**Background**

The *upper and lower Maslov dequantizations* are respectively the limits $h \downarrow 0$ and $h \uparrow 0$ of deformations of the semifield $(\mathbb{R}_+,+,\cdot)$ defined for $0 \ne h \in \mathbb{R}$ by the map
\begin{equation}
\Phi_h(x) := h \log x
\end{equation}
which sends $\mathbb{R}_+$ to $\mathbb{R} \cup \{-\sigma(h) \cdot \infty\}$.
Since the map $\Phi_h$ is invertible, it defines deformed addition and multiplication operations: i.e.,
\begin{equation}
\label{eq:MaslovArithmetic}
u +_h v := \Phi_h (\Phi_h^{-1}(u) + \Phi_h^{-1}(v)); \quad u \cdot_h v := \Phi_h (\Phi_h^{-1}(u) \cdot \Phi_h^{-1}(v)).
\end{equation}
Explicitly, we have
\begin{equation}
u +_h v := h \log \left ( e^{u/h} + e^{v/h} \right ); \quad u \cdot_h v := h \log \left ( e^{u/h} \cdot e^{v/h} \right ) \equiv u+v.
\end{equation}
Now writing $\lozenge$ for $\vee$ when $h > 0$ and $\wedge$ when $h < 0$, it is easy to see that $(\sum_{(h)})_{j \in [n]} u_j := u_1 +_h \dots +_h u_n = h \log \sum_{j \in [n]} e^{u_j/h} = \lozenge_j u_j + h \log \sum_j e^{(u_j-\lozenge_k u_k)/h}$. Clearly, as $h \rightarrow 0$, this tends to $\lozenge_j u_j$. This is manifested in the figure below:

There is a well-established theory of analysis in the dequantization limit $h \rightarrow 0$: this is essentially the *Ansatz*
\begin{equation}
^\lozenge : \sum \mapsto \lozenge; \quad ^\lozenge : \prod \mapsto \sum
\end{equation}
where again $\lozenge$ is either $\vee$ or $\wedge$, as appropriate.
The first of the correspondences above naturally leads to
\begin{equation}
^\lozenge : \int \mapsto \text{ext}^\lozenge,
\end{equation}
where here the *extremum* is given by $\text{ext}^\vee := \sup$ and $\text{ext}^\wedge := \inf$. This turns out to have a rigorous interpretation, so that, e.g.,
\begin{equation}
^\lozenge : \int K(t,u) f(u) du \mapsto \text{ext}^\lozenge_u \left ( K(t,u) + f(u) \right )
\end{equation}
is not only suggestive but correct, and in fact such integral transforms exhaust the class of linear operators. There is also a correspondence between the Fourier and Legendre transformations that is a bit trickier.

**Question**

Is there a reasonable notion of deformed integral for the case where $h$ is small but nonzero?

The *Ansatz* suggests something which diverges as $\Delta x \downarrow 0$ unless simultaneously $h \rightarrow 0$ in a controlled way, in which event the extremum results. For my purposes this is degenerate.