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Estimates for the composition of the carathéodory and homotopy operators
Journal of Inequalities and Applications volume 2012, Article number: 188 (2012)
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 norm estimates for the composition operators.
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 , . The homotopy operator T is widely used in the decomposition and the -theory of differential forms. And in [3], we have extended the homotopy operator to the domain that is deformed to every point. In the meanwhile, the carathéodory operator G form classic examples to discuss boundedness and continuity of nonlinear operators and play an important part in advanced functional analysis, and in [4] we have extended it to differential forms. In many situations, we need to estimate the various norms of the operators and their compositions.
Throughout this paper, we always assume that Ω is a bounded and convex domain and B is a ball in , . Let σB be the ball with the same center as B and with , . We do not distinguish the balls from cubes in this paper. For any subset , we use to denote the Lebesgue measure of E. In [2], we have the estimate for :
for all , where is bounded and convex. And for carathéodory operator, we obtain
With these estimates, we can obtain the estimates for the composition of them. Finally, we obtain the norm estimates for the composition operator.
The main theorems are proved by reference to Chap. 7 of [1].
2 Some preliminaries about differential forms
The majority of notations and preliminaries used throughout this paper can be found in [1]. For the sake of convenience, we list briefly them in this section.
Let denote the standard orthogonal basis of . Suppose that is the linear space of l-covectors, generated by the exterior products , corresponding to all ordered l-tuples , , . The Grassmann algebra is a graded algebra with respect to the exterior products. For and , the inner product in Λ is given by with summation over all l-tuples and all integrals . We define the Hodge star operator by
where is a k-form, is a permutation of and is the signature of the permutation. The norm of is given by the formula .
A differential l-form ω is a Schwartz distribution on Ω with values in . We use to denote the space of all differential l-forms, and to denote the l-forms
with all coefficients . Thus, , , is a Banach space with norm
The space is the subspace of with the condition
The Sobolev space of l-forms is . The norms are given by
We denote the exterior derivative by for , which means
Its formal adjoint operator is defined by
which is called the Hodge codifferential.
In [3], we define an operation for any and we construct a homotopy operator by averaging over all points :
where ψ in is normalized so that . We obtain the following decomposition for the operator T:
From [2], we know that for any differential form , , , we have
where is flatness of Ω (see [2]). See [5–13] for more details of differential forms and its applications.
Then we define the carathéodory conditions and carathéodory operator for differential forms (see [4]).
Definition 2.1 For a mapping , where Ω is an open set in , we say that f satisfies carathéodory conditions if
-
1.
For all most , is continuous with respect to ω, which means that f can be expanded as , where and is continuous about ω for all most ; and
-
2.
For any fixed , is measurable about s, which means that each coefficient function is measurable about s for any fixed .
Throughout this paper, we assume that satisfies the carathéodory condition (C-condition). Similarly, we can define the continuity of about .
Definition 2.2 Suppose that is a measurable set (), and . We define the carathéodory operator for differential forms by
For the carathéodory operator, we have the similar result for differential forms as for the functions (see [4]).
Theorem 2.1 The carathéodory operator G maps continuously and boundedlyinto, if and only if, there exists, , satisfying the following inequality:
Here, we suppose.
We define Muckenhoupt weights (see [1]).
Definition 2.3 A weight ω satisfies the -condition in a subset , where , and write when
where the supremum is over all balls .
The following class of two-weight or -weights appeared in [1] and [13].
Definition 2.4 A pair of weights satisfy the -condition in a set , write for some and with , if
for all balls .
In the present paper, we deal with the A-harmonic equations formulated by .
We also need the following weak reverse Hölder inequality (Lemma 3.1.1 of [1]).
Theorem 2.2 Let u be a solution of the nonhomogeneous A-harmonic equation in a domain Ω and, . Then there exists a constant C, independent of u, such that
for all balls B withfor some.
For -weights ω, we have the following reverse Hölder inequality (Lemma 1.4.7 of [1]).
Theorem 2.3 If, , then there exist constantsand C, independent of ω, such that
for all balls.
3 Main results and proofs
Theorem 3.1 Let, , , be a solution of the A-harmonic equation in domain Ω is bounded and convex andbe the homotopy operator. Assume thatandfor some. Then. Moreover, there exists a constant C, independent of u, such that
for all balls B withand any real number α with.
Proof We only need to prove the inequality holds. With (2.6) and (2.7), we have
Then just like the process of the proof for Theorem 7.3.14 in [1], we obtain the inequality.
We discuss the inequality with and separately. For , first we set . With Hölder inequality, we obtain
By (3.2), we obtain
Let , then . With (3.3) and (3.4) and using Theorem 2.2, we have
And using Hölder’s inequality again, we obtain
for all balls B with . With (3.5) and (3.6), we find that
As , we have
With (3.7) and (3.8), we have
for all balls B with . This is just (3.1) with . Then we prove the case of . First, with Theorem 2.3, we know
here and are all constants. Let , then we know and . With Hölder’s inequality (3.2), and (3.10), we obtain
Set . With Theorem 2.2, we have
And we use Hölder’s inequality again
With , we have
With (3.11)-(3.14), we have
for all balls B with . Thus, we complete the proof. □
Actually by the method developed in [1], for the two weight , we have the following inequality.
