Cauchy-Riemann orbifolds

Abstract

For any $CR$ orbifold $B$, of $CR$ dimension $n$, we build a vector bundle (in the sense of J. Girbau & M. Nicolau, [13]) $T_{1,0}(B)$ over $B$, so that $T_{1,0}(B)_{p} \approx \bm{C}^{n}/G_{x}$ at any singular point $p=\varphi(x)\in B$ (and the portion of $T_{1,0}(B)$ over the regular part of $B$ is an ordinary $CR$ stmcture), hence study the tangential Cauchy--Riemann equations on orbifolds. As an application, we build a two-sided parametrix for for the Kohn--Rossi laplacian $\Box_{\Omega}$ (on the domain $\Omega$ of a local uniformizing system $\{\Omega, G, \varphi\}$ of $B$) inverting $\Box_{\Omega}$ over the $G$-invariant $(0, q)$-forms $(1 \leq q \leq n-1)$ up to (smoothing) operators of type $1$ (in the sense of G. B. Folland & E. M. Stein, [12]).