JFS Journal of Function Spaces 2314-8888 2314-8896 Hindawi Publishing Corporation 943986 10.1155/2014/943986 943986 Research Article Norm Comparison Estimates for the Composite Operator Li Xuexin Wang Yong http://orcid.org/0000-0002-0525-9703 Xing Yuming Ding Shusen Department of Mathematics Harbin Institute of Technology Harbin 150001 China hit.edu.cn 2014 3032014 2014 12 11 2013 06 02 2014 20 02 2014 30 3 2014 2014 Copyright © 2014 Xuexin Li et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

This paper obtains the Lipschitz and BMO norm estimates for the composite operator 𝕄 s P applied to differential forms. Here, 𝕄 s is the Hardy-Littlewood maximal operator, and P is the potential operator. As applications, we obtain the norm estimates for the Jacobian subdeterminant and the generalized solution of the quasilinear elliptic equation.

1. Introduction

Since the differential form has been proposed by Elie Cartan, its development is obvious. Differential form theory has been applied to many fields, such as partial differential equations, nonlinear analysis, and control theory. Particularly, it is an important part in differential forms to research norm inequalities for an operator or composite operator. L p norm has been studied very thoroughly , so we try to establish the Lipschitz norm and BMO norm inequalities for the composition of the Hardy-Littlewood maximal operator and potential operator. It is useful to explore the properties of the weighted norms; see . So, we also research the weighted Lipschitz norm and the weighted BMO norm inequalities.

As the traditional notations, we write Ω for a bounded convex domain in n , n 2 , endowed with the usual Lebesgue measure denoted by | Ω | . B and σ B are concentric balls, with    diam ( σ B ) = σ diam ( B ) . ω denotes a weight defined by ω L loc ( n ) , and ω > 0 a.e.. The l -form , denoted by Λ l = Λ l ( n ) , is a l -vector , spanned by exterior products e I = e i 1 e i 2 e i l , for all ordered l -tuples I = ( i 1 , i 2 , , i l ) , 1 i 1 < i 2 < < i l n . The l -form u ( x ) = Σ I u I ( x ) d x I is called a differential l -form, if u I is differential. We use D ( Ω , Λ l ) to denote the differential l -form and L s ( Ω , Λ l ) to denote the l -form u ( x ) on Ω satisfying Ω | u I | s < . In particular, we know that the 0 -form is a function. We define the exterior derivative d : D ( Ω , Λ l ) D ( Ω , Λ l + 1 ) by (1) d ( u ( x ) ) = k = 1 n 1 i 1 < i 2 < < i l n u i 1 i 2 i l ( x ) x k × d x k d x i 1 d x i l , l = 0,1 , , n - 1 . We define by (2) u = sign ( π ) u i 1 i 2 i l ( x ) d x j 1 d x j n - l , where π = ( i 1 , , i l , j 1 , , j n - l ) is a permutation of ( 1 , , n ) and sign ( π ) is the signature of the permutation. Now, we can define the Hodge codifferential d like this (3) d = ( - 1 ) n l + 1 d : D ( Ω , Λ l + 1 ) D ( Ω , Λ l ) , l = 0,1 , , n - 1 . The differential form we research here satisfies the nonhomogeneous A -harmonic equation (4) d A ( x , d u ) = B ( x , d u ) , where A : Ω × Λ l ( n ) Λ l ( n ) and B : Ω × Λ l ( n ) Λ l - 1 ( n ) satisfy the conditions (5) | A ( x , ξ ) | a | ξ | p - 1 , A ( x , ξ ) · ξ | ξ | p , | B ( x , ξ ) | b | ξ | p - 1 for almost every x Ω and all ξ l ( n ) . Here a , b > 0 are constants and 1 < p < is a fixed exponent associated with (4).

2. Lipschitz and BMO Norm Inequalities

In this section, we introduce some definitions. Then, we give main lemmas used in theorems. Finally, we give the norm comparison estimates for the composite operator.

For u L loc 1 ( Ω , Λ l ) , l = 0,1 , , n , we write u loc Li p k ( Ω , Λ l ) , 0 k 1 , if (6) u loc Li p k , Ω = sup σ Q Ω | Q | - ( n + k ) / n u - u Q 1 , Q < for some σ > 1 .

