---
tag: C9Z0P713CF
kind: definition
name: Laplace transform on a convex cone
msc: [44A30, 44A10]
defines: MAKCYGGERD
uses: [6VE4KA12VD, QFY1JR1TK4, QT3M720TCV]
sources:
  - url: https://en.wikipedia.org/wiki/Laplace_transform
---

## Definition

Let $\Gamma \subseteq \R^n$ be a closed [[convex cone|6VE4KA12VD]] with nonempty interior, and let $\ip{\cdot}{\cdot}$ be the standard inner product on $\R^n$.
Let $\Gamma^* = \set{y \in \R^n : \ip{y}{x} \ge 0 \text{ for all } x \in \Gamma}$ be its [[dual cone|QFY1JR1TK4]], and let $\Omega$ be the interior of $\Gamma^*$.
Let $f \colon \Gamma \to \C$ be locally integrable.
The *Laplace transform* of $f$ on $\Gamma$ is the function
$$(\mathcal{L}_\Gamma f)(z) = \int_\Gamma e^{-\ip{z}{x}} f(x)\, dx, \qquad z = y + i\eta \in \C^n,$$
defined at those $z$ where the integral converges absolutely; here $\ip{z}{x} = \sum_j z_j x_j$.

## Remarks

If $y \in \Omega$, there is $c > 0$ with $\ip{y}{x} \ge c \abs{x}$ for all $x \in \Gamma$, by compactness of $\Gamma \cap \set{\abs{x} = 1}$.
So if $\abs{f(x)} \le C (1 + \abs{x})^N$, the transform converges absolutely on the tube $\Omega + i \R^n$ and is holomorphic there.

Example: $\Gamma = [0, \infty)^n$ is self-dual, and $\mathcal{L}_\Gamma$ is the [[multidimensional Laplace transform|QT3M720TCV]].
For $f = 1$ one gets $\mathcal{L}_\Gamma f(z) = 1/(z_1 \cdots z_n)$ on $\operatorname{Re} z_j > 0$.

Example: let $\Gamma$ be the cone of positive semidefinite real symmetric $m \times m$ matrices, with $\ip{y}{x} = \tr(yx)$.
Let $dx$ be the Lebesgue measure in the entries $x_{ij}$, $i \le j$.
The cone is self-dual, and $\Omega$ is the cone of [[positive definite matrices|0KX66CZMBB]].
For real $s > (m - 1)/2$ and $y \in \Omega$,
$$\int_\Omega e^{-\tr(yx)} (\det x)^{s - (m+1)/2}\, dx = \Gamma_m(s)\, (\det y)^{-s}, \qquad \Gamma_m(s) = \pi^{m(m-1)/4} \prod_{j=0}^{m-1} \Gamma\left(s - \frac{j}{2}\right).$$

Non-example: let $\Gamma$ contain a line $\R v$.
Then $\Gamma^*$ lies in the hyperplane $v^\perp$ and $\Omega$ is empty.
Since $\Gamma + \R v = \Gamma$, Fubini's theorem along the direction $v$ shows that even $f = 1$ has a divergent transform at every $z$.