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

For \(\nu \in \mathbb {R}\) $\nu \in \mathbb{R}$ , we consider the invariant Laplacians \(\Delta _{\nu }\) ${\mathrm{\Delta }}_{\nu }$ in the unit complex ball    \({\mathcal {B}}^{n}=(SU(n,1)/S(U(n)\times U(1))\) ${\mathcal{B}}^n=(SU(n,1)/S(U(n)\times U(1))$ \(\begin{aligned} \Delta _{\nu }= & {} 4(1-|z|^{2})\Bigg \{\sum _{i,j=1}^{n}(\delta _{ij}-z_{i}\bar{z_{j}})\dfrac{\partial ^{2}}{\partial z_{i}\partial \bar{z_{j}}}-\frac{\nu }{2}\sum _{j=1}^{n}z_{j}\dfrac{\partial }{\partial z_{j}}+\frac{\nu }{2}\sum _{j=1}^{n}\bar{z_{j}}\dfrac{\partial }{\partial \bar{z_{j}}}+\frac{\nu ^2}{4}\Bigg \} \end{aligned}\) $$\begin{array}{cc}\hfill {\mathrm{\Delta }}_{\nu }=& {4(1-|z|}^2)\{\sum _{i,j=1}^n({\delta }_{\mathit{ij}}-z_i\overline{z_j}){\displaystyle \frac{{\partial }^2}{\partial z_i\partial \overline{z_j}}}-\frac{\nu }{2}\sum _{j=1}^n{z}_j{\displaystyle \frac{\partial }{\partial z_j}}+\frac{\nu }{2}\sum _{j=1}^n\overline{z_j}{\displaystyle \frac{\partial }{\partial \overline{z_j}}}+\frac{{\nu }^2}{4}\}\hfill \end{array}$$ and the spectral projectors \({\mathcal {Q}}_{\lambda ,\nu }\) ${\mathcal{Q}}_{\lambda ,\nu }$ associated to \(\Delta _{\nu }\) ${\mathrm{\Delta }}_{\nu }$ defined by \(\begin{aligned} {\mathcal {Q}}_{\lambda ,\nu }f= & {} |{\textbf{c}}_{\nu }(\lambda )|^{-2}f*\varphi _{\lambda ,\nu }(z), \end{aligned}\) $$\begin{array}{cc}\hfill {\mathcal{Q}}_{\lambda ,\nu }f=& |{\mathbf{c}}_{\nu }{(\lambda )|}^{-2}f\ast {\phi }_{\lambda ,\nu }(z),\hfill \end{array}$$ where \(\varphi _{\lambda ,\nu }\) ${\phi }_{\lambda ,\nu }$ is the \(S(U(n)\times U(1))\) $S(U(n)\times U(1))$ -invariant eigenfunction of \(\Delta _{\nu }\) ${\mathrm{\Delta }}_{\nu }$ and \({\textbf{c}}_{\nu }(\lambda )\) ${\mathbf{c}}_{\nu }(\lambda )$ the Harish-Chandra function. The goal of this paper is to give an image characterization of \({\mathcal {Q}}_{\lambda ,\nu }\) ${\mathcal{Q}}_{\lambda ,\nu }$ of \({\mathcal {C}}_{c}^{\infty }({\mathcal {B}}^{n})\) ${\mathcal{C}}_c^{\infty }({\mathcal{B}}^n)$ and \(L^{2}({\mathcal {B}}^{n},(1-|z|^2)^{-n-1}dm(z))\) $L^2({\mathcal{B}}^n{,(1-|z|}^2{)}^{-n-1}dm(z))$ .