$ is the total-variation between $\mu$ and $\nu$
• $\|\nu\|_0 := \underset{\gamma \in \Phi_\nu}{\inf}\int_{X \times X}d(x,x')d\gamma(x,x')$ and $\Phi_\nu$ is the subset of nonnegative measures $\gamma$ on $X \times X$ such that $\gamma(X \times B) - \gamma(B \times X) = \nu(B)$ every Borell subset $B$ of $X$.

Let $\|f\|_L := \max(\|f\|_\infty, L(f))$ be the bounded-Lipschitz norm on $Lip(X)$. Then

Theorem. $(\mathcal M(X),\|\cdot\|_{KR})$ is a normed vector space and the dual pairing $$ \langle f, \mu\rangle := \int_{X} fd\mu,\;for\;(f,\mu) \in Lip(X) \times \mathcal M(X) $$ establishes an isometric isomorphism between $(Lip(X), \|.\|_L)$ and the topological dual of $(\mathcal M(X),\|\cdot\|_{KR})$.

This result goes back to Kantorovich and Rubenstein (hence "KR"), and is well-documented in Hanin '92 (see Theorem 0 there).

Answer 2

The space of Lipschitz functions is the space $B^1_{\infty,\infty}$, so its dual is basically the space of distributions $B^{-1}_{1,1}$. A natural example of an element in the dual that isn't represented by a Radon measure is white noise in dimension $1$. As a probabilist, I certainly wouldn't call that "crazy" so I am not sure how to interpret your question about the "less crazy situation" ;-)