L ϕ -Embedding inequalities for some operators on differential forms

Transcription

1 Shi et al. Journal of Inequalities and Applications :392 DOI /s x R S A R C H Open Access L -mbedding inequalities for some operators on differential forms Guannan Shi 1, Shusen Ding 2 and Yuming Xing 1* * Correspondence: xyuming@hit.edu.cn 1 Department of Mathematics, Harbin Institute of Technology, Harbin, , P.R. China Full list of author information is available at the end of the article Abstract In this paper, the local Poincaré inequality and embedding inequality are proved first. Then the global embedding inequality of composite operators for differential forms on L -averaging domains with L -norm is established. Some examples are also given to illustrate applications. Keywords: homotopy operator; Dirac operator; Green s operator; L -averaging domains 1 Introduction The purpose of this paper is to establish the global Sobolev embedding inequality with L -norm for the composite operators applied to differential forms on L -averaging domains. As extensions of functions, differential forms have been deeply studied and widely applied in many fields, such as global analysis, nonlinear analysis, PDs and differential geometry; see [1 10] for more information. The Sobolev embedding inequality, as a fundamental tool in the Sobolev space of functions, has been very well studied in [11]. The homotopy operator T, Dirac operator D, and Green s operator G are three key operators acting on differential forms and each of them has received much attention recently. However, the compositions of these operators are so complicated that the results concerned on them are quite few. ven though they are very complicated, the study for the composite operators has been increasing during the recent years; see [6 9, 12]. In many cases, we need to study or to use the compositions of operators. For example, we have to deal with the composition of the Dirac operator D and Green s operator G when we consider the Poisson equation D 2 Gu = u Hu, where D 2 = and H is the projection operator. In 2013, Ding and Liu initiated the study of the composition D G and obtained some basic inequalities in [2]. During our recent investigation of the operator theory of differential forms, we realized that for applying the decomposition theorem to D G, it is necessary to deal with the composition T D G.Hence,wearemotivatedtostudythecomposition T D G. We obtain some basic estimates for T D G and establish the global Sobolev embedding inequality with L -norm TDGu TDGu W 1, C u 1, L, where is an L -averaging domain [13] Shi et al. This article is distributed under the terms of the Creative Commons Attribution 4.0 International License which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original authors and the source, provide a link to the Creative Commons license, and indicate if changes were made.

2 Shi et al. Journal of Inequalities and Applications :392 Page 2 of 13 For convenience, we keep using the traditional notations and terminologies. xcept for special instructions, R n is a bounded domain, and denotes the Lebesgue measure of, n 2. Suppose that r x is a ball with a radius r, centeredatx. For any σ >0, and σ have the same center and satisfy diam σ =σ diam. Let l R n bethespaceofalll-forms in R n, which is expanded by the exterior product of e = e i 1 e i 2 e i l,where =i 1,...,i l, 1 i 1 < < i l n, l =1,2,...,n. C l is the space of a smooth l-form on.weused, l to denote the space of all differential l-forms on,thatis,uxbelongstod, l if and only if there exist some lth-differential functions u in such that ux= u x = u i1 i 2 i l x i1 i2 il. L p, l is a anach space with the norm equipped by u p, = ux p 1/p,where ux D, l and every coefficient function u L p, 0 < p <. Infact,ux on is the Schwarz distribution. If wx >0a.e.andwx L 1 loc Rm, wx is called a weight. Let dμ = wx, thenl p, l, w is a weighted anach space with the norm expressed by ux p,,w = ux p wx 1/p. In this notation, the exterior derivative is denoted by d and the Hodge codifferential operator is expressed by d. Refer to [1] for more details. Moreover, the Dirac operator, designated by D, was initially set forth by Paul Dirac, and we get its extensions, such as the Hodge-Dirac operator and the uclidean Dirac operator. In this paper, the Dirac operator we choose is the Hodge-Dirac operator D = d + d. The Green s operator is a bound self-adjoint linear operator. It is commonly used to define the Poisson equation for differential forms; here it is represented by G. The homotopy operator T [10], advanced by Iwaniec, is defined on C D, l and constructed as follows: Tu = ψyk y udy, D where the linear operator K y is denoted by K y ux; ξ 1,...,ξ l 1 = 1 0 tl 1 utx + y ty; x y, ξ 1,...,ξ l 1 dt,andψ from C 0 Disnormalizedsothat ψy dy =1. From [10], we obtain some good results. For any differential form u, the decomposition below holds: u = dtu+tdu. 1.1 Meanwhile, for u D, l, we define u = { 1 uy dy, l =0, dtu, l =1,2,...,n. 1.2 Then we know that, for any differential form u L s loc, l, l =1,...,n,1<s <, Tu s, C u s, and Tu s, C diam u s, 1.3 holds for any ball in. Concerning the homotopy operator T, according to [1],wecanobtainaniceproperty. That is, R n is the union of a collection of cube k, whose sides are parallel to the axes, whose interiors are mutually disjoint, and whose diameters are approximately proportional to their distances from F,whereF is the complement of in R n.indetail, R n can be expressed as follows:

