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

and Applied Analysis 3 Thus, observing (15) and (16), we immediately obtain that ‖T(G(u))‖ p,B,w ≤ C 6 |B| (1−β)/βp+(r/p) ‖u‖ p,B,w ≤ C 6 |D| (1−β)/βp+(r/p) ‖u‖ p,B,w ≤ C 7 ‖u‖ p,B,w . (18) HereC 7 is a constant independent of u.Thus we complete the proof of Theorem 5. Furthermore, if u is an A-harmonic tensor on D, ρ > 1 and 0 < s, t < ∞, then there exists a constantC, independent of u, such that ‖u‖ s,B ≤ C|B| (t−s)/ts ‖u‖ t,ρB (19) for all balls or cubs B with ρB ⊂ D (for more details aboutAharmonic tensors, see [10]). By the property of A-harmonic tensor, using the same method developed in the proof of Theorem 5, we can easily extend into the following A r (D)weighted version. Corollary 6. Let D ⊂ Rn be a bounded convex domain, n < p < ∞, u be an A-harmonic tensor, and T : Lp(D, ∧k) → L p (D, ∧ k−1 ) be the Homotopy operator, k = 1, 2, . . . , n. Then there exists a constant C, independent of u, such that ‖T (G (u))‖ p,B,w α ≤ C‖u‖ p,ρB,w α (20) for any ball B ⊂ D, w(x) ∈ A r (D), and 1 < r < p/n, 0 < α ≤ 1, ρ > 1. In order to obtain the boundedness of the composition T ∘ G, we need the following modified Whitney cover in [10] and see [11] for more details about Whitney cover. Lemma 7. Each open subset E ⊂ Rn has a modified Whitney cover of cubs W = {Q i } satisfying ⋃ i Q i = E and ∑ Q i ∈W χ √5/4Q i ≤ N ⋅ χ E (x), for all x ∈ Rn and some N > 1, where χ E (x) is the characteristic function for the set E. Theorem 8. Let D ⊂ Rn be a bounded convex domain, n < p < ∞. Then the composite operator T ∘ G : Lp(D, ∧k, w) → L p (D, ∧ k−1 , w) is bounded, k = 1, 2, . . . , n. Herew(x) ∈ A r (D) and 1 < r < p/n. Proof. From Lemma 7, we know that there exists a sequence of cubsW = {Q i } such that ⋃ i Q i = D and ∑ Q i ∈W χ √5/4Q i ≤ N⋅χ E (x) for all x ∈ D, whereN > 1 is some constant. Hence, for u ∈ Lp(D, ∧k, w), we have


Introduction
Our purpose is to study the   theory of the composition of the Homotopy  and Green's operator  acting on differential forms on a bounded convex domain.Both operators play an important role in many fields, including harmonic analysis, potential theory, and partial equations (see [1][2][3][4][5][6]).In the present paper, we will obtain some (, )-type norm inequalities for the composition of the Homotopy  and Green's operator  and also prove the   ()-weighted integral inequality on a bounded convex domain.These results will provide effective tools for studying behavior of solutions of -harmonic equations and related differential systems on manifolds.
We start this paper by introducing some notations and definitions.Let  be a Riemannian, compact, oriented, and  ∞ -smooth manifold without boundary on   and let Ω be an open subset of   .Also, we use  to denote Green's operator throughout this paper.Furthermore, we use  to denote a ball and  to denote the ball with the same center as  and with diameter () =  diameter ().We do not distinguish balls from cubs in this paper.

Boundedness of the Composition of the Homotopy and Green's Operator in 𝐿 𝑝 Space
In this section, we will prove the   ()-weighted norm inequality for the composition of the Homotopy  and Green's operator  on a bounded convex domain.Then using the Whitney cover, we develop the local result to the global domain.In [8], Gol' dshtein and Troyanov proved the following lemma.
Lemma 1.Let  ⊂   be a bounded convex domain.The operator  maps   (, ∧  ) continuously to   (, ∧ −1 ) in the following cases: From [3], we have the following lemma about  estimates for Green's operator.
For   () weight, we also need the following result which appears in [9].Lemma 4. If () ∈   (), then there exist constants  > 1 and , independent of , such that for all balls  ⊂ .
Furthermore, if  is an -harmonic tensor on ,  > 1 and 0 < ,  < ∞, then there exists a constant , independent of , such that for all balls or cubs  with  ⊂  (for more details about harmonic tensors, see [10]).By the property of -harmonic tensor, using the same method developed in the proof of Theorem 5, we can easily extend into the following   ()weighted version.
In order to obtain the boundedness of the composition  ∘ , we need the following modified Whitney cover in [10] and see [11] for more details about Whitney cover.Proof.From Lemma 7, we know that there exists a sequence of cubs  = {  } such that ⋃    =  and ∑   ∈  √5/4  ≤  ⋅   () for all  ∈ , where  > 1 is some constant.Hence, for  ∈   (, ∧  , ), we have where  = () and  2 =  1  is independent of  and each   .Thus, we complete the proof of Theorem 8.

Norm Estimates with Power-Type Weights
Let  ⊂   be a bounded domain and  be a nonempty of  =  ⋃ .If we use dist(, ) to denote the distance of the point  from the set , then () = (dist(, ))  for  ∈  is called power-type weight.In this section, we will establish some strong (, )-type norm inequalities with power-type weights for the composition of the Homotopy  and Green's operator  acting on differential form.In the following proof, we will use the following Lemma which appears in [8].
Next, we consider the following norm comparison equipped with power-type weights.
Note that, in the proof of Theorem 11, if we let the composite operator  ∘  act on the solution of nonhomogeneous harmonic equation, then we can drop lim  → 0 ℎ() = 0. Next, we state the result as follows.
It is easy to find that the above corollary does not hold for balls  ⊂  with  ⋂  ̸ = Φ but holds for those balls with  ⊂ .Next, we introduce the following singular integral inequality.