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?
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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.
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$\begingroup$ Also [Izvestiya Mathematics vol. 198 iss. 4] Antipova, Irina A - Inversion of many-dimensional Mellin transforms and solutions of algebraic equations (2007) [10.1070_SM2007v198n04ABEH003844] $\endgroup$ Commented Mar 24, 2022 at 0:23
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$\begingroup$ Also "Mellin Transforms and Algebraic Functions" by Antipovaa and Zykova (researchgate.net/publication/…). $\endgroup$ Commented Nov 17, 2022 at 17:07