Norm Comparison Estimates for the Composite Operator

Abstract

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 [1–3], 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 [4–7]. 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 ℝ^𝑛, 𝑛≥2, endowed with the usual Lebesgue measure denoted by |Ω|. 𝐵 and 𝜎𝐵 are concentric balls, with diam(𝜎𝐵)=𝜎diam(𝐵). 𝜔 denotes a weight defined by 𝜔∈𝐿_loc(ℝ^𝑛), and 𝜔>0 a.e.. The 𝑙-form, denoted by Λ^𝑙=Λ^𝑙(ℝ^𝑛), is a 𝑙-vector, spanned by exterior products 𝑒_𝐼=𝑒_𝑖1∧𝑒_𝑖2∧⋯∧𝑒_𝑖𝑙, for all ordered 𝑙-tuples 𝐼=(𝑖₁,𝑖₂,…,𝑖_𝑙), 1≤𝑖₁<𝑖₂<⋯<𝑖_𝑙≤𝑛. The 𝑙-form 𝑢(𝑥)=Σ_𝐼𝑢_𝐼(𝑥)𝑑𝑥_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 𝑑(𝑢(𝑥=))_{𝑛𝑘=11≤𝑖1<𝑖2<⋯<𝑖𝑙≤𝑛}𝜕𝑢_{𝑖1𝑖2⋯𝑖𝑙}(𝑥)𝜕𝑥_𝑘×𝑑𝑥_𝑘∧𝑑𝑥_𝑖1∧⋯∧𝑑𝑥_𝑖𝑙,𝑙=0,1,…,𝑛−1. (1) We define ⋆ by ⋆𝑢=sign(𝜋)𝑢_{𝑖1𝑖2⋯𝑖𝑙}(𝑥)𝑑𝑥_𝑗1∧⋯∧𝑑𝑥_{𝑗𝑛−𝑙}, (2) where 𝜋=(𝑖₁,…,𝑖_𝑙,𝑗₁,…,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 𝑑^⋆=(−1)^𝑛𝑙+1⋆𝑑⋆∶𝐷^′Ω,Λ^𝑙+1⟶𝐷^′Ω,Λ^𝑙,𝑙=0,1,…,𝑛−1. (3) The differential form we research here satisfies the nonhomogeneous 𝐴-harmonic equation 𝑑^⋆𝐴(𝑥,𝑑𝑢)=𝐵(𝑥,𝑑𝑢), (4) where 𝐴∶Ω×Λ^𝑙(ℝ^𝑛)→Λ^𝑙(ℝ^𝑛) and 𝐵∶Ω×Λ^𝑙(ℝ^𝑛)→Λ^𝑙−1(ℝ^𝑛) satisfy the conditions 𝐴(𝑥,𝜉)≤𝑎𝜉^𝑝−1,𝐴(𝑥,𝜉)⋅𝜉≥𝜉^𝑝,𝐵(𝑥,𝜉)≤𝑏𝜉^𝑝−1 (5) for almost every 𝑥∈Ω and all 𝜉∈∧^𝑙(ℝ^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∈𝐿¹_loc(Ω,Λ^𝑙),𝑙=0,1,…,𝑛, we write 𝑢∈locLip_𝑘(Ω,Λ^𝑙),0≤𝑘≤1, if ‖𝑢‖_{locLip𝑘,Ω}=sup_{𝜎𝑄⊂Ω}|𝑄|^{−(𝑛+𝑘)/𝑛}𝑢−𝑢_𝑄1,𝑄<∞ (6) for some 𝜎>1.

For 𝑢∈𝐿¹_loc(Ω,Λ^𝑙),𝑙=0,1,…,𝑛, we say 𝑢∈BMO(Ω,Λ^𝑙), if ‖𝑢‖_∗,Ω=sup_{𝜎𝑄⊂Ω}|𝑄|⁻¹𝑢−𝑢_𝑄1,𝑄<∞ (7) for some 𝜎>1.

