In this paper, we prove some results on bi-Hölder extensions not only for biholomorphisms but also for more general Kobayashi metric quasi-isometries between the domains. Furthermore, we establish the Gehring–Hayman type theorems on certain complex domains which play an important role through the paper. Then by applying the above results, we show the bi-Hölder equivalence between the Euclidean boundary and the Gromov boundary of bounded convex domains which are Gromov hyperbolic with respect to their Kobayashi metrics.

Let $X$ be an abstract not necessarily compact orientable CR manifold of dimension $2n-1$, $n\geqslant2$, and let $L^k$ be the $k$-th tensor power of a CR complex line bundle $L$ over $X$. Given $q\in\set{0,1,\ldots,n-1}$, let $\Box^{(q)}_{b,k}$ be the Gaffney extension of Kohn Laplacian for $(0,q)$ forms with values in $L^k$. For $\lambda\geq0$, let $\Pi^{(q)}_{k,\leq\lambda}:=E((-\infty,\lambda])$, where $E$ denotes the spectral measure of $\Box^{(q)}_{b,k}$. In this work, we prove that $\Pi^{(q)}_{k,\leq k^{-N_0}}F^*_k$, $F_k\Pi^{(q)}_{k,\leq k^{-N_0}}F^*_k$, $N_0\geq1$, admit asymptotic expansions with respect to $k$ on the non-degenerate part of the characteristic manifold of $\Box^{(q)}_{b,k}$, where $F_k$ is some kind of microlocal cut-off function. Moreover, we show that $F_k\Pi^{(q)}_{k,\leq0}F^*_k$ admits a full asymptotic expansion with respect to $k$ if $\Box^{(q)}_{b,k}$ has small spectral gap property with respect to $F_k$ and $\Pi^{(q)}_{k,\leq0}$ is $k$-negligible away the diagonal with respect to $F_k$. By using these asymptotics, we establish almost Kodaira embedding theorems on CR manifolds and Kodaira embedding theorems on CR manifolds with transversal CR $S^1$ action.

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Let $(X,T^{1,0}X)$ be a compact orientable embeddable three dimensional strongly pseudoconvex CR manifold and let ${\rm P\,}$ be the associated CR Paneitz operator. In this paper, we show that (I) ${\rm P\,}$ is self-adjoint and ${\rm P\,}$ has $L^2$ closed range. Let $N$ and $\Pi$ be the associated partial inverse and the orthogonal projection onto ${\rm Ker\,}{\rm P\,}$ 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 Szeg\"o projection and let $\tau$, $\tau_0$ be the orthogonal projections onto $\hat{\mathcal{P}}$ and $\hat{\mathcal{P}_0}$ respectively. Then, $\Pi=S+\ol S+F_0$, $\tau=S+\ol S+F_1$, $\tau_0=S+\ol S+F_2$, 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 $\hat{\mathcal{P}}^\perp\bigcap{\rm Ker\,}{\rm P\,}$, $\hat{\mathcal{P}_0}^\perp\bigcap\hat{\mathcal{P}}$, $\hat{\mathcal{P}_0}^\perp\bigcap{\rm Ker\,}{\rm P\,}$ are all finite dimensional subspaces of $C^\infty(X)$ (it is well-known that $\hat{\mathcal{P}_0}\subset\hat{\mathcal{P}}\subset{\rm Ker\,}{\rm P\,}$). (III) ${\rm Spec\,}{\rm P\,}$ is a discrete subset of $\Real$ and for every $\lambda\in{\rm Spec\,}{\rm P\,}$, $\lambda\neq0$, $\lambda$ is an eigenvalue of ${\rm P\,}$ and the associated eigenspace $H_\lambda({\rm P\,})$ is a finite dimensional subspace of $C^\infty(X)$.

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