A Generalization of Exponential Class and its Applications
aa r X i v : . [ m a t h . A P ] D ec A Generalization of Exponential Class and ItsApplications
GAO Hongya LIU Chao TIAN Hong
College of Mathematics and Computer Science, Hebei University, Baoding, 071002, China
Abstract . A function space, L θ, ∞ ) (Ω), 0 ≤ θ < ∞ , is defined. It is proved that L θ, ∞ ) (Ω)is a Banach space which is a generalization of exponential class. An alternative definitionof L θ, ∞ ) (Ω) space is given. As an application, we obtain weak monotonicity property forvery weak solutions of A -harmonic equation with variable coefficients under some suitableconditions related to L θ, ∞ ) (Ω), which provides a generalization of a known result dueto Moscariello. A weighted space L θ, ∞ ) w (Ω) is also defined, and the boundedness forthe Hardy-Littlewood maximal operator M w and a Calder´on-Zygmund operator T withrespect to L θ, ∞ ) w (Ω) are obtained. AMS Subject Classification:
Keywords:
Exponential class, weak monotonicity, very weak solution, A -harmonic equation, Hardy-Littlewood maximal operator, Calder´on-Zygmund op-erator. § For 1 < p < ∞ and a bounded open subset Ω ⊂ R n , the grand Lebesgue space L p ) (Ω)consists of all functions f ( x ) ∈ T <ε ≤ p − L p − ε (Ω) such that k f k p ) , Ω = sup <ε ≤ p − (cid:18) ε − Z Ω | f | p − ε dx (cid:19) p − ε < ∞ , (1 . Corresponding author: GAO Hongya, E-mail: [email protected], TEL: 863125079658, FAX: 863125079638. here − R Ω = | Ω | R Ω stands for the integral mean over Ω. The grand Sobolev space W ,p )0 (Ω)consists of all functions u ∈ T <ε ≤ p − W ,p − ε (Ω) such that k u k W ,p )0 = sup <ε ≤ p − (cid:18) ε − Z Ω |∇ f | p − ε dx (cid:19) p − ε < ∞ . (1 . L p (Ω) and W ,p (Ω), respectively, were introducedin the paper [1] by Iwaniec and Sbordone in 1992 where they studied the integrability ofthe Jacobian under minimal hypotheses. For p = n in [2] imbedding theorems of Sobolevtype were proved for functions f ∈ W ,n )0 (Ω). The small Lebesgue space L ( p (Ω) wasfound by Fiorenza [3] in 2000 as the associate space of the grand Lebesgue space L p ) (Ω).Fiorenza and Karadzhov gave in [4] the following equivalent, explicit expressions for thenorms of the small and grand Lebesgue spaces, which depend only on the non-decreasingrearrangement (provided that the underlying measure space has measure 1): k f k L ( p ≈ Z (1 − ln t ) − p (cid:18)Z t [ f ∗ ( s )] p ds (cid:19) p dtt , < p < ∞ , k f k L p ) ≈ sup EXP (Ω), the exponential class, consists of all measurable functions f such that Z Ω e λ | f | dx < ∞ λ > 0. It is a Banach space under the norm k f k EXP = inf (cid:26) λ > Z Ω e | f | /λ dx ≤ (cid:27) . In this section, we define a space L θ, ∞ ) (Ω), 0 ≤ θ < ∞ , which is a generalization of EXP (Ω), and prove that it is a Banach space. Definition 2.1. For θ ≥ , the space L θ, ∞ ) (Ω) is defined by L θ, ∞ ) (Ω) = f ( x ) ∈ \ ≤ p< ∞ L p (Ω) : sup ≤ p< ∞ p θ (cid:18) − Z Ω | f ( x ) | p dx (cid:19) p < ∞ . (2 . L θ, ∞ ) (Ω) = g ( x ) ∈ \ ≤ p< ∞ L p (Ω) : lim sup p →∞ p θ (cid:18) − Z Ω | g ( x ) | p dx (cid:19) p < ∞ . (2 . ′ There are two special