Hardy-Littlewood maximal operator is defined by 𝕄_𝑠(𝑢)=𝕄_𝑠𝑢(𝑥)=sup_𝑟>01|𝐵(𝑥,𝑟)|_{𝐵(𝑥,𝑟)}𝑢𝑦^𝑠𝑑𝑦^1/𝑠, (8) 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 [8], which is defined as follows: 𝑃𝑢=𝑃𝑢(𝑥)=_𝐼Ω𝐾𝑥,𝑦𝑢_𝐼𝑦𝑑𝑦𝑑𝑥_𝐼, (9) 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 [9]). 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 𝕄_𝑠(𝑢)_𝑡,Ω≤𝐶‖𝑢‖_𝑡,Ω. (10)

Lemma 2 (see [8]). 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 ‖𝑃(𝑢)‖_𝑠,Ω≤𝐶‖𝑢‖_𝑠,Ω, (11) where 1<𝑠<∞.

The following lemma was established by Iwaniec and Lutoborski in [10].

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 𝑢_Ω𝑠,Ω≤𝐶‖𝑢‖_𝑠,Ω, (12) 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 ℝ^𝑛, then 𝑓𝑔_𝑠,Ω≤𝑓_𝑞,Ω𝑔_𝑝,Ω (13) for any Ω⊂ℝ^𝑛.

We also need the Caccioppoli inequality for differential form.

Lemma 5 (see [11]). 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 ‖𝑑𝑢‖_𝑠,𝐵≤𝐶diam(𝐵)⁻¹‖𝑢−𝑐‖_{𝑠,𝜎𝐵} (14) for all balls with 𝜎𝐵⊂Ω and any closed form 𝑐, where 1<𝑠<∞, and 𝜎>1.

The following Poincaré inequality appears in [5].

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 𝑢−𝑢_{𝐵𝑠,𝐵}≤𝐶|𝐵|diam(𝐵)‖𝑑𝑢‖_𝑠,𝐵 (15) 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 𝕄_𝑠𝑃(𝑢)−𝕄_𝑠𝑃(𝑢)_Ω𝑡,Ω≤𝐶‖𝑢‖_𝑡,Ω. (16)

Proof. From Lemma 3 and Minkowski inequality, we have 𝑢−𝑢_Ω𝑡,Ω=_Ω𝑢(𝑥)−𝑢_Ω(𝑥)^𝑡𝑑𝑥^1/𝑡≤_Ω|𝑢(𝑥)|^𝑡𝑑𝑥^1/𝑡+_Ω𝑢_Ω(𝑥)^𝑡𝑑𝑥^1/𝑡≤_Ω|𝑢(𝑥)|^𝑡𝑑𝑥^1/𝑡+𝐶_1Ω|𝑢(𝑥)|^𝑡𝑑𝑥^1/𝑡≤𝐶₂‖𝑢‖_𝑡,Ω. (17) Replacing 𝑢 with 𝕄_𝑠𝑃(𝑢) and combining Lemmas 1 and 2, we obtain 𝕄_𝑠𝑃(𝑢)−𝕄_𝑠𝑃(𝑢)_Ω𝑡,Ω≤𝐶₂𝕄_𝑠𝑃(𝑢)_𝑡,Ω≤𝐶₃‖𝑃(𝑢)‖_𝑡,Ω≤𝐶₄‖𝑢‖_𝑡,Ω. (18) Theorem 7 has been completed.

Then, we estimate the Lipschitz norm of 𝕄_𝑠𝑃(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 𝕄_𝑠𝑃(𝑢)_{loc𝐿𝑖𝑝𝑘,Ω}≤𝐶𝑑𝕄_𝑠𝑃(𝑢)_𝑡,Ω, (19) where 𝑘 is a constant with 0≤𝑘≤1.

