Some strong (p,q)(p,q)-type inequalities for the homotopy operator

We prove the boundedness of the homotopy operator in the weighted LpLp spaces. As extensions of (p,p)(p,p)-type norm inequalities for the homotopy operator, we also establish some strong (p,q)(p,q)-type norm inequalities with ArAr weights and power-type weights for the homotopy operator acting on differential forms. Finally, we obtain some estimates for singular integrals.