For u L loc 1 ( Ω , Λ l ) , l = 0,1 , , n , we say u BMO ( Ω , Λ l ) , if (7) u * , Ω = sup σ Q Ω | Q | - 1 u - u Q 1 , Q < for some σ > 1 .

Hardy-Littlewood maximal operator is defined by (8) 𝕄 s ( u ) = 𝕄 s u ( x ) = sup r > 0 ( 1 | B ( x , r ) | B ( x , r ) | u ( y ) | s    d y ) 1 / s , where 1 s < , u is a locally L s -integrable form and B ( x , r ) is a ball with radius r .

Potential operator is first extended to differential form by Bi in , which is defined as follows: (9) P u = P u ( x ) = I Ω K ( x , y ) u I ( y ) d y d x I , where the kernel K ( x , y ) is a nonnegative measurable function defined for x y , u is a differential l -form, and the summation is over all ordered l -tuples I .

We need the following two lemmas for the Hardy-Littlewood maximal operator and potential operator to prove Theorem 7.

Lemma 1 (see [<xref ref-type="bibr" rid="B8">9</xref>]).

Let u L t ( Ω , Λ l ) , l = 0,1 , , n , be a differential form in a domain Ω and 𝕄 s be the Hardy-Littlewood maximal operator, 1 s < t < . Then, 𝕄 s ( u ) L t ( Ω ) and there exists a constant C , independent of u , such that (10) 𝕄 s ( u ) t , Ω C u t , Ω .

Lemma 2 (see [<xref ref-type="bibr" rid="B9">8</xref>]).

Let u D ( Ω , Λ l ) , l = 0,1 , , n , be a differential form in a domain Ω and let P be the potential operator. Then, there exists a constant C , independent of u , such that (11) P ( u ) s , Ω C u s , Ω , where 1 < s < .

The following lemma was established by Iwaniec and Lutoborski in .

Lemma 3.

Let u L s ( Ω , Λ l ) , l = 0,1 , , n , 1 < s < , be a differential form in a domain Ω . Then, there exists a constant C , independent of u , such that (12) u Ω s , Ω C u s , Ω , where u Ω is a closed form.

We will use the following generalized Hölder inequality repeatedly in this paper.

Lemma 4.

Let 0 < q < , 0 < p < , and s - 1 = q - 1 + p - 1 . If f and g are measurable functions on n , then (13) f g s , Ω f q , Ω g p , Ω for any Ω n .

We also need the Caccioppoli inequality for differential form.

Lemma 5 (see [<xref ref-type="bibr" rid="B11">11</xref>]).

Let u D ( Ω , Λ l ) , l = 0,1 , , n , be a solution of the nonhomogeneous A-harmonic equation (4) in Ω . Then, there exists a constant C , independent of u , such that (14) d u s , B C diam ( B ) - 1 u - c s , σ B for all balls with σ B Ω and any closed form c , where 1 < s < , and σ > 1 .

The following Poincaré inequality appears in .

Lemma 6.

Let u D ( Ω , Λ l ) , l = 0,1 , , n - 1 , be a solution of the nonhomogeneous A-harmonic equation (4) in Ω . Then, there exists a constant C , independent of u , such that (15) u - u B s , B C | B | diam ( B ) d u s , B for any ball B in Ω .

First, we establish a Poincaré-type inequality for the composite operator 𝕄 s P .

Theorem 7.

Let u L t ( Ω , Λ l ) , l = 1,2 , , n , 1 s < t < , be a differential form in a domain Ω . Then, there exists a constant C , independent of u , such that (16) 𝕄 s P ( u ) - ( 𝕄 s P ( u ) ) Ω t , Ω C u t , Ω .

Proof.

From Lemma 3 and Minkowski inequality, we have (17) u - u Ω t , Ω = ( Ω | u ( x ) - u Ω ( x ) | t d x ) 1 / t ( Ω | u ( x ) | t d x ) 1 / t + ( Ω | u Ω ( x ) | t d x ) 1 / t ( Ω | u ( x ) | t d x ) 1 / t + C 1 ( Ω | u ( x ) | t d x ) 1 / t C 2 u t , Ω . Replacing u with 𝕄 s P ( u ) and combining Lemmas 1 and 2, we obtain (18) 𝕄 s P ( u ) - ( 𝕄 s P ( u ) ) Ω t , Ω C 2 𝕄 s P ( u ) t , Ω C 3 P ( u ) t , Ω C 4 u t , Ω . Theorem 7 has been completed.

