What does it mean when one says the inequality must be understood in the barrier sense, when necessary?
I encountered this notion when I was reading Perelman's paper "The entropy formula for the Ricci flow and its geometric applications" (lemma 8.3(a)).
Definition 1.2 in On the Distributional Hessian of the Distance Function explains the difference: an inequality can hold in distributional sense, in barrier sense, or in viscosity sense.
Given a Riemannian manifold $X$, a point $x_0\in X$ and a continuous function $\phi:X\rightarrow \mathbb{R}$, we say that $\Delta\phi\leq 0$ at $x_0$ in barrier sense if for any $\epsilon>0$, there is a $\mathcal{C}^2$ function $\psi_{x_0,\epsilon}$ on a neighborhood of $x_0$ such that $\psi_{x_0,\epsilon}(x_0)=\phi(x_0)$, $\Delta\psi_{x_0,\epsilon}<\epsilon$, and $\psi_{x_0,\epsilon}\geq\phi$.
A functional inequality that holds in the barrier sense need not hold as a classic inequality, which is why the words "when necessary" are used in the quote of the OP, to weaken the inequality statement and make it apply when a classic inequality fails.
