2017/04/13 by Séverine Biard, Biard, Séverine, Emil J. Sträube +1 · 1 citation
Mathematics · #32V20 #32W10 #Advanced Harmonic Analysis Research #Complex Variables (math.CV) #FOS: Mathematics #Geometric Analysis and Curvature Flows #Holomorphic and Operator Theory
paper · pdf · doi:10.48550/arxiv.1704.04212
openalex publication_date 2017/04/13 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Let M be a pseudoconvex, oriented, bounded and closed CR submanifold of\n\ℂn of hypersurface type. We show that Sobolev estimates for the\ncomplex Green operator hold simultaneously for forms of symmetric bidegrees,\nthat is, they hold for (p,q)--forms if and only if they hold for\n(m-p,m-1-q)--forms. Here m equals the CR dimension of M plus one.\nSymmetries of this type are known to hold for compactness estimates. We further\nshow that with the usual microlocalization, compactness estimates for the\npositive part percolate up the complex, i.e. if they hold for (p,q)--forms,\nthey also hold for (p,q+1)--forms. Similarly, compactness estimates for the\nnegative part percolate down the complex. As a result, if the complex Green\noperator is compact on (p,q1)--forms and on (p,q2)--forms (q1\≤\nq2), then it is compact on (p,q)--forms for q1\≤ q\≤ q2. It\nis interesting to contrast this behavior of the complex Green operator with\nthat of the \\∂--Neumann operator on a pseudoconvex domain.\n