3 Shi et al. Journal of Inequalities and Applications :392 Page 3 of 13 1 = k=1 k; 2 0 i 0 j =,ifj i; 3 there exist two constants C 1 >0and C 2 >0,suchthat C 1 diam k distance k, F C 2 diam k. ased on this fact, the homotopy operator T can be extended to any domain R n.for any x, x k for some k, lett k be the homotopy operator defined on k,whichis the convex and bounded set. So, the following definition for T on any domain holds: T = + k=1 T k χ k, where χ k is the characteristic function defined on. Further, it is essential to recall some well-known results as regards the A-harmonic equation for differential forms, which appeared in [1]. To be more precise, we here consider the non-homogeneous A-harmonic equation as follows: d Ax, du=x, du, 1.4 where A : l R n l R n and : l R n l 1 R n satisfy the conditions Ax, ξ a ξ p 1, Ax, ξ ξ ξ p and x, ξ b ξ p 1 for almost every x and all ξ l R n. Meanwhile, the parameters a, b >0and1<p < associated with 1.4 are constants and a fixed exponential, respectively. If =0,the equation d Ax, du = 0 is called a homogeneous A-harmonic equation. In addition, for proving the global embedding inequality on the L -averaging domain, we also need the following definitions and notations. A function x is called an Orlicz function, if x satisfies:1 x is continuously increasing; 2 0 = 0. Furthermore, if the Orlicz function x is a convex function, x is named a Young function. Therefore, based on this type of functions, the Orlicz norm for differential forms can be denoted as follows. Take x as a Orlicz function, and R n as a bounded domain, for any ux L p loc, l, l =0,1,2,...,n, the Orlicz norm for the differential form is equipped with { u = inf L λ >0: u λ } dμ 1, where the measure μ is expressed by dμ = wx, wx isaweight. It is easy to prove that L is a anach space. Actually, provided that x istakenas x =x s s >0,thenx is an Orlicz function, and it trivially corresponds to a L s μ space. As a result of that, we can say that L is the generalization of Ls. Similarly, we intend to give L -averaging domains. ased on the Orlicz norm above, the Orlicz-Sobolev space of l-form is denoted by W 1,, l, with the norm u W 1, = u W 1,, l = diam 1 u L + u L. 1.5