Then, we estimate the Lipschitz norm of 𝕄 s P ( u ) in terms of L t -norm.

Theorem 8.

Let u L t ( Ω , Λ l ) , l = 1,2 , , n , 1 s < t < , be a solution of the nonhomogeneous A-harmonic equation (4) in Ω . Then, there exists a constant C , independent of u , such that (19) 𝕄 s P ( u ) loc Lip k , Ω C d 𝕄 s P ( u ) t , Ω , where k is a constant with 0 k 1 .

Proof.

Setting u = 𝕄 s P ( u ) in Lemma 6 we have (20) 𝕄 s P ( u ) - ( 𝕄 s P ( u ) ) B t , B C 1 | B | diam ( B ) d 𝕄 s P ( u ) t , B for all balls B with B Ω . Using Hölder inequality, we have (21) 𝕄 s P ( u ) - ( 𝕄 s P ( u ) ) B 1 , B ( B | 𝕄 s P ( u ) - ( 𝕄 s P ( u ) ) B | t    d x ) 1 / t × ( B 1 t / ( t - 1 )    d x ) ( t - 1 ) / t | B | ( t - 1 ) / t 𝕄 s P ( u ) - ( 𝕄 s P ( u ) ) B t , B = | B | 1 - 1 / t 𝕄 s P ( u ) - ( 𝕄 s P ( u ) ) B t , B | B | 1 - 1 / t C 1 | B | diam ( B ) d 𝕄 s P ( u ) t , B C 2 | B | 2 - 1 / t + 1 / n d 𝕄 s P ( u ) t , B . From the definition of Lipschitz norm and (21), it follows that (22) 𝕄 s P ( u ) loc Li p k , Ω = sup σ B Ω | B | - ( n + k ) / n 𝕄 s P ( u ) - ( 𝕄 s P ( u ) ) B 1 , B = sup σ B Ω | B | - 1 - k / n 𝕄 s P ( u ) - ( 𝕄 s P ( u ) ) B 1 , B sup σ B Ω | B | - 1 - k / n C 2 | B | 2 - 1 / t + 1 / n d 𝕄 s P ( u ) t , B = sup σ B Ω C 2 | B | 1 - k / n - 1 / t + 1 / n d 𝕄 s P ( u ) t , B sup σ B Ω C 2 | Ω | 1 - k / n - 1 / t + 1 / n d 𝕄 s P ( u ) t , B C 3 sup σ B Ω d 𝕄 s P ( u ) t , B C 3 d 𝕄 s P ( u ) t , Ω . Theorem 8 has been completed.

Now, we establish the norm comparison theorems of the composite operator between Lipschitz norm and BMO norm.

Theorem 9.

Let u L s ( Ω , Λ l ) , l = 1,2 , , n , 1 s < , be a differential form in Ω . Then, there exists a constant C , independent of u , such that (23) 𝕄 s P ( u ) * , Ω C 𝕄 s P ( u ) loc Lip k , Ω , where k is a constant with 0 k 1 .

Proof.

From the definition of BMO norm, we have (24) 𝕄 s P ( u ) * , Ω = sup σ B Ω | B | - 1 𝕄 s P ( u ) - ( 𝕄 s P ( u ) ) B 1 , B = sup σ B Ω | B | k / n | B | - ( n + k ) / n 𝕄 s P ( u ) - ( 𝕄 s P ( u ) ) B 1 , B sup σ B Ω | Ω | k / n | B | - ( n + k ) / n 𝕄 s P ( u ) - ( 𝕄 s P ( u ) ) B 1 , B | Ω | k / n sup σ B Ω | B | - ( n + k ) / n 𝕄 s P ( u ) - ( 𝕄 s P ( u ) ) B 1 , B C sup σ B Ω | B | - ( n + k ) / n 𝕄 s P ( u ) - ( 𝕄 s P ( u ) ) B 1 , B C 𝕄 s P ( u ) loc Li p k , Ω . Theorem 9 has been completed.

Theorem 10.

