Spectral Projections and Paley–Wiener Theorem for the Unit Ball in $$\mathbb {C}^{n}$$

Abstract

For \(\nu \in \mathbb {R}\), we consider the invariant Laplacians \(\Delta _{\nu }\) in the unit complex ball \({\mathcal {B}}^{n}=(SU(n,1)/S(U(n)\times U(1))\)

$$\begin{aligned} \Delta _{\nu }= & {} 4(1-|z|^{2})\Bigg \{\sum _{i,j=1}^{n}(\delta _{ij}-z_{i}\bar{z_{j}})\dfrac{\partial ^{2}}{\partial z_{i}\partial \bar{z_{j}}}-\frac{\nu }{2}\sum _{j=1}^{n}z_{j}\dfrac{\partial }{\partial z_{j}}+\frac{\nu }{2}\sum _{j=1}^{n}\bar{z_{j}}\dfrac{\partial }{\partial \bar{z_{j}}}+\frac{\nu ^2}{4}\Bigg \} \end{aligned}$$

and the spectral projectors \({\mathcal {Q}}_{\lambda ,\nu }\) associated to \(\Delta _{\nu }\) defined by

$$\begin{aligned} {\mathcal {Q}}_{\lambda ,\nu }f= & {} |{\textbf{c}}_{\nu }(\lambda )|^{-2}f*\varphi _{\lambda ,\nu }(z), \end{aligned}$$

where \(\varphi _{\lambda ,\nu }\) is the \(S(U(n)\times U(1))\)-invariant eigenfunction of \(\Delta _{\nu }\) and \({\textbf{c}}_{\nu }(\lambda )\) the Harish-Chandra function. The goal of this paper is to give an image characterization of \({\mathcal {Q}}_{\lambda ,\nu }\) of \({\mathcal {C}}_{c}^{\infty }({\mathcal {B}}^{n})\) and \(L^{2}({\mathcal {B}}^{n},(1-|z|^2)^{-n-1}dm(z))\).