4 Shi et al. Journal of Inequalities and Applications :392 Page 4 of 13 From the definition of W 1, 1,,weknowthatW is equal to L L 1,.Indetail,ifx is expressed by x=t s s > 0, we can conclude the norm of W 1,s u W 1,s = u W 1,s, l = diam 1 u L s + u L s. for the differential form: esides that, we will show another definition, which was initially introduced by Ding. Definition 1.1 [13] Letx be a Young function, the proper domain R n is called the L -averaging domain, if μ< and there exists a constant C > 0 such that for any 0 and u L 1 loc, μ, u satisfies 1 τ u u 0 1 dμ C sup σ u u dμ, μ 4 μ where the measure μ is denoted by dμ = wx, wx isaweight,σ and τ are constants with 0 < τ, σ < 1, and the supremum is over all balls with 4. Using the same analysis method as of the L -norm, we can conclude that L -averaging domains are the generalization of L s -averaging domains. 2 Main results efore the main results given, we will make some restrictions to the Young function x. Here, we let the Young function xbelongtothegp, q, C-class 1 p < q <, C 1, that is, for any t > 0, the Young function xsatisfies: 1 1/C t 1/p /f t C; 2 1/C t 1/q /gt C, where f and g are increasingly convex and concave functions defined on [0, ], respectively. Now, we establish four important theorems based on the above-mentioned conditions. Theorem 2.1 Let T be the homotopy operator, D be the Dirac operator, and G be the Green s operator. Meanwhile, we assume that u L 1 loc, u C l is a solution of the non-homogeneous A-harmonic equation, the Young function imposed a doubling property x belongs to the Gp, q, C-class, and the bounded subset R n is the L -averaging domain. Then, for any ball, we get TDGu TDGu L where σ andσ >1is a constant. C diam u L σ, Theorem 2.2 Let T be the homotopy operator, D be the Dirac operator, and G be the Green s operator. Meanwhile, we assume that u L 1 loc, u C l, is a solution of the non-homogeneous A-harmonic equation, the Young function x having imposed a doubling property belongs to the Gp, q, C-class and the bounded subset R n is the L -averaging domain. Then, for any ball, we get TDGu TDGu 1, W where σ andσ >1is a constant. C diam u 1, L, σ

5 Shi et al. Journal of Inequalities and Applications :392 Page 5 of 13 ased on the above theorem, we cannot only establish the following global embedding inequality on L -averaging domains, but we also get the global Poincaré-type inequality fortunately. Theorem 2.3 Let T be the homotopy operator, D be the Dirac operator, and G be the Green s operator. Additionally, we assume that u L 1 loc, u C l, is a solution of the non-homogeneous A-harmonic equation, the Young function x having imposed a doubling property belongs to the Gp, q, C-class and the bounded subset R n is the L -averaging domain. Then, for any ball, we have TDGu TDGu W 1, C u 1, L. Theorem 2.4 Let T be the homotopy operator, D be the Dirac operator, and G be the Green s operator. Additionally, we assume that u L 1 loc, u C l, is a solution of the non-homogeneous A-harmonic equation, the Young function x having imposed a doubling property belongs to the Gp, q, C-class and the bounded subset R n is the L -averaging domain. Then, for any ball, we have TDGu TDGu L C u, 0 L where 0 isafixedball. 3 Preliminary results For proving the theorems in Section 2, we shall show and demonstrate some lemmas in this section. Lemma 3.1 [1] Let 0<p, q <, and 1 t = 1 p + 1, if f and g are the measurable functions q defined on R n, then fg t,i f p,i g q,i for any I R n. The inequality in Lemma 3.1 is actually the generalized Hölder inequality. Specifically, if t = 1and1 < p, q <, the inequality above is the classical Hölder inequality. Lemma 3.2 [1] Let u be a solution of the non-homogeneous A-harmonic equation in adomain, and 0<s, t <, then there exists a constant C, independent of u, such that u s, C t s st u t,σ for all balls with σ, where σ >1is some constant. In fact, we can get a valuable result; if u satisfies the inequality in Lemma 3.2, wesayu belongs to the WRH-class.