Let u L s ( Ω , Λ l ) , l = 1,2 , , n , 1 < s < , be a solution of the nonhomogeneous A-harmonic equation (4) in Ω . Then, there exists a constant C , independent of u , such that (25) 𝕄 s P ( u ) loc Lip k , Ω C 𝕄 s P ( u ) * , Ω , where k is a constant with 0 k 1 .

Proof.

From (21), we obtain (26) 𝕄 s P ( u ) - ( 𝕄 s P ( u ) ) B 1 , B C 1 | B | 2 - 1 / s + 1 / n d 𝕄 s P ( u ) s , B . Based on Lemma 5, we get (27) d 𝕄 s P ( u ) s , B C 2 diam ( B ) - 1 𝕄 s P ( u ) - ( 𝕄 s P ( u ) ) B s , σ B . From the weak reverse Hölder inequality, it follows that (28) 𝕄 s P ( u ) - ( 𝕄 s P ( u ) ) B s , σ B C 3 | B | ( 1 - s ) / s 𝕄 s P ( u ) - ( 𝕄 s P ( u ) ) B 1 , σ B , where σ > σ > 1 . So, we get (29) 𝕄 s P ( u ) - ( 𝕄 s P ( u ) ) B 1 , B C 1 | B | 2 - 1 / s + 1 / n d 𝕄 s P ( u ) s , B C 4 | B | 𝕄 s P ( u ) - ( 𝕄 s P ( u ) ) B 1 , σ B . Let σ ′′ > σ ; we obtain (30) 𝕄 s P ( u ) loc Li p k , Ω = sup σ ′′ B Ω | B | - ( n + k ) / n 𝕄 s P ( u ) - ( 𝕄 s P ( u ) ) B 1 , B sup σ ′′ B Ω C 4 | B | 1 - ( k / n ) | B | - 1 𝕄 s P ( u ) - ( 𝕄 s P ( u ) ) B 1 , σ B C 5 sup σ ′′ B Ω C 4 | B | - 1 𝕄 s P ( u ) - ( 𝕄 s P ( u ) ) B 1 , σ B C 6 𝕄 s P ( u ) * , Ω . Theorem 10 has been completed.

Combining Theorems 8 and 9, we obtain the following result easily.

Corollary 11.

Let u L t ( Ω , Λ l ) , l = 1,2 , , n , 1 s < t < , be a solution of the nonhomogeneous A-harmonic equation (4) in Ω . Then, there exists a constant C , independent of u , such that (31) 𝕄 s P ( u ) * , Ω C d 𝕄 s P ( u ) t , Ω .

3. The Weighted Lipschitz and BMO Norm Inequalities

In this section, we obtain some weighted inequalities for the composition operator 𝕄 s P .

The weight function we use here is A ( α , β , γ , Ω ) weight. The A ( α , β , γ , Ω ) -class is a new weight class which was first proposed by Xing in . It contains the well-known A r ( Ω ) -weight as a proper subset.

Definition 12.

One says that a measurable function ω ( x ) defined on a subset Ω n satisfies the A ( α , β , γ , Ω ) -condition for some positive constants α , β , γ , if ω ( x ) > 0 a.e., and (32) sup B Ω ( 1 | B | B ω α d x ) ( 1 | B | B ω - β d x ) γ / β < . Let u L loc 1 ( Ω , Λ l , ω ) , l = 0,1 , 2 , , n . We say u loc Li p k ( Ω , Λ l , ω ) , 0 k 1 , if (33) u loc Li p k , Ω , ω = sup σ Q Ω ( μ ( Q ) ) - ( n + k ) / n u - u Q 1 , Q , ω < for some σ > 1 , where Ω is bounded in n , ω is a weight, and μ is Radon measure defined by d μ = ω ( x ) d x .

Let u L loc 1 ( Ω , Λ l , ω ) , l = 0,1 , 2 , , n . We say u BMO ( Ω , Λ l , ω ) , if (34) u * , Ω , ω = sup σ Q Ω ( μ ( Q ) ) - 1 u - u Q 1 , Q , ω < for some σ > 1 , where Ω is bounded in n , ω is a weight, and μ is Radon measure defined by d μ = ω ( x ) d x .

Now, we estimate the weighted Lipschitz and BMO norm for the composition operator 𝕄 s P .

Theorem 13.

