On CR Paneitz operators and CR pluriharmonic functions

Abstract

Let $(X,T^{1,0}X)$ be a compact orientable embeddable three dimensional strongly pseudoconvex CR manifold and let $\mathrm{P}$ be the associated CR Paneitz operator. In this paper, we show that (I) $\mathrm{P}$ is self-adjoint and $\mathrm{P}$ has $L^2$ closed range. Let $N$ and $\Pi$ be the associated partial inverse and the orthogonal projection onto $\mathrm{Ker}\mathrm{P}$ respectively, then $N$ and $\Pi$ enjoy some regularity properties. (II) Let $\hat{\mathcal{P}}$ and $\widehat{{\mathcal{P}}_0}$ be the space of $L^2$ CR pluriharmonic functions and the space of real part of $L^2$ global CR functions respectively. Let $S$ be the associated Szegö projection and let $\tau$, ${\tau }_0$ be the orthogonal projections onto $\hat{\mathcal{P}}$ and $\widehat{{\mathcal{P}}_0}$ respectively. Then, $Π=S+\ol S+F_0$, $τ=S+\ol S+F_1$, $τ_0=S+\ol S+F_2$, where $F_0,F_1,F_2$ are smoothing operators on $X$. In particular, $\Pi$, $\tau$ and ${\tau }_0$ are Fourier integral operators with complex phases and ${\hat{\mathcal{P}}}^{\perp }\bigcap \mathrm{Ker}\mathrm{P}$, ${\widehat{{\mathcal{P}}_0}}^{\perp }\bigcap \hat{\mathcal{P}}$, ${\widehat{{\mathcal{P}}_0}}^{\perp }\bigcap \mathrm{Ker}\mathrm{P}$ are all finite dimensional subspaces of $C^{\infty }(X)$ (it is well-known that $\widehat{{\mathcal{P}}_0}\subset \hat{\mathcal{P}}\subset \mathrm{Ker}\mathrm{P}$). (III) $\mathrm{Spec}\mathrm{P}$ is a discrete subset of $\Real$ and for every $\lambda \in \mathrm{Spec}\mathrm{P}$, $\lambda \ne 0$, $\lambda$ is an eigenvalue of $\mathrm{P}$ and the associated eigenspace $H_{\lambda }(\mathrm{P})$ is a finite dimensional subspace of $C^{\infty }(X)$.

23 pages