For ν∈R\documentclass[12pt]{minimal}
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\begin{document}$$\nu \in \mathbb {R}$$\end{document}, we consider the invariant Laplacians Δν\documentclass[12pt]{minimal}
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\begin{document}$$\Delta _{\nu }$$\end{document} in the unit complex ball Bn=(SU(n,1)/S(U(n)×U(1))\documentclass[12pt]{minimal}
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\begin{document}$${\mathcal {B}}^{n}=(SU(n,1)/S(U(n)\times U(1))$$\end{document}Δν=4(1-|z|2){∑i,j=1n(δij-zizj¯)∂2∂zi∂zj¯-ν2∑j=1nzj∂∂zj+ν2∑j=1nzj¯∂∂zj¯+ν24}\documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned} \Delta _{\nu }= & {} 4(1-|z|^{2})\Bigg \{\sum _{i,j=1}^{n}(\delta _{ij}-z_{i}\bar{z_{j}})\dfrac{\partial ^{2}}{\partial z_{i}\partial \bar{z_{j}}}-\frac{\nu }{2}\sum _{j=1}^{n}z_{j}\dfrac{\partial }{\partial z_{j}}+\frac{\nu }{2}\sum _{j=1}^{n}\bar{z_{j}}\dfrac{\partial }{\partial \bar{z_{j}}}+\frac{\nu ^2}{4}\Bigg \} \end{aligned}$$\end{document}and the spectral projectors Qλ,ν\documentclass[12pt]{minimal}
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\begin{document}$${\mathcal {Q}}_{\lambda ,\nu }$$\end{document} associated to Δν\documentclass[12pt]{minimal}
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\begin{document}$$\Delta _{\nu }$$\end{document} defined by Qλ,νf=|cν(λ)|-2f∗φλ,ν(z),\documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned} {\mathcal {Q}}_{\lambda ,\nu }f= & {} |{\textbf{c}}_{\nu }(\lambda )|^{-2}f*\varphi _{\lambda ,\nu }(z), \end{aligned}$$\end{document}where φλ,ν\documentclass[12pt]{minimal}
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\begin{document}$$\varphi _{\lambda ,\nu }$$\end{document} is the S(U(n)×U(1))\documentclass[12pt]{minimal}
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\begin{document}$$S(U(n)\times U(1))$$\end{document}-invariant eigenfunction of Δν\documentclass[12pt]{minimal}
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\begin{document}$$\Delta _{\nu }$$\end{document} and cν(λ)\documentclass[12pt]{minimal}
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\begin{document}$${\textbf{c}}_{\nu }(\lambda )$$\end{document} the Harish-Chandra function. The goal of this paper is to give an image characterization of Qλ,ν\documentclass[12pt]{minimal}
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\begin{document}$${\mathcal {Q}}_{\lambda ,\nu }$$\end{document} of Cc∞(Bn)\documentclass[12pt]{minimal}
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\begin{document}$${\mathcal {C}}_{c}^{\infty }({\mathcal {B}}^{n})$$\end{document} and L2(Bn,(1-|z|2)-n-1dm(z))\documentclass[12pt]{minimal}
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\begin{document}$$L^{2}({\mathcal {B}}^{n},(1-|z|^2)^{-n-1}dm(z))$$\end{document}.
