Complex-weighted Plancherel formula and Jacobi transform (Q1919011)

Let \(D\) be the unit disk in the complex plane \(\mathbb{C}\). Let \({\mathcal D} (D)\) be the space of \(C^\infty\)-functions on \(D\) having compact supports and \({\mathcal D}^\# (D)\) the space of radial functions in \({\mathcal D} (D)\). For \(\nu \in \mathbb{C}\), the representation \(T^\nu\) of the Möbius group \(SU (1,1)\) on \({\mathcal D} (D)\) is defined by \[ \begin{aligned} T^\nu (g) f(z) & = f \left( {\alpha z + \beta \over \overline \beta z + \overline \alpha} \right) (\overline \beta z + \overline \alpha)^{- \nu}, \\ g & = \left( \begin{matrix} \alpha & \beta \\ \overline \beta & \overline \alpha \end{matrix} \right) \in SU(1,1). \end{aligned} \] Let \(L_\nu\) be the invariant Laplacian induced from the representation \(T^\nu\) given by \[ L_\nu = - 4 \bigl( 1 - |z |^2 \bigr)^2 {\partial^2 \over \partial \overline z \partial z} + 4 \nu \bigl( 1 - |z |^2 \bigr) \overline z {\partial \over \partial \overline z} - \nu^2 + 2 \nu \] and \(\varphi^\nu_\lambda (z)\) the radial eigenfunction of \(L_\nu\) satisfying \(\varphi^\nu_\lambda (0) = 1\) and \(L_\nu \varphi^\nu_\lambda (z) = (1 + \lambda^2) \varphi^\nu_\lambda (z)\). For \(f \in {\mathcal D}^\# (D)\), we define the spherical transform \(\widehat f (\lambda)\) by \[ \widehat f (\lambda) = \int_D f(z) \varphi^\nu_\lambda (z) d \mu_\nu (z), \quad \lambda \in \mathbb{C}, \] where \[ d \mu_\nu (z) = \bigl( 1 - |z |^2 \bigr)^{\nu - 2} dxdy, \quad z = x + iy. \] The main results of this paper are analogs of the Paley-Wiener theorem, the inversion formula, and the Plancherel formula for the spherical transform \(\widehat f (\lambda)\). The inversion formula and the Plancherel formula for the case \(\nu = 0\) and the case \(\nu \in \mathbb{R}\) have been established by \textit{S. Helgason} [Topics in harmonic analysis on homogeneous spaces, Progr. Math., vol. 13 (Boston 1981; Zbl 0467.43001)] and \textit{H. Liu} and \textit{L. Peng} [Math. Scand. 72, 99-119 (1993; Zbl 0785.22018)] respectively. In this paper the authors generalize these results to \(\nu \in \mathbb{C}\) by using the Jacobi transform.