# What does this notation (inner product with differential) mean?

The following symbols are used in On optimal transport of matrix-valued measures by Yann Brenier and Dmitry Vorotnikov (cf. p. 3):

• $$A : B := \text{tr}(A B^{\mathsf{T}}) = \sum_{i, j = 1}^{d} a_{i, j} b_{i, j}$$ is the Frobenius inner product of two real $$d \times d$$ matrices $$A, B \in \mathbb R^{d \times d}$$.
• $$\mathcal P^+$$ is the subspace of real symmetric positive semidefinite $$d \times d$$ matrices.
• $$\mathbb P^+$$ is the set of $$\mathcal P^+$$-valued Radon measures $$P$$ on $$\mathbb R^d$$ with finite tr$$(\text{d}P(\mathbb R^d))$$.

Question 1. What is $$\text{d} P$$? Is $$\text{d}P(\mathbb R^d) = P(\mathbb R^d) \in \mathcal P^+$$? In the scalar case $$d = 1$$, I would expect $$\text{d} P$$ to be some sort of unnormalized density (with respect to the Lebesgue measure). All $$\mathcal P^+$$-valued measures $$P$$ have a density $$P'$$ with respect to the associated scalar trace measure $$\text{tr}(P)$$ (cf. p. 1-2 of Duran and Lopez-Rodriguez's The $$L^p$$ space of a positive definite matrix of measures and density of matrix polynomials in $$L^1$$), so is $$\text{d}P$$ maybe the density of $$P$$ with respect to the trace measure?

Question 2. I was wondering how to interpret (cf. p. 4) the real number (?) $$\label{eq:star} \tag{\star} \int_{\mathbb R^d} \Phi : (\text{d} G_1 - \text{d} G_0)$$ for $$G_0, G_1 \in \mathbb P^+$$ and a bounded and Lipschitz continuous function $$\Phi$$ on $$\mathbb R^d$$ (probably mapping into $$\mathbb R$$, but this is not indicated), since the differential appears in the inner product. E.g. if $$G_k = P_k \delta_{x_k}$$ for $$P_k \in \mathcal P^+$$ and $$x_k \in \mathbb R^d$$ for $$k \in \{ 0, 1 \}$$, then I would suspect $$G_0, G_1 \in \mathbb P^+$$, but I don't know how to calculate \eqref{eq:star} for any $$\Phi$$.

If e.g. $$G_0$$ had a density $$g_0$$ with respect to some scalar measure $$\lambda$$, then I can imagine that $$\int_{\mathbb R^d} \Phi : \text{d}G_0 = \int_{\mathbb R^d} \Phi(x) : g_0(x) \; \text{d} \lambda(x).$$

I have encountered a similar notation (inner product / dual pairing with a differential) in the abstract of A simple proof of Singer's theorem by Hensgen and also in equation (7) of L. Ning and T. T. Georgiou. Metrics for matrix-valued measures via test functions.

I also searched for this notation in Diestel and Uhl's "Vector measures", bit didn't find it used.

## Update

Since the entries $$(G_0 - G_1)_{i, j}$$ of $$G_0 - G_1$$ are scalar measures in their own right, can we interpret \eqref{eq:star} as the following sum of integrals of scalar functions against scalar measures? $$\int_{\mathbb R^d} \sum_{i, j = 1}^{d} [\Phi(x)]_{i, j} \; \text{d}[(G_0 - G_1)_{i, j}](x)$$

Question 1: For any measure $$\mu$$, we often use the notation $$\mathrm d\mu$$ to mean the same thing as "$$\mu$$." In your context, $$P$$ (or $$\mathrm dP$$) is a matrix of measures, something like $$P = \begin{pmatrix} \mu_{11} & \dots & \mu_{1d} \\ \vdots & \ddots & \vdots\\ \mu_{d1} & \dots & \mu_{dd} \end{pmatrix},$$ so $$P(\mathbb R^d)$$ or $$\mathrm dP(\mathbb R^d)$$ (either of them) means the matrix of numbers $$P(\mathbb R^d) = \begin{pmatrix} \mu_{11}(\mathbb R^d) & \dots & \mu_{1d}(\mathbb R^d) \\ \vdots & \ddots & \vdots\\ \mu_{d1}(\mathbb R^d) & \dots & \mu_{dd}(\mathbb R^d) \end{pmatrix}.$$

