## On the generalisation of the Laplace transform

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I consider a measure transform $A$ given by $$A\mu(x) = \int\limits_{\mathbb{R}^n_{+}} e^{-g(x,y)} \mu(dy)$$ where $g(x,y)$ is some positive smooth function, $\mu$ is a Borel measure. Is it a well-known transform? Where can I read about it? I'm interested in conditions of injectivity and in description of it's image.

If it is very general we can consider only homogeneous of order $1$ and concave functions $g(x,y)$ such that $g(x,y) = g(x_1 y_1,x_2 y_2, \ldots, x_n y_n)$.

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There is a such a large literature on Laplace and related transforms on cones (e.g. the positive orthant) that it is hard to know where to start. The book of Stein and Weiss might be one place. There is a TAMS paper of O. Rothaus entitled "Some properties of Laplace transforms of measures" which directly addresses some of your questions, although surely there are other sources. – Dan Fox Jan 27 2012 at 8:20
I think this is too general to have any useful answer, without extra conditions on $g$; although I'd be glad to be wrong, since any nontrivial general results on this would be very interesting. If you know $g$ satisfies a differential equation, for example, you could start to do stuff. – Zen Harper Jun 4 at 9:12
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