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 $g(x,y) = g(x_1 y_1,x_2 y_2, \ldots, x_n y_n)$.