Theorem 3.2 Let, , , be a solution of the A-harmonic equation in domain Ω is bounded and convex andbe the homotopy operator. Assume thatandfor some, . Then, . Moreover, there exists a constant C, independent of u, such that
for all balls B withand any real number α with.
Proof Let . As , with Hölder inequality, we have
for all balls . Then, from (3.2), we obtain
Let , then we know . With (3.17) and (3.18) and Theorem 2.2, we have
Then by the generalized Hölder’s inequality, we have
for all balls B with , where we use . Then with (3.19) and (3.20), we obtain
Then, with , we have
With (3.21) and (3.22), we have
for all balls B with . □
For the compositions of the gradient operator ∇, the homotopy operator T, the carathéodory operator G, , we obtain the local Sobolev-Poincaré embedding theorem.
Theorem 3.3 Let, , , be a solution of the A-harmonic equation in bounded and convex domain Ω, be the homotopy operator, ∇ be the gradient operator and G be the carathéodory operator. Thenand. Moreover, there exists a constant C, independent of u, such that
and
Proof Actually, we only need to prove (3.16) and (3.17). From these two inequalities, the remaining part of the theorem follows. From (2.4), we obtain
for any . Let , we have
With the definition of norm, (3.2), and (3.27), we have
Thus, we obtain the inequality. □
Using the same method as in the proof of Theorem 3.1, we obtain the weighted inequality for .
Corollary 3.4 Let, , , be a solution of the A-harmonic equation in bounded and convex domain Ω, be the homotopy operator, ∇ be the gradient operator and G be the carathéodory operator. Assume thatandfor some. Then. Moreover, there exists a constant C, independent of u, such that
For , we also have the similar result.
Corollary 3.5 Let, , , be a solution of the A-harmonic equation in a bounded, convex domain Ω andbe the homotopy operator. Assume thatandfor some. Then. Moreover, there exists a constant C, independent of u, such that
for all balls B withand any real number α with.
Proof If (3.22) holds, then follows. Hence, we only need to prove (3.22). From (2.6) and (2.7), we have
Using the method in the proof of Theorem 3.1, we obtain the inequality. □
Actually for two weight , for some and , we have the similar inequalities, with the method developed in the proof of Theorem 3.2.
Corollary 3.6 Let, , , be a differential form satisfying A-harmonic equation in a bounded, convex domainandbe the homotopy operator defined in (2.2). Assumeandfor someand. Then there exists a constant C, independent of u, such that
for all balls B withand all real number α with.
The above inequality is an extension of the usual inequality of -weights. If we choose and in the two weighted inequalities, we obtain the weight case.
References
Agarwal PR, Ding S, Nolder CA: Inequalities for Differential Forms. Springer, New York; 2009.
Iwaniec T, Lutoborski A: Integral estimates for null Lagrangians. Arch. Ration. Mech. Anal. 1993, 125: 25–79. 10.1007/BF00411477
Tang Z, Zhu J, Huang J, Li J: An extension of the Poincaré lemma of differential forms. 2. In World Congress on Engineering and Technology. IEEE, New York; 2011:403–406.
Tang Z, Zhu J: Carathéodory operator of differential forms. J. Inequal. Appl. 2011., 2011: Article ID 88. doi:10.1186/1029–242X-2011–88
Zhu J, Li J: Some priori estimates about solutions to nonhomogeneous A-harmonic equations. J. Inequal. Appl. 2010., 2010: Article ID 520240
Ding S: Two-weight Caccioppoli inequalities for solutions of nonhomogeneous A-harmonic equations on Riemannian manifolds. Proc. Am. Math. Soc. 2004, 132: 2367–2375. 10.1090/S0002-9939-04-07347-2
Ding S: Local and global norm comparison theorems for solutions to the nonhomogeneous A-harmonic equation. J. Math. Anal. Appl. 2007, 335: 1274–1293. 10.1016/j.jmaa.2007.02.048
Ding S, Zhu J:Poincaré-type inequalities for the homotopy operator with -norms. Nonlinear Anal., Theory Methods Appl. 2011, 74(11):3728–3735. 10.1016/j.na.2011.03.018
Nolder CA: Hardy-Littlewood theorems for A-harmonic tensors. Ill. J. Math. 1999, 43: 613–631.
Nolder CA: Global integrability theorems for A-harmonic tensors. J. Math. Anal. Appl. 2000, 247: 236–247. 10.1006/jmaa.2000.6850
Nolder CA: Conjugate harmonic functions and Clifford algebras. J. Math. Anal. Appl. 2005, 302: 137–142. 10.1016/j.jmaa.2004.08.008
Serrin J: Local behavior of solutions of quasi-linear equations. Acta Math. 1964, 111: 247–302. 10.1007/BF02391014
Neugebauer CJ:Inserting -weights. Proc. Am. Math. Soc. 1983, 87: 644–648.
Zhu, J, Ding, S, Tang, Z: The reverse Hölder and Caccioppoli type inequalities for generalized A-harmonic equations (in press)
Acknowledgement
The research of the first author was supported by the NSF of Hunan Province (No. 11JJ3004) and NSF of NUDT (No. JC10-02-02).
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Professor JZ gave the ideas. ZT gave the proofs and completed the manuscript.
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Tang, Z., Zhu, J. Estimates for the composition of the carathéodory and homotopy operators. J Inequal Appl 2012, 188 (2012). https://doi.org/10.1186/1029-242X-2012-188
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DOI: https://doi.org/10.1186/1029-242X-2012-188