6 Shi et al. Journal of Inequalities and Applications :392 Page 6 of 13 Lemma 3.3 [2] Suppose that the differential form u L p, l, l =1,...,n, and G is the Green s operator, then there exists a constant C, independent of u, such that dd Gu+d dgu+ d + d Gu+Gu p, C u p, for any. ecause = d d + dd and D = d + d, by using Lemma 3.3,itimpliesthat Gu p, C u p, and DGu p, C u p,. Lemma 3.4 [10] If u L p D, l, 1 < p <, then u D L p D, l and u D p,d A p nμd u D. According to the above results, we can prove a very useful lemma as follows. Lemma 3.5 Let T be the homotopy operator, D be the Dirac operator, and G be the Green s operator. Additionally, we assume that u C l is a solution of the non-homogeneous A-harmonic equation, the Young function x belongs to the Gp, q, C-class, and the bounded subset R n is the L -averaging domain. Then, for any ball, we have TDGu TDGu for any, where s >1. s, C diam u s, Proof Notice that TDGu is at least differential 1-form, therefore, by using 1.1 andre- placing u by TDGu, we have TDGu=Td TDGu + dt TDGu. It follows from 1.2thatdTTDGu = TDGu,thatis, TDGu TDGu s, = Td TDGu s,. Applying 1.3, Lemma 3.3, and Lemma 3.4,weobtain TDGu TDGu s, = Td TDGu s, C 1 diam d TDGu s, C 1 diam DGu C 2 diam DGu s, C 3 diam u s,. s, Thus,wefinishtheproofofthislemma.

7 Shi et al. Journal of Inequalities and Applications :392 Page 7 of 13 Note y observing the proving process, we can get a valuable result as follows: Td TDGu s, C diam u s,. Lemma 3.6 Let T be the homotopy operator, DbetheDiracoperator, and G be the Green s operator. Additionally, we assume that u C l is a solution of the non-homogeneous A-harmonic equation, the Young function x having imposed a doubling property belongs to the Gp, q, C-class and the bounded subset R n is the L -averaging domain. Then, for any ball, we have TDGu L C diam u, L σ where the constant σ >1and σ. Proof From 1.3, we have TDGu s, C 1 diam DGu s, C 2 diam u s,. 3.1 ecause belongs to the Gp, q, C-class, we have TDGu = g g 1 g g C 2 TDGu g 1 TDGu C 1 TDGu q C 1 TDGu 1 q q. Since is doubling, according to 3.1and Lemma 3.2,wehave TDGu C 2 C 3 1+ n 1 u q, 1 C 4 1+ n 1 p 1 q 1 p u p σ C 5 f p1+ 1n p q 1 u p σ C 6 f p 1+p q 1 + n 1 + u p C 7 σ σ C 8 1+ q 1 + n 1 p 1 u

8 Shi et al. Journal of Inequalities and Applications :392 Page 8 of 13 C 9 1+ q 1 + n 1 p 1 u C 10 n 1 σ σ u. 3.2 Therefore, by using 1 n = C 11 diam, we can get the following result: TDGu L C diam u, L σ where σ >1. This is the end of the proof of Lemma 3.6. Lemma 3.7 Let T be the homotopy operator, DbetheDiracoperator, and G be the Green s operator. Additionally, we assume that u L 1 loc and u C l is a solution of the non-homogeneous A-harmonic equation, the Young function x having imposed a doubling property belongs to the Gp, q, C-classand is a bounded domain. Then we get Td TDGu L C diam u L σ for all balls with σ, where σ >1is some constant. Proof According to the elementary inequality 1.3 of the homotopy operator T and Lemma 3.4,wehave Td TDGu p, C 1 d TDGu p, C 2 DGu p,. 3.3 Applying 3.3 and Lemma 3.3,weget Td TDGu p, C 2 diam u p,. Using the same method as used in Lemma 3.6, the following inequality holds: Td TDGu L C diam u. L σ The proof of Lemma 3.7 is completed. Lemma 3.8 Covering lemma [1] ach domain has a modified Whitney cover of cubes V = {Q i } such that Q i =, i χ 5 Nχ 4 Q i Q i V and some N >1,and if Q i Q j, then there exists a cube R this cube does not need to be a member of V in Q i Q j such that Q i Q j NR. Moreover, if is δ-john, then there is a distinguished cube Q 0 V which can be connected with every cube Q V by a chain of cubes Q 0, Q 1,...,Q k = QfromV and such that Q ρq i, i =0,1,2,...,k, for some ρ = ρn, δ.

