Résolution de \({\bar \partial}\) sur des portions de sphères ou d'anneaux. (Resolution of \({\bar \partial}\) on portions of spheres or rings) (Q755942)

This paper pursues an approach initiated by \textit{J.-P. Rosay} [Ann. Sc. Norm. Super. Pisa, Cl. Sci., IV. Ser. 13, 225-243 (1986; Zbl 0633.32007)] for using integral kernels to study local solvability of the \({\bar \partial}_ b\)-equation. Consider in \({\mathbb{C}}^ n\) the portion of the sphere \(\sum^{n}_{j=1}| z_ j|^ 2=1\) cut out by the inequality \(\sum^{k}_{j=1}| z_ j|^ 2<1/2\). (The case \(k=1\) was considered by Rosay.) Let g be a smooth \({\bar \partial}_ b\)-closed (0,q)-form on this portion of the sphere, where \(1\leq q\leq n-3.\) The question is whether the equation \({\bar \partial}_ bu=g\) admits a smooth solution u. The answer is affirmative as long as q is not one of the three values \(n-k-1,\) \(n-k,\) and \(n-k+1.\) When \(q=n-k-1,\) the equation is in general not solvable; when \(q=n-k+1,\) it is solvable on compact subsets; when \(q=n-k\) the question is left open. A completely analogous result holds for solvability of the \({\bar \partial}\)-equation on the part of the annulus \(1<\sum^{n}_{j=1}| z_ j|^ 2<4\) cut out by the inequality \(\sum^{k}_{j=1}| z_ j|^ 2<1/2\).