The Free Fisher Information along the Free Heat Flow: Bounds, Lower Semicontinuity in Time and Integrability

In the setting of Noncommutative Laws, Couplings and the Wasserstein Distance: Standing Notation, let $\boxplus$ be the free additive convolution, $\mathrm{sc}_{d,t}$ the semicircular law of variance $t$ and $\Phi^{*}$ the free Fisher information; rays such as $(0,\infty)$, integrability on a Borel subset of $\mathbb{R}$ and $\int_{B}f(t)\,dt$ are as in Rays, and Integrability and the Integral of a Real Function on a Borel Subset of the Real Line §rays and Rays, and Integrability and the Integral of a Real Function on a Borel Subset of the Real Line §integral, and $(0,1)$ is an open interval. Let $\lambda\in\Sigma_{d}$. For real $t>0$ the law $\lambda\boxplus\mathrm{sc}_{d,t}$ has conjugate variables by Conjugate Variables along the Free Heat Flow: the Projected Semicircular Increment and the Bound d/t on the Free Fisher Information §conjugate; let

$$f_{\lambda}(t)=\Phi^{*}(\lambda\boxplus\mathrm{sc}_{d,t}),\qquad g_{\lambda}(t)=\frac{d}{1+t}-f_{\lambda}(t)\qquad(t\in(0,\infty)).$$

1. (Bounds)¶ For every real $t>0$, $M(\lambda)+td>0$ and

$$\frac{d^{2}}{M(\lambda)+td}\le f_{\lambda}(t)\le\frac{d}{t}.$$

2. (Lower semicontinuity)¶ For every real $a$, the set $\{t\in(0,\infty):\ f_{\lambda}(t)>a\}$ is open in $\mathbb{R}$ for the absolute value metric. Consequently the extensions of $f_{\lambda}$ and $g_{\lambda}$ by $0$ to $\mathbb{R}$ are measurable with respect to the Borel $\sigma$-algebra.

3. (Integrability)¶ $g_{\lambda}$ is integrable on $[1,\infty)$. Moreover $g_{\lambda}$ is integrable on $(0,\infty)$ if and only if $f_{\lambda}$ is integrable on $(0,1)$.