Let u L q ( Ω , Λ l , μ ) , l = 1,2 , , n , 1 s < q < , be a solution of the nonhomogeneous A-harmonic equation (4) in Ω . Radon measure μ is defined by ω ( x ) d x = d μ , and ω ( x ) A ( α , β , γ , Ω ) for some α > 1 , where 1 < p < , β = α q / ( α p - p - α q ) , γ = α q / p , and α p - p - α q > 0 . Then, there exists a constant C , independent of u , such that (35) 𝕄 s P ( u ) loc Lip k , Ω , ω C d 𝕄 s P ( u ) p , Ω , ω .

Proof.

Using Hölder inequality with 1 / q = 1 / α q + ( α - 1 ) / α q , we have (36) ( B | 𝕄 s P ( u ) - ( 𝕄 s P ( u ) ) B | q ω ( x )    d x ) 1 / q = ( B | 𝕄 s P ( u ) - ( 𝕄 s P ( u ) ) B ω ( x ) 1 / q | q d x ) 1 / q ( B | 𝕄 s P ( u ) - ( 𝕄 s P ( u ) ) B | α q / ( α - 1 ) d x ) ( α - 1 ) / α q × ( B ( ω ( x ) 1 / q ) α q d x ) 1 / α q C 1 | B | diam ( B ) ( B | d 𝕄 s P ( u ) | α q / ( α - 1 ) d x ) ( α - 1 ) / α q × ( B ω ( x ) α d x ) 1 / α q . Using Hölder inequality with ( α - 1 ) / α q = ( 1 / p ) + ( α p - p - α q ) / α q p , we obtain (37) ( B | d 𝕄 s P ( u ) | α q / ( α - 1 )    d x ) ( α - 1 ) / α q = ( B | d 𝕄 s P ( u ) ω ( x ) 1 / p ω ( x ) - 1 / p | α q / ( α - 1 ) d x ) ( α - 1 ) / α q ( B | d 𝕄 s P ( u ) ω ( x ) 1 / p | p d x ) 1 / p × ( B ( ω ( x ) - 1 / p ) α q p / ( α p - p - α q ) d x ) ( α p - p - α q ) / α q p = ( B | d 𝕄 s P ( u ) | p ω ( x ) d x ) 1 / p × ( B ( 1 ω ( x ) ) α q / ( α p - p - α q ) d x ) ( α p - p - α q ) / α q p . Since ω ( x ) A ( α , α q / ( α p - p - α q ) , α q / p , Ω ) , we get (38) ( B ω ( x ) α d x ) 1 / α q ( B ( 1 ω ( x ) ) α q / ( α p - p - α q ) d x ) ( α p - p - α q ) / α q p = ( × ( B ( 1 ω ( x ) ) α q / ( α p - p - α q )    d x ) ( α p - p - α q ) / p ( B ω ( x ) α d x ) × ( B ( 1 ω ( x ) ) α q / ( α p - p - α q ) d x ) ( α p - p - α q ) / p ) 1 / α q = | B | 2 / α q × ( ( B ( 1 ω ( x ) ) α q / ( α p - p - α q ) d x ) ( α p - p - α q ) / p ( 1 | B | ( ω ( x ) ) α d x ) × ( B ( 1 ω ( x ) ) α q / ( α p - p - α q ) d x 1 | B | × B ( 1 ω ( x ) ) α q / ( α p - p - α q ) d x ) ( α p - p - α q ) / p ) 1 / α q C 2 . From (36), (37), and (38), we know that (39) ( B | 𝕄 s P ( u ) - ( 𝕄 s P ( u ) ) B | q ω ( x ) d x ) 1 / q C 3 | B | diam ( B ) ( B | d 𝕄 s P ( u ) | p ω ( x ) d x ) 1 / p . So, we obtain (40) 𝕄 s P ( u ) - ( 𝕄 s P ( u ) ) B 1 , B , ω = B | 𝕄 s P ( u ) - ( 𝕄 s P ( u ) ) B | d μ ( B | 𝕄 s P ( u ) - ( 𝕄 s P ( u ) ) B | q ω ( x ) d x ) 1 / q × ( B 1 q / ( q - 1 )    d μ ) ( q - 1 ) / q = ( μ ( B ) ) ( q - 1 ) / q 𝕄 s P ( u ) - ( 𝕄 s P ( u ) ) B q , B , ω C 4 ( μ ( B ) ) ( q - 1 ) / q | B | diam ( B ) d 𝕄 s P ( u ) p , B , ω . Based on the definition of the weighted Lipschitz norm and (40), we have (41) 𝕄 s P ( u ) loc Li p k , Ω , ω = sup σ B Ω ( μ ( B ) ) - ( n + k ) / n 𝕄 s P ( u ) - ( 𝕄 s P ( u ) ) B 1 , B , ω C 5 sup σ B Ω ( μ ( B ) ) - ( n + k ) / n + ( q - 1 ) / q + 1 + 1 / n × d 𝕄 s P ( u ) p , B , ω C 5 sup σ B Ω ( μ ( Ω ) ) - ( n + k ) / n + ( q - 1 ) / q + 1 + 1 / n × d 𝕄 s P ( u ) p , B , ω C d 𝕄 s P ( u ) p , Ω , ω . Theorem 13 has been completed.