Proof. Setting 𝑢=𝕄_𝑠𝑃(𝑢) in Lemma 6 we have 𝕄_𝑠𝑃(𝑢)−𝕄_𝑠𝑃(𝑢)_{𝐵𝑡,𝐵}≤𝐶₁|𝐵|diam(𝐵)𝑑𝕄_𝑠𝑃(𝑢)_𝑡,𝐵 (20) for all balls 𝐵 with 𝐵⊂Ω. Using Hölder inequality, we have 𝕄_𝑠𝑃(𝑢)−(𝕄_𝑠𝑃(𝑢))_𝐵1,𝐵≤_𝐵𝕄_𝑠𝑃(𝑢)−𝕄_𝑠𝑃(𝑢)_𝐵^𝑡𝑑𝑥^1/𝑡×_𝐵1^{𝑡/(𝑡−1)}𝑑𝑥^{(𝑡−1)/𝑡}≤|𝐵|^{(𝑡−1)/𝑡}𝕄_𝑠𝑃(𝑢)−𝕄_𝑠𝑃(𝑢)_{𝐵𝑡,𝐵}=|𝐵|^1−1/𝑡𝕄_𝑠𝑃(𝑢)−𝕄_𝑠𝑃(𝑢)_{𝐵𝑡,𝐵}≤|𝐵|^1−1/𝑡𝐶₁|𝐵|diam(𝐵)𝑑𝕄_𝑠𝑃(𝑢)_𝑡,𝐵≤𝐶₂|𝐵|^{2−1/𝑡+1/𝑛}𝑑𝕄_𝑠𝑃(𝑢)_𝑡,𝐵. (21) From the definition of Lipschitz norm and (21), it follows that 𝕄_𝑠𝑃(𝑢)_{locLip𝑘,Ω}=sup_{𝜎𝐵⊂Ω}|𝐵|^{−(𝑛+𝑘)/𝑛}𝕄_𝑠𝑃(𝑢)−𝕄_𝑠𝑃(𝑢)_𝐵1,𝐵=sup_{𝜎𝐵⊂Ω}|𝐵|^{−1−𝑘/𝑛}𝕄_𝑠𝑃(𝑢)−𝕄_𝑠𝑃(𝑢)_𝐵1,𝐵≤sup_{𝜎𝐵⊂Ω}|𝐵|^{−1−𝑘/𝑛}𝐶₂|𝐵|^{2−1/𝑡+1/𝑛}𝑑𝕄_𝑠𝑃(𝑢)_𝑡,𝐵=sup_{𝜎𝐵⊂Ω}𝐶₂|𝐵|^{1−𝑘/𝑛−1/𝑡+1/𝑛}𝑑𝕄_𝑠𝑃(𝑢)_𝑡,𝐵≤sup_{𝜎𝐵⊂Ω}𝐶₂|Ω|^{1−𝑘/𝑛−1/𝑡+1/𝑛}𝑑𝕄_𝑠𝑃(𝑢)_𝑡,𝐵≤𝐶₃sup_{𝜎𝐵⊂Ω}𝑑𝕄_𝑠𝑃(𝑢)_𝑡,𝐵≤𝐶₃𝑑𝕄_𝑠𝑃(𝑢)_𝑡,Ω. (22) 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 𝕄_𝑠𝑃(𝑢)_∗,Ω≤𝐶𝕄_𝑠𝑃(𝑢)_{loc𝐿𝑖𝑝𝑘,Ω}, (23) where 𝑘 is a constant with 0≤𝑘≤1.