9 Shi et al. Journal of Inequalities and Applications :392 Page 9 of 13 4 Demonstration of main results According to the above definitions and lemmas, we will prove four theorems in detail. First of all, let us prove Theorem 2.1. Proof of Theorem 2.1 Firstofall,similartotheproofofLemma3.5,wehave TDGu TDGu L = Td TDGu L. 4.1 This means that the key point is to prove the following inequality: Td TDGu L C u. L σ According to the properties of the Gp, q, C-class, we have Td TDGu = g g 1 g g C 2 Td TDGu g 1 Td TDGu C 1 Td TDGu q C 1 Td TDGu 1 q q. ecause is doubling, using Lemma 3.5,wehave Td TDGu C 2 C 3 1+ n 1 u q, 1 C 4 1+ n 1 p 1 q 1 p u p σ C 5 f p1+ 1n p q 1 u p σ C 6 f p 1+p q 1 + n 1 u p σ C 7 C 8 1+ q 1 + n 1 p 1 u σ C 9 1+ q 1 + n 1 p 1 u σ C 10 n 1 u. 4.2 σ Combining 4.1with4.2andusing n 1 = C 11 diamyield TDGu TDGu C 12 diam u. σ

10 Shi et al. Journal of Inequalities and Applications :392 Page 10 of 13 As a result, we obtain TDGu TDGu C diam λ σ u λ for any with σ and any constant λ > 0. It means that the following inequality on L -averaging domains holds: TDGu TDGu L C diam u L σ. The proof of this theorem is finished. Remark From the proof of Theorem 2.1, it is easy to derive the following inequality: Td TDGu L C diam u. 4.3 L σ We shall prove Theorem 2.2by using Theorem 2.1and Lemma 3.7. Proof for Theorem 2.2 Using 1.1repeatedly,we have TDGu TDGu 1, W = Td TDGu 1, W. 4.4 Combining 4.4with1.5, and applying Theorem 2.1and Lemma3.7,wehave TDGu TDGu 1, W = diam 1 Td TDGu L + Td TDGu L diam 1 C1 diam u L + C2 u σ 1 L σ 2 C 1 u L σ 1 + C 1 u L σ 2 C u L σ, where σ = max{σ 1, σ 2 } and σ > 1, for all balls with σ. This is the end of the proof of Theorem 2.2. Proof for Theorem 2.3 From the covering lemma, Lemma 3.8, and Lemma 3.7,wehave Td TDGu L Td TDGu L V V C1 u L σ C 2 N u L C 3 u. 4.5 L Similarly, using the covering lemma, Lemma 3.8,and4.3implies Td TDGu L Td TDGu L V V C4 diam u L σ

11 Shi et al. Journal of Inequalities and Applications :392 Page 11 of 13 Thus, from 1.5, 4.5, and 4.6, we obtain C 5 diamn u L C 6 diam u. 4.6 L TDGu TDGu W 1, = Td TDGu W 1, = diam 1 Td TDGu L + Td TDGu L diam 1 C6 diam u L + C3 u L C 7 u. L We have completed the proof of Theorem 2.3. Next, we will prove Theorem 2.4 by using Definition 1.1 and Theorem 2.1. Proof of Theorem 2.4 According to Definition 1.1,we have 1 TDGu TDGu sup 1 TDGu TDGu 0 4 sup 1 C diam u. 4 ecause sup u does not depend on,weobtain 1 TDGu TDGu sup 1 C diam u. 0 4 Therefore, we have λ 1 TDGu TDGu C λ 1 u. 0 σ We finish the proof of Theorem 2.4. In addition, we can obtain a global estimate about the composite operators using the same method as of Theorem 2.1. Corollary 4.1 Let T be the homotopy operator, D be the Dirac operator, and G be the Green s operator. Additionally, we assume that u C l is a solution of the nonhomogeneous A-harmonic equation, the Young function x belongsto the Gp, q, C-class, and the bounded subset R n is the L -averaging domain. Then, for any ball, we have TDGu L C diam u. L Remark If we choose x=x s,wehave TDGu s, C diam u s,.

