Theorems on Changing the Order of Integration in Laplace Transform over Classical Domain of the Second Type

The problem of changing the order of integration in the Laplace transform over classical domains of the second type is investigated. These domains belong to the class of Cartan classical domains and consist of complex symmetric matrices satisfying certain positivity conditions. Matrix-valued original functions defined on the class of symmetric matrices and their Laplace transforms with respect to matrix arguments are considered. Sufficient conditions for interchanging the order of integration in the Laplace transform integral are established. The obtained theorems provide a rigorous justification for changing the order of integration in integral representations associated with matrix Laplace transforms and can be applied in the theory of holomorphic functions on classical domains as well as in the study of integral transforms involving symmetric matrices.

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