Proof. From the definition of BMO norm, we have 𝕄_𝑠𝑃(𝑢)_∗,Ω=sup_{𝜎𝐵⊂Ω}|𝐵|⁻¹𝕄_𝑠𝑃(𝑢)−𝕄_𝑠𝑃(𝑢)_𝐵1,𝐵=sup_{𝜎𝐵⊂Ω}|𝐵|^𝑘/𝑛|𝐵|^{−(𝑛+𝑘)/𝑛}𝕄_𝑠𝑃(𝑢)−𝕄_𝑠𝑃(𝑢)_𝐵1,𝐵≤sup_{𝜎𝐵⊂Ω}|Ω|^𝑘/𝑛|𝐵|^{−(𝑛+𝑘)/𝑛}𝕄_𝑠𝑃(𝑢)−𝕄_𝑠𝑃(𝑢)_𝐵1,𝐵≤|Ω|^𝑘/𝑛sup_{𝜎𝐵⊂Ω}|𝐵|^{−(𝑛+𝑘)/𝑛}𝕄_𝑠𝑃(𝑢)−𝕄_𝑠𝑃(𝑢)_𝐵1,𝐵≤𝐶sup_{𝜎𝐵⊂Ω}|𝐵|^{−(𝑛+𝑘)/𝑛}𝕄_𝑠𝑃(𝑢)−𝕄_𝑠𝑃(𝑢)_𝐵1,𝐵≤𝐶𝕄_𝑠𝑃(𝑢)_{locLip𝑘,Ω}. (24) 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 𝕄_𝑠𝑃(𝑢)_{loc𝐿𝑖𝑝𝑘,Ω}≤𝐶𝕄_𝑠𝑃(𝑢)_∗,Ω, (25) where 𝑘 is a constant with 0≤𝑘≤1.

Proof. From (21), we obtain 𝕄_𝑠𝑃(𝑢)−𝕄_𝑠𝑃(𝑢)_𝐵1,𝐵≤𝐶₁|𝐵|^{2−1/𝑠+1/𝑛}𝑑𝕄_𝑠𝑃(𝑢)_𝑠,𝐵. (26) Based on Lemma 5, we get 𝑑𝕄_𝑠𝑃(𝑢)_𝑠,𝐵≤𝐶₂diam(𝐵)⁻¹𝕄_𝑠𝑃(𝑢)−𝕄_𝑠𝑃(𝑢)_{𝐵𝑠,𝜎𝐵}. (27) From the weak reverse Hölder inequality, it follows that 𝕄_𝑠𝑃(𝑢)−𝕄_𝑠𝑃(𝑢)_{𝐵𝑠,𝜎𝐵}≤𝐶₃|𝐵|^{(1−𝑠)/𝑠}𝕄_𝑠𝑃(𝑢)−𝕄_𝑠𝑃(𝑢)_{𝐵1,𝜎′𝐵}, (28) where 𝜎^′>𝜎>1. So, we get 𝕄_𝑠𝑃(𝑢)−𝕄_𝑠𝑃(𝑢)_𝐵1,𝐵≤𝐶₁|𝐵|^{2−1/𝑠+1/𝑛}𝑑𝕄_𝑠𝑃(𝑢)_𝑠,𝐵≤𝐶₄|𝐵|𝕄_𝑠𝑃(𝑢)−𝕄_𝑠𝑃(𝑢)_{𝐵1,𝜎′𝐵}. (29) Let 𝜎^′′>𝜎^′; we obtain 𝕄_𝑠𝑃(𝑢)_{locLip𝑘,Ω}=sup_{𝜎′′𝐵⊂Ω}|𝐵|^{−(𝑛+𝑘)/𝑛}𝕄_𝑠𝑃(𝑢)−𝕄_𝑠𝑃(𝑢)_𝐵1,𝐵≤sup_{𝜎′′𝐵⊂Ω}𝐶₄|𝐵|^{1−(𝑘/𝑛)}|𝐵|⁻¹𝕄_𝑠𝑃(𝑢)−𝕄_𝑠𝑃(𝑢)_{𝐵1,𝜎′𝐵}≤𝐶₅sup_{𝜎′′𝐵⊂Ω}𝐶₄|𝐵|⁻¹𝕄_𝑠𝑃(𝑢)−𝕄_𝑠𝑃(𝑢)_{𝐵1,𝜎′𝐵}≤𝐶₆𝕄_𝑠𝑃(𝑢)_∗,Ω. (30) 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 𝕄_𝑠𝑃(𝑢)_∗,Ω≤𝐶𝑑𝕄_𝑠𝑃(𝑢)_𝑡,Ω. (31)

