ESTIMATES FOR THE COMPLEX GREEN OPERATOR: SYMMETRY, PERCOLATION, AND INTERPOLATION Academic Article

abstract

• 2018 American Mathematical Society. Let M be a pseudoconvex, oriented, bounded, and closed CR-submanifold of C n of hypersurface type. We show that Sobolev estimates for the complex Green operator hold simultaneously for forms of symmetric bidegrees; that is, they hold for (p, q)forms if and only if they hold for (m p, m 1 q)forms. Here m equals the CR-dimension of M plus one. Symmetries of this type are known to hold for compactness estimates. We further show that with the usual microlocalization, compactness estimates for the positive part percolate up the complex; i.e., if they hold for (p, q)forms, they also hold for (p, q + 1)forms. Similarly, compactness estimates for the negative part percolate down the complex. As a result, if the complex Green operator is compact on (p, q1)forms and on (p, q2)forms (q1 q2), then it is compact on (p, q)forms for q1 q q2. It is interesting to contrast this behavior of the complex Green operator with that of the -Neumann operator on a pseudoconvex domain.

authors

• Straube, Emil

published proceedings

• TRANSACTIONS OF THE AMERICAN MATHEMATICAL SOCIETY

author list (cited authors)

• Biard, S., & Straube, E. J.

citation count

• 1

complete list of authors

• Biard, Severine||Straube, Emil J

publication date

• October 2019

publisher

• American Mathematical Society (AMS)  Publisher

published in

• Transactions of the American Mathematical Society  Journal

keywords

• (partial Derivative)over-bar(m)
• Compactness Estimates
• Complex Green Operator
• Cr-submanifolds Of Hypersurface Type
• Form Level Symmetry And Percolation
• Sobolev Estimates

Digital Object Identifier (DOI)

• 10.1090/tran/7385

start page

• 2003

end page

• 2020

volume

• 371

issue

• 3

URL

• http://dx.doi.org/10.1090/tran/7385