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We establish the Ar(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 Lp norms of T(G(u)) or (T(G(u)))B in terms of Lq norms of u or du.

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