cases of L θ, ∞ ) (Ω) that are worth mentioning since they coincidewith two known spaces. Case 1 : θ = 0. In this case, L , ∞ ) (Ω) = f ( x ) ∈ \ ≤ p< ∞ L p (Ω) : sup ≤ p< ∞ (cid:18) − Z Ω | f ( x ) | p dx (cid:19) p < ∞ . From the fact (see [8, P12]) L ∞ (Ω) = f ∈ \ ≤ p< ∞ L p (Ω) : lim p →∞ k f k p < ∞ , we get L , ∞ ) (Ω) = L ∞ (Ω). Case 2 : θ = 1. The following proposition shows that L θ, ∞ (Ω) can be regarded as ageneralization of EXP (Ω). Proposition 2.1 L , ∞ ) (Ω) = EXP (Ω). Proof. In order to realize that a function in the L , ∞ ) (Ω) space is in EXP (Ω), it issufficient to read the last lines of [2]. The vice-versa is also true, see e.g. [9, Chap. VI,4xercise no. 17].It is clear that for any 0 ≤ θ < θ ′ ≤ ∞ and any q < ∞ , we have the inclusions L ∞ (Ω) ⊂ L θ, ∞ ) (Ω) ⊂ L θ ′ , ∞ ) (Ω) ⊂ L q (Ω) . (2 . θ > 0, then L θ, ∞ ) (Ω) is slightly larger than L ∞ (Ω). Theorem 2.1. For θ > , the space L ∞ (Ω) is a proper subspace of L θ, ∞ ) (Ω) .Proof. In the proof of Theorem 2.1 we always assume θ > 0. Let f ( x ) ∈ L ∞ (Ω), thenthere exists a constant M < ∞ , such that | f ( x ) | ≤ M , a.e. Ω. Thus,sup ≤ p< ∞ p θ (cid:18) − Z Ω | f ( x ) | p dx (cid:19) p ≤ sup ≤ p< ∞ Mp θ = M < ∞ , which implies f ( x ) ∈ L θ, ∞ ) (Ω).The following example shows that L ∞ (Ω) ⊂ L θ, ∞ ) (Ω) is a proper subset. Since wehave the inclusion (2.2), then it is no loss of generality to assume that θ ≤ 1. Considerthe function f ( x ) = ( − ln x ) θ defined in the open interval (0 , f ( x ) / ∈ L ∞ (0 , f ( x ) ∈ L θ, ∞ ) (0 , m a positive integer,integration by parts yields Z ( − ln x ) m dx = x ( − ln x ) m | − Z xd ( − ln x ) m = − lim x → + x ( − ln x ) m + m Z ( − ln x ) m − dx. (2 . x → + x ( − ln x ) m = lim x → + ( − ln x ) m x = lim x → + m ( − ln x ) m − x = · · · = m ! lim x → + x = 0 . Z ( − ln x ) m dx = m Z ( − ln x ) m − dx. By induction, Z f m ( x ) dx = m Z ( − ln x ) m − dx = · · · = m ! Z dx = m ! . (2 . p (cid:18) − Z Ω | f ( x ) | p dx (cid:19) p is non-decreasing, thus (2.4) yieldssup ≤ p< ∞ p θ (cid:18) − Z | f ( x ) | p dx (cid:19) p = sup ≤ p< ∞ " p (cid:18)Z ( − ln x ) pθ dx (cid:19) pθ θ ≤ sup ≤ p< ∞ " p (cid:18)Z ( − ln x ) [ pθ ]+1 dx (cid:19) pθ ]+1 θ = sup ≤ p< ∞ " ([ pθ ] + 1)! pθ ]+1 p θ ≤ sup ≤ p< ∞ (cid:20) [ pθ ] + 1 p (cid:21) θ ≤ , where we have used the assumption θ ≤ 1, and [ pθ ] is the integer part of pθ . The proofof Theorem 2.1 has been completed.For functions f ( x ) , f ( x ) ∈ L θ, ∞ ) (Ω) and α ∈ R, the addition f ( x ) + f ( x ) and themultiplication αf ( x ) are defined as usual. Theorem 2.2. L θ, ∞ ) (Ω) is a linear space on R.Proof. This theorem is easy to prove, we omit the details.For f ( x ) ∈ L θ, ∞ ) (Ω), we define k f k θ, ∞ ) , Ω = sup ≤ p< ∞ p θ (cid:18) − Z Ω | f ( x ) | p dx (cid:19) p . (2 . k · k θ, ∞ ) , Ω when there is no possibility of confusion. Theorem 2.3. k · k θ, ∞ ) is a norm.Proof. (1) It is obvious that k f k θ, ∞ ) ≥ k f k θ, ∞ ) = 0 if and only if f = 0 a.e. Ω;(2) For any f ( x ) , f ( x ) ∈ L θ, ∞ ) (Ω), Minkowski inequality in L p (Ω) yields k f + f k θ, ∞ ) = sup ≤ p< ∞ p θ (cid:18) − Z Ω | f + f | p dx (cid:19) p ≤ sup ≤ p< ∞ p θ "(cid:18) − Z Ω | f | p dx (cid:19) p + (cid:18) − Z Ω | f | p dx (cid:19) p ≤ sup ≤ p< ∞ p θ (cid:18) − Z Ω | f | p dx (cid:19) p + sup ≤ p< ∞ p θ (cid:18) − Z Ω | f | p dx (cid:19) p = k f k θ, ∞ ) + k f k θ, ∞ ) ;(3) For all λ ∈ R and all f ( x ) ∈ L θ, ∞ ) (Ω), it is obvious that k λf k θ, ∞ ) = | λ |k f k θ, ∞ ) . Theorem 2.4. (cid:0) L θ, ∞ ) (Ω) , k · k θ, ∞ ) (cid:1) is a Banach space.Proof. Suppose that { f n } ∞ n =1 ⊂ L θ, ∞ ) (Ω), and for any positive integer p , k f n + p − f n k θ, ∞ ) → , n → ∞ . (2 . σ -finite, then Ω = S ∞ m =1 Ω m with | Ω m | < ∞ . It is no loss of generality toassume that the Ω m s are disjoint. (2.4) implies that for any positive integer p , Z Ω m | f n + p ( x ) − f n ( x ) | dx → , n → ∞ . Thus, by the completeness of L (Ω m ), there exists f ( m ) ( x ) ∈ L (Ω m ), such that f n ( x ) → f ( m ) ( x ) , n → ∞ , in L (Ω m ) . (2 . m , there exists a subsequence { f ( m ) n ( x ) } of { f m − n ( x ) } ,7 f (0) n ( x ) } = { f n ( x ) } , such that f ( m ) n ( x ) → f ( m ) ( x ) , n → ∞ , a.e. x ∈ Ω m . If we let f ( x ) = f ( m ) ( x ) , x ∈ Ω m , m = 1 , , · · · , then f ( n ) n ( x ) → f ( x ) , n → ∞ , a.e. x ∈ Ω . It is no loss of generality to assume that the subsequence { f ( n ) n ( x ) } of { f n ( x ) } is itself,thus f n ( x ) → f ( x ) , n → ∞ , a.e. x ∈ Ω . We now prove f ( x ) ∈ L θ, ∞ ) (Ω) and k f n − f k θ, ∞ ) → 0, ( n → ∞ ). In fact, by (2.6), forany ε > 0, there exists N = N ( ε ), such that if n > N , thensup ≤ q< ∞ q θ (cid:18) − Z Ω | f n + p ( x ) − f n ( x ) | q dx (cid:19) q < ε. Let p → ∞ , one has sup ≤ q< ∞ q θ (cid:18) − Z Ω | f n ( x ) − f ( x ) | q dx (cid:19) q < ε, n > N. Hence f ( x ) ∈ L θ, ∞ ) (Ω), and k f n ( x ) − f ( x ) k θ, ∞ ) → n → ∞ . This completes the proofof Theorem 2.4. Definition 2.2. The grand Sobolev space W θ, ∞ ) (Ω) consists of all functions f belongingto T ≤ p< ∞ W , ∞ ) (Ω) and such that ∇ f ∈ L θ, ∞ ) (Ω) . That is, W θ, ∞ ) (Ω) = f ∈ \ ≤ p< ∞ W , ∞ ) (Ω) : ∇ f ∈ L θ, ∞ ) (Ω) . This definition will be used in Section 4.8 L θ, ∞ (Ω) In this section, we give an alternative definition of L θ, ∞ ) (Ω) in terms of weak Lebesguespaces. Let us first recall the definition of weak L p (0 < p < ∞ ) spaces, or the Marcinkiewiczspaces, L pweak (Ω), see [10, Chapter 1, Section 2], [11, Chapter 2, Section 5] or [12, Chapter2, Section 18]. Definition 3.1. Let < p < ∞ . We say that f ∈ L pweak (Ω) if and only if there exists apositive constant k = k ( f ) such that f ∗ ( t ) = |{ x ∈ Ω : | f ( x ) | > t }| ≤ kt p (3 . for every t > , where | E | is the n -dimensional Lebesgue measure of E ⊂ R n , and f ∗ ( t ) = |{ x ∈ Ω : | f ( x ) | > t }| denotes the distribution function of f . For p > 1, we recall that if f ∈ L pweak (Ω), then f ∈ L q (Ω) for every 1 ≤ q < p , and f ∈ L pweak (Ω) if and only if for every measurable set E ⊂ Ω, the following inequalityholds Z E | f ( x ) | dx ≤ c | E | p − p for some constant c > M p ( f ) = (cid:20) | Ω | sup t> t p f ∗ ( t ) (cid:21) p < ∞ . (3 . Z Ω | f ( x ) | s dx = s Z ∞ t s − f ∗ ( t ) dt < ∞ . (3 . efinition 3.2. For