Permutation operators and harmonic analysis associated with partial differential operators (Q1119816)

We consider the partial differential operators: \[ D_ 1=\partial /\partial \theta;\quad D_ 2=\partial^ 2/\partial y^ 2+[(2\alpha +1)\coth y+th y]\partial /\partial y-(1/ch^ 2 y)(\partial^ 2/\partial \theta \quad^ 2)+(\alpha +1)^ 2 \] with \((y,\theta)\in]0,\infty [\times {\mathbb{R}}\) and \(\alpha\in {\mathbb{R}}\), \(\alpha\geq 0.\) We determine permutation operators of \(D_ 1\), \(D_ 2\) into \(\partial /\partial \theta\), \(\partial^ 2/\partial y^ 2\), and we establish their relation with the Gegenbauer integral transforms, the generalized Riemann-Liouville and the Weyl integral transforms associated with differential operators on bounded and unbounded intervals (also called \textit{J. L. Lions}' transmutation operators [see Math. Rep. 4, No.1, 1-282 (1988)]). Next we study a harmonic analysis and the almot-periodic functions associated with the operators \(D_ 1\), \(D_ 2\).