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 [7]. 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⊂ℝ^𝑛 satisfies the 𝐴(𝛼,𝛽,𝛾,Ω)-condition for some positive constants 𝛼,𝛽,𝛾, if 𝜔(𝑥)>0 a.e., and sup_𝐵⊂Ω1|𝐵|_𝐵𝜔^𝛼1𝑑𝑥|𝐵|_𝐵𝜔^−𝛽𝑑𝑥^𝛾/𝛽<∞. (32) Let 𝑢∈𝐿¹_loc(Ω,Λ^𝑙,𝜔),𝑙=0,1,2,…,𝑛. We say 𝑢∈locLip_𝑘(Ω,Λ^𝑙,𝜔),0≤𝑘≤1, if ‖𝑢‖_{locLip𝑘,Ω,𝜔}=sup_{𝜎𝑄⊂Ω}𝜇(𝑄)^{−(𝑛+𝑘)/𝑛}𝑢−𝑢_{𝑄1,𝑄,𝜔}<∞ (33) for some 𝜎>1, where Ω is bounded in ℝ^𝑛, 𝜔 is a weight, and 𝜇 is Radon measure defined by 𝑑𝜇=𝜔(𝑥)𝑑𝑥.
Let 𝑢∈𝐿¹_loc(Ω,Λ^𝑙,𝜔),𝑙=0,1,2,…,𝑛. We say 𝑢∈BMO(Ω,Λ^𝑙,𝜔), if ‖𝑢‖_∗,Ω,𝜔=sup_{𝜎𝑄⊂Ω}𝜇(𝑄)⁻¹𝑢−𝑢_{𝑄1,𝑄,𝜔}<∞ (34) for some 𝜎>1, where Ω is bounded in ℝ^𝑛, 𝜔 is a weight, and 𝜇 is Radon measure defined by 𝑑𝜇=𝜔(𝑥)𝑑𝑥.

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)𝑑𝑥=𝑑𝜇, and 𝜔(𝑥)∈𝐴(𝛼,𝛽,𝛾,Ω) for some 𝛼>1, where 1<𝑝<∞, 𝛽=𝛼𝑞/(𝛼𝑝−𝑝−𝛼𝑞), 𝛾=𝛼𝑞/𝑝, and 𝛼𝑝−𝑝−𝛼𝑞>0. Then, there exists a constant 𝐶, independent of 𝑢, such that 𝕄_𝑠𝑃(𝑢)_{loc𝐿𝑖𝑝𝑘,Ω,𝜔}≤𝐶𝑑𝕄_𝑠𝑃(𝑢)_𝑝,Ω,𝜔. (35)

