Lipschitz and BMO norm inequalities for the composite operator on differential forms

In this paper, we obtain Poincaré-type inequalities for the composite operator acting on differential forms and establish the Lp$L^{p}$, Lipschitz, and BMO norm estimates. We also give the weighted versions of the comparison theorems for the Lp$L^{p}$, Lipschitz, and BMO norms.


Introduction
Differential forms are a generalization of the traditional functions. In recent years, differential forms have been widely used in physics systems, differential geometry, and PDEs. In this paper, we are interested in the properties of the composite operator acting on differential forms. Operator theory plays a critical role in investigating the properties of the solutions to partial differential equations. Many questions in partial differential equations involve estimating various norms of operators. The operator theory for functions has been very well developed in recent years. However, compared to the function cases, the operator theory for differential forms is more complicated, so we need some advanced methods to deal with operators. This paper contributes to derive the properties of the composite operator M s • D • G on differential forms, where M s is the general sharp maximal operator defined by Here d is the exterior differential operator on differential forms, and d * is the formal adjoint operator of d. See [] for more details. The operator G is the well-known Green's operator satisfying the equation where H is the harmonic projection operator. See [-] for more results and applications for the sharp maximal operator, the Dirac operator, and Green's operator.
In the following, M stands for a bounded convex domain in R n , n ≥ . The Lebesgue measure of a measurable set E ⊆ R n is denoted by |E|. We use B and σ B to denote concentric balls such that diam(σ B) = σ diam(B). By l = l (R n ) we denote the linear space of all l-vectors spanned by the exterior products e I = e i  ∧ e i  ∧ · · · ∧ e i l for all ordered l-tuples I = (i  , i  , . . . , i l ),  ≤ i  < i  < · · · < i l ≤ n. The l-form u(x) = I u I (x) dx I is a linear combination of the standard basis dx I = dx i  ∧ · · · ∧ dx i l for all ordered l-tuples I. If the coefficient u I is differential, we say that u is a differential l-form. By D (M, l ) we denote the space of all differential l-forms. Similarly, we write L s (M, l ) for the l-form u(x) on M with u I satisfying M |u I | s < ∞.
A differential l-form u ∈ D (M, l ) is called a closed form if du =  in M. From the Poincaré lemma ddu =  we know that du is a closed form. The module of a differential form u is given by |u|  = * (u ∧ * u) ∈ D (M,  ).
A very important operator, the homotopy operator T : is normalized by M ϕ(y) dy = , and K y is the liner operator defined by For the homotopy operator T, we have the following decomposition, which will be used repeatedly in this paper: for any differential form u. A closed form u M is defined by u M = d(Tu); in particular, when u is a -form, u M = |M| - M u(y) dy. In regard to Green's operator, we need the following results in []: , and dG(u) s,B = Gd(u) s,B for any differential form u in M and  < s < ∞.

Poincaré-type inequality
In this section, we give a Poincaré-type inequality for the composite operator M s • D • G, which will be used in the estimates for the L p , Lipschitz, and BMO norms. We will need the following lemmas.
The following estimate for the homotopy operator T appears in [].
Lemma . Let u ∈ L t loc , l = , , . . . , n,  < t < ∞, be a differential form in M, and T be the homotopy operator defined on differential forms. Then there exists a constant C, independent of u, such that We will use the generalized Hölder inequality repeatedly.

Lemma . []
Let  < q < ∞,  < p < ∞, and s - = q - + p - . If f and g are measurable functions on R n , then The following lemma appears in [].
First, we establish the boundedness for the composite operator M s • D • G.
Proof For a ball B in M, using Lemma . for any B (x,r) ⊂ B and the decomposition theorem, we have Since  + /n -/s > , taking the supremum over r, we get Using the generalized Hölder inequality, we find Combining () and (), we obtain The proof of Lemma . is completed.
Replacing u by M s DG(u) and using Lemma ., we get The proof of Theorem . is completed.

Lipschitz and BMO norm inequalities
In this section, we compare the L p norm, Lipschitz norm, and BMO norm of the composite operator M s • D • G applied to differential forms. Especially, when we estimate the Lipschitz norm in terms of the BMO norm, we need the differential form to satisfy some versions of harmonic equations. We first introduce some definitions. We call an equation a nonhomogeneous A-harmonic equation if where the operators A : M × l (R n ) → l (R n ) and B : M × l (R n ) → l- (R n ) satisfy for all balls B with B ⊂ M. Using the Hölder inequality, we have where k is a constant with  ≤ k ≤ , and C is a constant independent of u.
Proof From () we have Using the reverse Hölder inequality and Lemma ., we get where σ > . So, we have Letting σ > σ , we have This ends the proof of Theorem ..
By Theorem . and Theorem . we can easily estimate the BMO norm of the composite operator M s • D • G.
Corollary . Let u ∈ L t (M, l ), l = , , . . . , n,  ≤ s < t < ∞, be a differential form in M. Then, where C is a constant independent of u.

The weighted norm inequalities
In this section, we consider the weighted situation. The weight function we select is A(α, β, γ , M)-weight, which contains the well-known A r (M)-weight. We will use the Radon measure to deal with the A (α, β, γ , M)-weight in the proof.

Definition . []
We say that a measurable function w(x) defined on a subset M ⊂ R n satisfies the A(α, β, γ , M)-condition for some positive constants α, β, γ if w(x) >  a.e. and Now, we give estimates for the weighted Lipschitz and BMO norms.
where C is a constant independent of u.
The following corollary can be obtained by combining Theorem . and Corollary ..
Corollary . Let u, μ, w(x), and p be as in Theorem .. Then, where C is a constant independent of u.

Applications
In this section, we apply our results to some differential forms.