We first establish the L p-norm inequalities for the composition of Green's operator and the potential operator. Then we develop the L ϕ-norm inequalities for the composition in the L ϕ-averaging domains. Finally, we display some examples for applications.

The purpose of this paper is to derive some Coifman type inequalities for the fractional convolution operator applied to differential forms. The Lipschitz norm and BMO norm estimates for this integral type operator acting on differential forms are also obtained.

In this paper, we establish BMO and Lipschitz norm inequalities for the composition of Green's operator and the potential operator. We also investigate the relationship among the Lipschitz norm, the BMO norm and the L p -norm. Finally, we display some examples for applications. MSC: Primary 35J60; secondary 31B05; 58A10; 46E35 Keywords: differential forms; Lipschitz norm; BMO norm; Green's operator; potential operator IntroductionDifferential forms are extensions of functions and can be used to describe various systems in partial differential equations (or PDEs), physics, theory of elasticity, quasiconformal analysis, etc. Differential forms have become invaluable tools for many fields of sciences and engineering; see [, ] for more details. Now we introduce some notations and definitions. Let be an open subset of R n (n ≥ )and O be a ball in R n . Let ρO denote the ball with the same center as O and (x) is differential on , then we call (x) a differential -form on and use D ( , ∧ ) to denote the space of all differential -forms on . C ∞ ( , ∧ ) denotes the space of smooth -forms on . We denote the exterior derivative by d, and thewhere is the Hodge star operator. where T is called a homotopy operator. For the homotopy operator, we know thatIn this paper, we focus on a class of differential forms satisfying the well-known nonhomogeneous A-harmonic equationwhere where the kernel K(x, y) is a non-negative measurable function defined for x = y, J (x) is defined on ⊂ R n and the summation is over all ordered -tuples J. For more results

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