Higher integrability of Green's operator and homotopy operator (Q321815)

The aim of this paper is to prove the higher integrability and higher order imbedding theorems of the composition of the homotopy operator \(T\) and Green's operator \(G\) applied to differential forms. The authors study analogues of the homotopy operator and Green's operator, respectively.NEWLINENEWLINEThe paper is organized as follows. The first section is an introduction to the subject. Section 2 concerns local higher integrability. The authors prove the local higher integrability of the composite operator \(ToP\) and higher order imbedding theorems for this composite operator. Section 2 concerns global higher integrability. In this section, the authors prove global higher integrability theorems and higher order imbedding theorems for operators \(T\), \(G\) and the composition \(ToG\). Section 3 is reserved for applications of the results obtained in this paper. Usually it is very hard to obtain the upper bounds or the higher integrability of the composite operator. The new results proved in this paper provide an efficient way to estimate the upper bounds for the norms of the composite operator and prove the higher integrability of the operator.