Proof. Using Hölder inequality with 1/𝑞=1/𝛼𝑞+(𝛼−1)/𝛼𝑞, we have _𝐵𝕄_𝑠𝑃(𝑢)−𝕄_𝑠𝑃(𝑢)_𝐵^𝑞𝜔(𝑥)𝑑𝑥^1/𝑞=_𝐵𝕄_𝑠𝑃(𝑢)−𝕄_𝑠𝑃(𝑢)_𝐵𝜔(𝑥)^1/𝑞𝑞𝑑𝑥^1/𝑞≤_𝐵𝕄_𝑠𝑃(𝑢)−𝕄_𝑠𝑃(𝑢)_𝐵^{𝛼𝑞/(𝛼−1)}𝑑𝑥^{(𝛼−1)/𝛼𝑞}×_𝐵𝜔(𝑥)^{1/𝑞𝛼𝑞}𝑑𝑥^1/𝛼𝑞≤𝐶₁|𝐵|diam(𝐵)_𝐵𝑑𝕄_𝑠𝑃(𝑢)^{𝛼𝑞/(𝛼−1)}𝑑𝑥^{(𝛼−1)/𝛼𝑞}×_𝐵𝜔(𝑥)^𝛼𝑑𝑥^1/𝛼𝑞. (36) Using Hölder inequality with (𝛼−1)/𝛼𝑞=(1/𝑝)+(𝛼𝑝−𝑝−𝛼𝑞)/𝛼𝑞𝑝, we obtain _𝐵𝑑𝕄_𝑠𝑃(𝑢)^{𝛼𝑞/(𝛼−1)}𝑑𝑥^{(𝛼−1)/𝛼𝑞}=_𝐵𝑑𝕄_𝑠𝑃(𝑢)𝜔(𝑥)^1/𝑝𝜔(𝑥)^{−1/𝑝𝛼𝑞/(𝛼−1)}𝑑𝑥^{(𝛼−1)/𝛼𝑞}≤_𝐵𝑑𝕄_𝑠𝑃(𝑢)𝜔(𝑥)^1/𝑝𝑝𝑑𝑥^1/𝑝×_𝐵𝜔(𝑥)^{−1/𝑝𝛼𝑞𝑝/(𝛼𝑝−𝑝−𝛼𝑞)}𝑑𝑥^{(𝛼𝑝−𝑝−𝛼𝑞)/𝛼𝑞𝑝}=_𝐵𝑑𝕄_𝑠𝑃(𝑢)^𝑝𝜔(𝑥)𝑑𝑥^1/𝑝×_𝐵1𝜔(𝑥)^{𝛼𝑞/(𝛼𝑝−𝑝−𝛼𝑞)}𝑑𝑥^{(𝛼𝑝−𝑝−𝛼𝑞)/𝛼𝑞𝑝}. (37) Since 𝜔(𝑥)∈𝐴(𝛼,𝛼𝑞/(𝛼𝑝−𝑝−𝛼𝑞),𝛼𝑞/𝑝,Ω), we get _𝐵𝜔(𝑥)^𝛼𝑑𝑥^1/𝛼𝑞_𝐵1𝜔(𝑥)^{𝛼𝑞/(𝛼𝑝−𝑝−𝛼𝑞)}𝑑𝑥^{(𝛼𝑝−𝑝−𝛼𝑞)/𝛼𝑞𝑝}=_𝐵𝜔(𝑥)^𝛼×𝑑𝑥_𝐵1𝜔(𝑥)^{𝛼𝑞/(𝛼𝑝−𝑝−𝛼𝑞)}𝑑𝑥^{(𝛼𝑝−𝑝−𝛼𝑞)/𝑝1/𝛼𝑞}=|𝐵|^2/𝛼𝑞×1|𝐵|(𝜔(𝑥))^𝛼×1𝑑𝑥|𝐵|×_𝐵1𝜔(𝑥)^{𝛼𝑞/(𝛼𝑝−𝑝−𝛼𝑞)}𝑑𝑥^{(𝛼𝑝−𝑝−𝛼𝑞)/𝑝1/𝛼𝑞}≤𝐶₂. (38) From (36), (37), and (38), we know that _𝐵𝕄_𝑠𝑃(𝑢)−𝕄_𝑠𝑃(𝑢)_𝐵^𝑞𝜔(𝑥)𝑑𝑥^1/𝑞≤𝐶₃|𝐵|diam(𝐵)_𝐵𝑑𝕄_𝑠𝑃(𝑢)^𝑝𝜔(𝑥)𝑑𝑥^1/𝑝. (39) So, we obtain 𝕄_𝑠𝑃(𝑢)−𝕄_𝑠𝑃(𝑢)_{𝐵1,𝐵,𝜔}=_𝐵𝕄_𝑠𝑃(𝑢)−𝕄_𝑠𝑃(𝑢)_𝐵≤𝑑𝜇_𝐵𝕄_𝑠𝑃(𝑢)−𝕄_𝑠𝑃(𝑢)_𝐵^𝑞𝜔(𝑥)𝑑𝑥^1/𝑞×_𝐵1^{𝑞/(𝑞−1)}𝑑𝜇^{(𝑞−1)/𝑞}=𝜇(𝐵)^{(𝑞−1)/𝑞}𝕄_𝑠𝑃(𝑢)−𝕄_𝑠𝑃(𝑢)_{𝐵𝑞,𝐵,𝜔}≤𝐶₄𝜇(𝐵)^{(𝑞−1)/𝑞}|𝐵|diam(𝐵)𝑑𝕄_𝑠𝑃(𝑢)_{𝑝,𝐵,𝜔}. (40) Based on the definition of the weighted Lipschitz norm and (40), we have 𝕄_𝑠𝑃(𝑢)_{locLip𝑘,Ω,𝜔}=sup_{𝜎𝐵⊂Ω}𝜇(𝐵)^{−(𝑛+𝑘)/𝑛}𝕄_𝑠𝑃(𝑢)−𝕄_𝑠𝑃(𝑢)_{𝐵1,𝐵,𝜔}≤𝐶₅sup_{𝜎𝐵⊂Ω}𝜇(𝐵)^{−(𝑛+𝑘)/𝑛+(𝑞−1)/𝑞+1+1/𝑛}×𝑑𝕄_𝑠𝑃(𝑢)_{𝑝,𝐵,𝜔}≤𝐶₅sup_{𝜎𝐵⊂Ω}𝜇(Ω)^{−(𝑛+𝑘)/𝑛+(𝑞−1)/𝑞+1+1/𝑛}×𝑑𝕄_𝑠𝑃(𝑢)_{𝑝,𝐵,𝜔}≤𝐶𝑑𝕄_𝑠𝑃(𝑢)_𝑝,Ω,𝜔. (41) 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)𝑑𝑥=𝑑𝜇, and 𝜔(𝑥)∈𝐴(𝛼,𝛽,𝛾,Ω). Then, there exists a constant 𝐶, independent of 𝑢, such that 𝕄_𝑠𝑃(𝑢)_∗,Ω,𝜔≤𝐶𝕄_𝑠𝑃(𝑢)_{loc𝐿𝑖𝑝𝑘,Ω,𝜔}, (42) where 𝛼,𝛽,𝛾 are some positive constants.

