Some weighted norm estimates for the composition of the homotopy and Green's operator (Q1725331)

Summary: We establish the \(A_r(D)\)-weighted integral inequality for the composition of the Homotopy \(T\) and Green's operator \(G\) on a bounded convex domain and also motivated it to the global domain by the Whitney cover. At the same time, we also obtain some \((p, q)\)-type norm inequalities. Finally, as applications of above results, we obtain the upper bound for the \(L^p\) norms of \(T(G(u))\) or \((T(G(u)))_B\) in terms of \(L^q\) norms of \(u\) or \(d u\).