Higher integrability of iterated operators on differential forms (Q326783)

In this paper, the authors first prove the local higher integrability and higher order imbedding theorems for the iterated operators \(D^kG^k(u)\) and \(D^{k+1}G^k(u)\), where \(k\) is a positive integer, \(G\) is the Green's operator and \(D=d+d^*\) is the Hodge-Dirac operator applied to differential forms. It is proved that both composite operators \(D^kG^k(u)\) and \(D^{k+1}G^k(u)\) are of higher integrability than that of \(u\). Then, the authors prove the global higher integrability and higher order imbedding inequalities for these operators. The upper bound estimates for the \(L^s\)-norms of \(D^kG^k(u)\) and \(D^{k+1}G^k(u)\) in terms of the \(L^p\)-norms of \(u\) are obtained, where the positive integral exponent \(s\) could be much larger than the positive integral exponent \(p\). In the last Section, some applications of these results are described on concrete examples.