Proof. From the definition of the weighted BMO norm, we have 𝕄_𝑠𝑃(𝑢)_∗,Ω,𝜔=sup_{𝜎𝐵⊂Ω}𝜇(𝐵)⁻¹𝕄_𝑠𝑃(𝑢)−𝕄_𝑠𝑃(𝑢)_{𝐵1,𝐵,𝜔}=sup_{𝜎𝐵⊂Ω}𝜇(𝐵)^𝑘/𝑛𝜇(𝐵)^{−(𝑛+𝑘)/𝑛}×𝕄_𝑠𝑃(𝑢)−𝕄_𝑠𝑃(𝑢)_{𝐵1,𝐵,𝜔}≤sup_{𝜎𝐵⊂Ω}𝜇(Ω)^𝑘/𝑛𝜇(𝐵)^{−(𝑛+𝑘)/𝑛}×𝕄_𝑠𝑃(𝑢)−𝕄_𝑠𝑃(𝑢)_{𝐵1,𝐵,𝜔}≤𝜇(Ω)^𝑘/𝑛sup_{𝜎𝐵⊂Ω}𝜇(𝐵)^{−(𝑛+𝑘)/𝑛}×𝕄_𝑠𝑃(𝑢)−𝕄_𝑠𝑃(𝑢)_{𝐵1,𝐵,𝜔}≤𝐶sup_{𝜎𝐵⊂Ω}𝜇(𝐵)^{−(𝑛+𝑘)/𝑛}×𝕄_𝑠𝑃(𝑢)−𝕄_𝑠𝑃(𝑢)_{𝐵1,𝐵,𝜔}≤𝐶𝕄_𝑠𝑃(𝑢)_{locLip𝑘,Ω,𝜔}. (43) 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)𝑑𝑥=𝑑𝜇, and 𝜔(𝑥)∈𝐴(𝛼,𝛽,𝛾,Ω) for some 𝛼>1, where 1<𝑝<∞, 𝛽=𝛼𝑞/(𝛼𝑝−𝑝−𝛼𝑞), 𝛾=𝛼𝑞/𝑝, and 𝛼𝑝−𝑝−𝛼𝑞>0. Then, there exists a constant 𝐶, independent of 𝑢, such that 𝕄_𝑠𝑃(𝑢)_∗,Ω,𝜔≤𝐶𝑑𝕄_𝑠𝑃(𝑢)_𝑝,Ω,𝜔. (44)