Theorem 14.

Let u L s ( Ω , Λ l , μ ) , l = 1,2 , , n , 1 s < , be a differential form in Ω . Radon measure μ is defined by ω ( x ) d x = d μ , and ω ( x ) A ( α , β , γ , Ω ) . Then, there exists a constant C , independent of u , such that (42) 𝕄 s P ( u ) * , Ω , ω C 𝕄 s P ( u ) loc Lip k , Ω , ω , where α , β , γ are some positive constants.

Proof.

From the definition of the weighted BMO norm, we have (43) 𝕄 s P ( u ) * , Ω , ω = sup σ B Ω ( μ ( B ) ) - 1 𝕄 s P ( u ) - ( 𝕄 s P ( u ) ) B 1 , B , ω = sup σ B Ω ( μ ( B ) ) k / n ( μ ( B ) ) - ( n + k ) / n × 𝕄 s P ( u ) - ( 𝕄 s P ( u ) ) B 1 , B , ω sup σ B Ω ( μ ( Ω ) ) k / n ( μ ( B ) ) - ( n + k ) / n × 𝕄 s P ( u ) - ( 𝕄 s P ( u ) ) B 1 , B , ω ( μ ( Ω ) ) k / n sup σ B Ω ( μ ( B ) ) - ( n + k ) / n × 𝕄 s P ( u ) - ( 𝕄 s P ( u ) ) B 1 , B , ω C sup σ B Ω ( μ ( B ) ) - ( n + k ) / n × 𝕄 s P ( u ) - ( 𝕄 s P ( u ) ) B 1 , B , ω C 𝕄 s P ( u ) loc Li p k , Ω , ω . Theorem 14 has been completed.

Combining Theorems 13 and 14, we obtain the following corollary.

Corollary 15.

Let u L q ( Ω , Λ l , μ ) , l = 1,2 , , n , 1 s < q < , be a solution of the nonhomogeneous A-harmonic equation (4) in Ω . Radon measure μ is defined by ω ( x ) d x = d μ , and ω ( x ) A ( α , β , γ , Ω ) for some α > 1 , where 1 < p < , β = α q / ( α p - p - α q ) , γ = α q / p , and α p - p - α q > 0 . Then, there exists a constant C , independent of u , such that (44) 𝕄 s P ( u ) * , Ω , ω C d 𝕄 s P ( u ) p , Ω , ω .

4. Applications

In this section, we use the theorems we obtain to estimate the norms of Jacobian subdeterminant and the generalized solution of the quasilinear elliptic equation.

Example 16.

Let u = ( u 1 , , u n ) be a map from Ω to n . J ( x , u ) is the Jacobian determinant. Now choosing the subdeterminant of the Jacobian determinant, (45) J ( x j 1 , x j 2 ; u i 1 , u i 2 ) = | u i 1 x j 1 u i 1 x j 2 u i 2 x j 1 u i 2 x j 2 | . We know that J ( x j 1 , x j 2 ; u i 1 , u i 2 ) d x j 1 d x j 2 is a 2 -form. Let v = J ( x j 1 , x j 2 ; u i 1 , u i 2 ) d x j 1 d x j 2 . If v L s ( Ω , Λ 2 ) , 1 s < , from Theorems 9 and 14, we obtain the following results: (46) 𝕄 s P ( v ) * , Ω C 𝕄 s P ( v ) loc Li p k , Ω , 𝕄 s P ( v ) * , Ω , ω C 𝕄 s P ( v ) loc Li p k , Ω , ω , where C is a constant, 0 k 1 , 𝕄 s is the Hardy-Littlewood maximal operator, P is the potential operator, and ω is a A ( α , β , γ , Ω ) weight satisfying α > 1 , β = α q / ( α p - p - α q ) , γ = α q / p , and α p - p - α q > 0 .

