Consider an integral transform of Borel measures supported on $\mathbb{R}^n_+$ given by $$ f(z) =\int\limits_{\mathbb{R}^n_+} x^{z}\frac{\mu(dx)}{x} $$ where $z = (z_1,...,z_n) \in \mathbb{C}^n$, $x^z = x_1^{z_1}...x_n^{z_n}$ and $\frac{1}{x} = \frac{1}{x_1...x_n}$. This transform generalizes the classical Mellin transform. Is there some literature where I can read about it? Is there an inversion theorem for this case?

I found the following paper dealing with multidimensional Mellin inversion (it is not open access, though): https://iopscience.iop.org/article/10.1070/RM2007v062n05ABEH004459/pdf. It gives an inversion theorem for suitable classes of functions.