Abstract
In the present paper, we deal with the composition of carathéodory and homotopy operators for differential forms satisfying the A-harmonic equation in the bounded and convex domain. We obtain estimates for the composition and the form of inequalities with weights. Moreover, we also obtain the composition for the gradient, carathéodory, and homotopy operators. Then we obtain the W 1 , p norm estimates for the composition operators.
Highlights
1 Introduction The purpose of this paper is to establish the inequalities for the composition of the homotopy operator T and the carathéodory operator G applied to differential forms in Rn, n ≥
The homotopy operator T is widely used in the decomposition and the Lp-theory of differential forms
In [ ], we have extended the homotopy operator to the domain that is deformed to every point
Summary
The purpose of this paper is to establish the inequalities for the composition of the homotopy operator T and the carathéodory operator G applied to differential forms in Rn, n ≥. The homotopy operator T is widely used in the decomposition and the Lp-theory of differential forms. In [ ], we have extended the homotopy operator to the domain that is deformed to every point. We need to estimate the various norms of the operators and their compositions. Throughout this paper, we always assume that is a bounded and convex domain and B is a ball in Rn, n ≥. We obtain the W ,p norm estimates for the composition operator. The main theorems are proved by reference to Chap. of [ ]
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