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About representation as convolution of the operator, permutable with the operator quasiregular representations of group of Lorentz. (Russian) Zbl 1089.22500

The article is devoted to a proof of the following auxiliary statement, which is analogous to those given in [V. P.Bur’skij, Ukr.Math.J.51, No. 2, 172–184 (1999); translation from Ukr.Mat.Zh.51, No. 2, 158–169 (1999; Zbl 0936.35049)] and [V. P.Burskyj and T. V.Shtepina, Ukr.Math.J.52, No. 11, 1679–1690 (2000); translation from Ukr.Mat.Zh.52, No. 11, 1473–1483 (2000; Zbl 0970.35083)]. The space of all bounded linear operators acting on the intersection of augmentations of spaces of infinitely differentiable functions on the \((n-1)\)-dimensional Lobachevskij space \(S_H\) with integrable modulus and square integrable modulus, which commute with all operators \(T(g)\), \(g \in G\), of a quasiregular representation of the Lie group \(G = \text{SO}_0(n-1,1)\), consists of convolution operators, i.e., for any such operator \(A\) and any admissible function \(\Phi\), there exists the function \(\Psi_A \in L_2(G)\) such that the following identity holds: \[ [\tilde{A} \tilde{\Phi}](g)=\int_{g_1 \in G}^{}\tilde{\Phi}(g_1)\Psi_A(g_{1}^{-1}g)dg_{1}. \] Here \(dg_1\) is the Haar measure on \(G\) and the objects with \(\sim\) are lifts of the objects on SO\(_0(n-1,1)\) using the exact sequence \[ 1\rightarrow \text{SO}(n-1)\rightarrow \text{SO}_0(n-1,1)\rightarrow S_H \rightarrow1. \]

MSC:

22E43 Structure and representation of the Lorentz group
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