Example 17.

Let u ( x ) = ( u 1 , u 2 , , u n ) : Ω n be a K -quasiregular mapping, K 1 . Then, (47) v = u i ( x ) ( i = 1,2 , , n ) or (48) v = log | u ( x ) | is a generalized solution of the following equation: (49) div A ( x , v ) = 0 , A = ( A 1 , A 2 , , A n ) . Here, A i ( x , η ) can be expressed as (50) A i ( x , η ) = η i ( i , j = 1 n α i , j ( x ) η i η j ) n / 2 . α i , j in the formula above are some functions, which can be expressed in terms of the differential matrix D u ( x ) , and satisfy (51) C 1 ( K ) | η | 2 i , j = 1 n α i , j η i η j C 2 ( K ) | η | 2 , where C 1 ( K ) , C 2 ( K ) > 0 . If we assume that 𝕄 s is the Hardy-Littlewood maximal operator, P is the potential operator, 0 k 1 , and ω is a A ( α , β , γ , Ω ) weight satisfying α > 1 , β = α q / ( α p - p - α q ) , γ = α q / p , and α p - p - α q > 0 , according to Theorem 13 and Corollary 15, we obtain the following inequalities: (52) 𝕄 s P ( v ) loc Li p k , Ω , ω C d 𝕄 s P ( v ) p , Ω , ω , 𝕄 s P ( v ) * , Ω , ω C 𝕄 s P ( v ) p , Ω , ω , where C is a constant.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Scott C. L p -theory of differential forms on manifolds Transactions of the American Mathematical Society 1995 347 6 2075 2096 10.2307/2154923 MR1297538 ZBL0849.58002 Ling Y. Bao G. Some local Poincaré inequalities for the composition of the sharp maximal operator and the Green's operator Computers & Mathematics with Applications 2012 63 3 720 727 10.1016/j.camwa.2011.11.036 MR2871672 ZBL1238.42006 Dai Z. Xing Y. Ding S. Wang Y. Inequalities for the composition of Green's operator and the potential operator Journal of Inequalities and Applications 2012 2012 article 271 13 10.1186/1029-242X-2012-271 Neugebauer C. J. Inserting A p -weights Proceedings of the American Mathematical Society 1983 87 4 644 648 10.2307/2043351 MR687633 Agarwal R. P. Ding S. Nolder C. Inequalities for Differential Forms 2009 New York, NY, USA Springer 10.1007/978-0-387-68417-8 MR2552910 Bao G. A r ( λ ) -weighted integral inequalities for A -harmonic tensors Journal of Mathematical Analysis and Applications 2000 247 2 466 477 10.1006/jmaa.2000.6851 MR1769089 Xing Y. A new weight class and Poincaré inequalities with the Radon measure Journal of Inequalities and Applications 2012 2012 article 32 11 10.1186/1029-242X-2012-32 MR2897464 ZBL1279.26038 Bi H. Weighted inequalities for potential operators on differential forms Journal of Inequalities and Applications 2010 2010 13 713625 10.1155/2010/713625 MR2600211 ZBL1193.47054 Xing Y. Ding S. Norms of the composition of the maximal and projection operators Nonlinear Analysis: Theory, Methods & Applications 2010 72 12 4614 4624 10.1016/j.na.2010.02.038 MR2639209 ZBL1189.35036 Iwaniec T. Lutoborski A. Integral estimates for null Lagrangians Archive for Rational Mechanics and Analysis 1993 125 1 25 79 10.1007/BF00411477 MR1241286 ZBL0793.58002 Ding S. Two-weight Caccioppoli inequalities for solutions of nonhomogeneous A -harmonic equations on Riemannian manifolds Proceedings of the American Mathematical Society 2004 132 8 2367 2375 10.1090/S0002-9939-04-07347-2 MR2052415