Fundamental solutions and optimal estimates for the tangential Cauchy-Riemann operator (Q956721)

The authors consider a generic \(q\)-concave CR variety \(M\) in \(\mathbb C^n\) of real codimension \(k\) and construct an integral kernel \(R_M\) with analogous properties as the Bochner-Martinelli kernel in \(\mathbb C^n\). In order to obtain optimal estimates for the solution of the tangential Cauchy-Riemann equation \(\overline \partial_b u=f \) and corresponding regularity results. As an important result of the regularity properties of the tangential Cauchy-Riemann operator, they prove the following corollary: let \(M\) be a generic \(1\)-concave CR subvariety of class \({\mathcal C}^{l+3}\) of a complex analytic variety and let \(T\) be a distribution of order \(l\) on \(M\). If \(\overline \partial_b T\) is a \((0,1)\)-form of class \({\mathcal C}^l\) on \(M\), then \(T\) is in fact a function of class \({\mathcal C}^{l+1/2}\) on \(M\).