Question 2: For two matrices $$\Phi,\Psi$$, the notation $$\Phi:\Psi$$ means the same thing as $$\mathrm{Tr}(\Phi\Psi^T)$$ (this is implicitly defined at the top of p. 3).

So if $$\Phi$$ is a matrix-valued function and $$\mu$$ is a matrix-valued measure, to understand what $$\int \Phi:\mathrm d\mu$$ means, it is probably simplest to think about what it means when $$\Phi$$ is a simple function, i.e., takes only finitely many values. Say $$\Phi = \sum_i A_i\mathbf 1_{E_i}$$ for some matrices $$A_i$$ and some measurable sets $$E_i\subset \mathbb R^d$$. Then the only sensible meaning for the integral should be $$\int \Phi:\mathrm d\mu = \sum_i A_i:\mu(E_i).$$ (Keep in mind how $$\mu(E_i)$$ is a matrix of numbers for each $$i$$, so we can fall back on the notation $$\Phi:\Psi$$ when $$\Phi,\Psi$$ are both matrices of numbers.) To generalize to integrable functions $$\Phi$$ should now be a routine application of the standard machinery of Lebesgue integration.

Lastly, if the preceding is understood, $$\int \Phi:(\mathrm d\mu_1 - \mathrm d\mu_2)$$ should be understood in the analogous way as usual integration against a signed measure.

• Thank you for your answer. About your first sentence for question 2: in the definition of the Frobenius norm the authors use $| \Phi | := \sqrt{\Phi : \Phi}$ for $\Phi \in \mathbb R^{n \times n}$ but the same symbol in (1.6) for the norm of a vector $q \in \mathbb R^n$, right? But you are saying that because of the definition $\| \phi \|_{\infty} := \sup | \phi |$ one should suspect that $\phi$ is matrix-valued? Jan 17 at 20:26
• If I understand you answer correctly, the interpretation I gave in the update section at the bottom of my question is correct? Jan 17 at 20:28
• @Ramanujan: So, in an expression like $\sup_{\|\Phi\|_{\mathrm{Lip}}\le 1}|\int \Phi: (dG_1-dG_0)|$, we understand that the "Lip-norm" is of a matrix, with its corresponding definition, and that's what I was talking about in my answer. It can also be the Lip norm of a real-valued function—I will likely edit that out of my answer, since the interpretation is actually flexible. I was fixating on the notation in the context of the integrals $\int \Phi: d\mu$. Jan 17 at 20:36
• @Ramanujan: As for your second comment, yes I believe the interpretation you gave is correct. There is just a little care that needs to be taken because expressions like $\int \Phi:d\mu$ in general have to be interpreted as Lebesgue integrals, but at least formally, the expression you gave in the update is exactly what $\int \Phi:d\mu$ is. And I am fairly confident it can be justified by appealing to the standard machinery of Lebesgue integration as well. (Check it for simple functions, then use monotone convergence and decompositions into positive and negative parts.) Jan 17 at 20:39
• Thank you. Am I right in assuming that in (2.2) for $\Phi \in \mathcal C_b^1([0, 1] \times \mathbb R^n; \mathcal S)$, the symbol $\Phi_{\tau}$ for $\tau \in [0, 1]$ is the function $\mathbb R^n \ni x \mapsto \Phi(\tau, x) \in \mathcal S$ and that for $\tau \in [0, 1]$, $\partial_{\tau} \Phi_{\tau}$ is the function $\mathbb R^n \ni x \mapsto \frac{\partial}{\partial \tau} \Phi(\tau, x)$? Jan 17 at 22:37