12 Shi et al. Journal of Inequalities and Applications :392 Page 12 of 13 5 Applications In this section, we will discuss the applications of the results obtained. xample 5.1 Let r >0and = {x 1, x 2, x 3 :x x2 2 + x2 3 r2 } R 3. Consider the 1-form ux 1, x 2, x 3 = x 1 1+x x2 2 + x x 2 1+x x2 2 + x x 3 1+x x2 2 + x defined in.thenu is a solution of the non-homogeneous A-harmonic equation for any operators A and satisfying the required conditions. As the application of the obtained theorems, our goal is to get the upper bound of TDGu satisfying the above conditions. Normally, we would consider calculating the integral of TDGu, however, one will see that TDGuwiththeL -normisverycomplicated. Inthiscase,wecanuseTheorem2.3 to get the estimate of TDGu withthel -norm in W 1,. Initially, according to the condition and the expression of u in xample 5.1,weseethat ux1, x 2, x 3 2 x 2 1 = 1 + x x2 2 + x x x x2 2 + x x x x2 2 + x Furthermore, because u is defined in and x x2 2 + x2 3 2 >0,weget ux 1, x 2, x As a result, by using Theorem 2.3 and 5.1, we have TDGu TDGu 1, W C 1 u L C 1 1 L C 2r 3. The above example can be extended to the case of R n. Particularly, we can check that the 1-form defined in R n, ux 1,...,x n = n i=1 x i 1+x x2 n 2 i, is a solution of the non-homogeneous A-harmonic equation for any operators A and satisfying the required conditions. So, we can also apply Theorem 2.3 to this extended case as we did in xample 5.1. To end this section, we take a 3-form in R 3 as an example. xample 5.2 Let k, m > 0 be constants. Consider a differential 3-form in R 3, ux, y, z= k dy dz. m + x 2 + y 2 + z2 Then u is a solution of the non-homogeneous A-harmonic equation for any operators A and satisfying the required conditions.

13 Shi et al. Journal of Inequalities and Applications :392 Page 13 of 13 Similarly, applying the same method, we can obtain the following result as regards TDGuinanyboundedset R 3 : and ux, y, z k m TDGu TDGu 1, W C u L C k/m L. Competing interests The authors declare that they have no competing interests. Authors contributions All results and investigations of this manuscript were due to the joint efforts of all authors. GS mainly studied the theoretical proofs and drafted the manuscript. SD and YX conceived of the initial idea of this manuscript and improved the final version. All authors read and approved the final manuscript. Author details 1 Department of Mathematics, Harbin Institute of Technology, Harbin, , P.R. China. 2 Department of Mathematics, Seattle University, Seattle, WA 98122, USA. Acknowledgements The authors express their deep appreciation to the referees efforts and suggestions, which highly improved the quality of this paper. Received: 14 August 2015 Accepted: 6 November 2015 References 1. Agarwal, RP, Ding, S, Nolder, C: Inequalities for Differential Forms. Springer, New York Ding, S, Liu, : Norm inequalities for composition of the Dirac and Green s operator. J. Inequal. Appl doi: / x Wang, Y, Wu, C: Sobolev embedding theorems and Poincaré inequalities for Green s operator on solution of the non-homogeneous A-harmonic equation. Comput. Math. Appl. 47, Liu, : A λ r -Weighted embedding inequalities for A-harmonic tensors. J. Math. Anal. Appl. 273, Xing, Y: Two-weight embedding inequalities for solutions to the A-harmonic equation. J. Math. Anal. Appl. 307, i, H: Weighted inequalities for potential operators on differential forms. J. Inequal. Appl. 2010, Article ID i, H, Ding, S: Some strong p, q-type inequalities for the homotopy operator. Comput. Math. Appl. 62, Ding, S, Liu, : A singular integral of the composite operator. Appl. Math. Lett. 22, Ding,S,Liu,:Globalestimatesforsingularintegralsofthecompositeoperator.Ill.J.Math.53, Iwaniec, T, Lutoborski, A: Integral estimates for null Lagrangians. Arch. Ration. Mech. Anal. 125, Adams, RA, Fourier, JJF: Sobolev Spaces, 2nd edn. lsevier, Amsterdam Scott, C: L p -Theory of differential forms on manifolds. Trans. Am. Math. Soc. 347, Ding, S: L μ-averaging domains and the quasi-hyperbolic metric. Comput. Math. Appl. 47,