Some Weighted Norm Estimates for the Composition of the Homotopy and Green’s Operator

Abstract

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 $du$.