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 $$\hat{\mathcal {P}}_{0}$$ 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 Szego projection and let $$\tau $$ , $$\tau _0$$ be the orthogonal projections onto $$\hat{\mathcal {P}}$$ and $$\hat{\mathcal {P}}_{0}$$ respectively. Then, $$\Pi =S+\overline{S}+F_0$$ , $$\tau =S+\overline{S}+F_1$$ , $$\tau _0=S+\overline{S}+F_2$$ , 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 $$\hat{\mathcal {P}}^\perp \bigcap {\hbox { Ker }}\mathrm{P}$$ , $$\hat{\mathcal {P}}_{0}^\perp \bigcap \hat{\mathcal {P}}$$ , $$\hat{\mathcal {P}}_{0}^\perp \bigcap {\hbox { Ker }}\mathrm{P}$$ are all finite dimensional subspaces of $$C^\infty (X)$$ (it is well-known that $$\hat{\mathcal {P}}_{0}\subset \hat{\mathcal {P}}\subset {\hbox { Ker }}\mathrm{P}$$ ). (III) $$\hbox {Spec }\mathrm{P}$$ is a discrete subset of $${\mathbb {R}}$$ and for every $$\lambda \in \hbox {Spec }\mathrm{P}$$ , $$\lambda \ne 0$$ , $$\lambda $$ is an eigenvalue of $$\mathrm{P}$$ and the associated eigenspace $$H_\lambda (\mathrm{P})$$ is a finite dimensional subspace of $$C^\infty (X)$$ .