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=(𝑢¹,…,𝑢^𝑛) be a map from Ω to ℝ^𝑛. 𝐽(𝑥,𝑢) is the Jacobian determinant. Now choosing the subdeterminant of the Jacobian determinant, 𝐽𝑥_𝑗1,𝑥_𝑗2;𝑢^𝑖1,𝑢^𝑖2=𝜕𝑢^𝑖1𝜕𝑥_𝑗1𝜕𝑢^𝑖1𝜕𝑥_𝑗2𝜕𝑢^𝑖2𝜕𝑥_𝑗1𝜕𝑢^𝑖2𝜕𝑥_𝑗2. (45) We know that 𝐽(𝑥_𝑗1,𝑥_𝑗2;𝑢^𝑖1,𝑢^𝑖2)𝑑𝑥_𝑗1∧𝑑𝑥_𝑗2 is a 2-form. Let 𝑣=𝐽(𝑥_𝑗1,𝑥_𝑗2;𝑢^𝑖1,𝑢^𝑖2)𝑑𝑥_𝑗1∧𝑑𝑥_𝑗2. If 𝑣∈𝐿^𝑠(Ω,Λ²), 1≤𝑠<∞, from Theorems 9 and 14, we obtain the following results: 𝕄_𝑠𝑃(𝑣)_∗,Ω≤𝐶𝕄_𝑠𝑃(𝑣)_{locLip𝑘,Ω},𝕄_𝑠𝑃(𝑣)_∗,Ω,𝜔≤𝐶𝕄_𝑠𝑃(𝑣)_{locLip𝑘,Ω,𝜔}, (46) 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=𝛼𝑞/(𝛼𝑝−𝑝−𝛼𝑞), 𝛾=𝛼𝑞/𝑝, and 𝛼𝑝−𝑝−𝛼𝑞>0.

Example 17. Let 𝑢(𝑥)=(𝑢¹,𝑢²,…,𝑢^𝑛)∶Ω→ℝ^𝑛 be a 𝐾-quasiregular mapping, 𝐾≥1. Then, 𝑣=𝑢^𝑖(𝑥)(𝑖=1,2,…,𝑛) (47) or 𝑣=log|𝑢(𝑥)| (48) is a generalized solution of the following equation: div𝐴(𝑥,∇𝑣)=0,𝐴=𝐴¹,𝐴²,…,𝐴^𝑛. (49) Here, 𝐴^𝑖(𝑥,𝜂) can be expressed as 𝐴^𝑖=𝜕𝑥,𝜂𝜕𝜂_𝑖^𝑛_𝑖,𝑗=1𝛼_𝑖,𝑗(𝑥)𝜂_𝑖𝜂_𝑗^𝑛/2. (50)𝛼_𝑖,𝑗 in the formula above are some functions, which can be expressed in terms of the differential matrix 𝐷𝑢(𝑥), and satisfy 𝐶₁(𝐾)𝜂²≤^𝑛_𝑖,𝑗=1𝛼_𝑖,𝑗𝜂_𝑖𝜂_𝑗≤𝐶₂(𝐾)𝜂², (51) where 𝐶₁(𝐾),𝐶₂(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=𝛼𝑞/(𝛼𝑝−𝑝−𝛼𝑞), 𝛾=𝛼𝑞/𝑝, and 𝛼𝑝−𝑝−𝛼𝑞>0, according to Theorem 13 and Corollary 15, we obtain the following inequalities: 𝕄_𝑠𝑃(𝑣)_{locLip𝑘,Ω,𝜔}≤𝐶𝑑𝕄_𝑠𝑃(𝑣)_𝑝,Ω,𝜔,𝕄_𝑠𝑃(𝑣)_∗,Ω,𝜔≤𝐶𝕄_𝑠𝑃(𝑣)_𝑝,Ω,𝜔, (52) where 𝐶 is a constant.

Conflict of Interests

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