Global estimates for compositions of operators applied to differential forms (Q1770706)

The authors present some global estimates, namely. Let \(\nabla\) be the gradient operator, \(T\) be the homotopy operator, and \(G\) Green's operator. The main result is: there exists a global constant \(C\), independent of \(\Omega\), \(u\) such that: for every bounded open domain \(\Omega\subset\mathbb{R}^n\) \[ \|\nabla\circ T\circ G(u)\|_\Omega\leq C|\Omega|\,\| u\|_\Omega \] for every \(A\) harmonic tensor \(\mu\). Analogous estimates are obtained for the composition of operators \(T\circ d\circ G\), where \(d\) is the differential operator. See also the papers of \textit{C. A. Nolder} [Ill. J. Math. 43, No. 4, 613--632 (1999; Zbl 0957.35046)] and \textit{C. Scott} [Trans. Am. Math. Soc. 347, No. 6, 2075--2096 (1995; Zbl 0849.58002).