θ ≥ , the weak space L θ, ∞ weak (Ω) is defined by L θ, ∞ weak (Ω) = f ∈ \ ≤ p< ∞ L pweak (Ω) : sup ≤ p< ∞ M p ( f ) p θ < ∞ . (3 . L θ, ∞ weak (Ω) = L θ, ∞ ) (Ω), thus L θ, ∞ weak (Ω) can be re-garded as an alternative definition of the space L θ, ∞ ) (Ω). Theorem 3.1. L θ, ∞ weak (Ω) = L θ, ∞ ) (Ω) . Proof. We divided the proof into two steps. Step 1 L θ, ∞ weak (Ω) ⊂ L θ, ∞ ) (Ω).If 1 ≤ s < p , for each a > 0, one can split the integral in the right-hand side of (3.3)to obtain Z Ω | f | s dx = s Z a t s − f ∗ ( t ) dt + s Z ∞ a t s − f ∗ ( t ) dx ≤ | Ω | a s + sa s − p p − s | Ω | M pp ( f ) . The second integral has been estimated by the inequality f ∗ ( t ) ≤ | Ω | t − p M pp ( f ), which is adirect consequence of the definition of the constant M p ( f ) (see (3.2)). Setting a = M p ( f )we arrive at − Z Ω | f | s dx ≤ M sp ( f ) + sp − s M sp ( f ) = pp − s M p ( f ) . This implies 1 s θ (cid:18) − Z Ω | f | s dx (cid:19) s ≤ s θ (cid:18) pp − s (cid:19) s M p ( f ) . (3 . ≤ s< ∞ s θ (cid:18) − Z Ω | f | s dx (cid:19) s = max ( sup ≤ s< s θ (cid:18) − Z Ω | f | s dx (cid:19) s , sup ≤ s< ∞ s θ (cid:18) − Z Ω | f | s dx (cid:19) s ) ≤ max ( k f k , sup ≤ s = p − < ∞ s θ (cid:18) − Z Ω | f | s dx (cid:19) s ) ≤ max (cid:26) k f k , sup ≤ s< ∞ s θ ( s + 1) s M s +1 ( f ) (cid:27) ≤ max (cid:26) k f k , ≤ s< ∞ M s ( f ) s θ (cid:27) < ∞ , here we have used (3.6) and the definition of L ∞ weak (Ω). Step 2 L θ, ∞ ) (Ω) ⊂ L ∞ weak (Ω).Since for any t > t p f ∗ ( t ) = t p Z { x ∈ Ω: | f ( x ) | >t } dx ≤ Z { x ∈ Ω: | f ( x ) | >t } | f | p dx ≤ Z Ω | f | p dx, then sup t> t p f ∗ ( t ) ≤ Z Ω | f | p dx. This implies M p ( f ) = (cid:20) | Ω | sup t> t p f ∗ ( t ) (cid:21) p ≤ (cid:18) − Z Ω | f | p dx (cid:19) p . Thus sup ≤ p< ∞ M p ( f ) p θ ≤ sup ≤ p< ∞ p θ (cid:18) − Z Ω | f | p dx (cid:19) p < ∞ . The proof of Theorem 3.1 has been completed.11 In this section, we give an application of the space L θ, ∞ (Ω) to monotonicity propertyof very weak solutions of the A -harmonic equationdiv A ( x, ∇ u ( x )) = 0 , (4 . A : Ω × R n → R n be a mapping satisfying the following assumptions:(1) the mapping x 7→ A ( x, ξ ) is measurable for all ξ ∈ R n ,(2) the mapping ξ 7→ A ( x, ξ ) is continuous for a.e. x ∈ R n ,for all ξ ∈ R n , and a.e. x ∈ R n ,(3) hA ( x, ξ ) , ξ i ≥ γ ( x ) | ξ | p , (4) |A ( x, ξ ) | ≤ τ ( x ) | ξ | p − , where 1 < p < ∞ , 0 < γ ( x ) ≤ τ ( x ) < ∞ , a.e. Ω.Conditions (1) and (2) insure that the composed mapping x 7→ A ( x, g ( x )) is measur-able whenever g is measurable. The degenerate ellipticity of the equation is describedby condition (3). Finally, condition (4) guarantees that, for any 0 ≤ θ < ∞ and any ε > A ( x, ∇ u ) can be integrated for u ∈ W θ,p (Ω) against functions in W , p − ε − pε (Ω) withcompact support. Definition 4.1. A function u ∈ W ,rloc (Ω) , max { , p − } < r ≤ p , is called a very weaksolution of (4.1), if Z Ω hA ( x, ∇ u ( x )) , ∇ ϕ ( x ) i dx = 012 or all ϕ ∈ W , rr − p +1 (Ω) . A fruitful idea in dealing with the continuity properties of Sobolev functions is thenotion of monotonicity. In one dimension a function u : Ω → R is monotone if itsatisfies both a maximum and minimum principle on every subinterval. Equivalently, wehave the oscillation bounds osc I u ≤ osc ∂I u for every interval I ⊂ Ω. The definition ofmonotonicity in higher dimensions closely follows this observation.A continuous function u : Ω → R n defined in a domain Ω ⊂ R n is monotone ifosc B u ≤ osc ∂B u for every ball B ⊂ R n . This definition in fact goes back to Lebesgue [13] in 1907 wherehe first showed the relevance of the notion of monotonicity in the study of elliptic PDEsin the plane. In order to handle very weak solutions of A -harmonic equation, we needto extend this concept, dropping the assumption of continuity. The following definitioncan be found in [14], see also [6, 7]. Definition 4.2. A real-valued function u ∈ W , loc (Ω) is said to be weakly monotone if,for every ball B ⊂ Ω and all constants m ≤ M such that | M − u | − | u − m | + 2 u − m − M ∈ W , ( B ) , (4 . we have m ≤ u ( x ) ≤ M (4 . for almost every x ∈ B . For continuous functions (4.2) holds if and only if m ≤ u ( x ) ≤ M on ∂B . Then (4.3)says we want the same condition in B , that is the maximum and minimum principles.13anfredi’s paper [14] should be mentioned as the beginning of the systematic studyof weakly monotone functions. Koskela, Manfredi and Villamor obtained in [15] that A -harmonic functions are weakly monotone. In [16], the first author obtained a resultwhich states that very weak solutions u ∈ W ,p − εloc (Ω) of the A -harmonic equation areweakly monotone provided ε is small enough. The objective of this section is to extendthe operator A to spaces slightly larger than L p (Ω). Theorem 4.1. Let γ ( x ) > , a.e. Ω , τ ( x ) ∈ L θ , ∞ ) (Ω) . If u ∈ W θ ,p ) (Ω) is a very weaksolution to (4.1), then it is weakly monotone in Ω provided that θ + θ < .Proof. For any ball B ⊂ Ω and 0 < ε < 1, let ψ = ( u − M ) + − ( m − u ) + ∈ W ,p − ε ( B ) . It is obvious that ∇ ψ = , for m ≤ u ( x ) ≤ M, ∇ u, otherwise, say, on a set E ⊂ B. Consider the Hodge decomposition (see [6]), |∇ ψ | − pε ∇ ψ = ∇ ϕ + h. The following estimate holds k h k p − ε − pε ≤ Cε k∇ ψ k − pεp − ε . (4 . ϕ acting as a test function yields Z E hA ( x, ∇ u ) , |∇ u | − pε ∇ u i dx = Z E hA ( x, ∇ u ) , h i dx. (4 . Z E γ ( x ) |∇ u | p (1 − ε ) dx ≤ Z E hA ( x, ∇ u ) , |∇ u | − pε ∇ u i dx = Z E hA ( x, ∇ u ) , h i dx ≤ Z E τ ( x ) |∇ u | p − | h | dx ≤ k τ k p − ε ( p − ε k∇ u k p − p − ε k h k p − ε − pε ≤ Cε k τ k p − ε ( p − ε k∇ u k p (1 − ε ) p − ε = C | E | ε · ε − θ (1 − ε ) (cid:20) p − ε ( p − ε (cid:21) θ (cid:20) ( p − εp − ε (cid:21) θ (cid:18) − Z E | τ | p − ε ( p − ε dx (cid:19) ( p − εp − ε ×× ε θ (1 − ε ) (cid:18) − Z E |∇ u | p − ε (cid:19) p (1 − ε ) p − ε . (4 . τ ∈ L θ , ∞ ) (Ω) implieslim ε → + (cid:20) ( p − εp − ε (cid:21) θ (cid:18) − Z E | τ | p − ε ( p − ε dx (cid:19) ( p − εp − ε ≤ k τ k θ , ∞ ) < ∞ . (4 . u ∈ W θ ,p ) (Ω), thenlim ε → + ε θ (1 − ε ) (cid:18) − Z E |∇ u | p − ε (cid:19) p (1 − ε ) p − ε ≤ k∇ u k pθ ,p ) < ∞ , (4 . θ + θ < 1, we havelim ε → + ε · ε − θ (1 − ε ) (cid:20) p − ε ( p − ε (cid:21) θ = (cid:18) pp − (cid:19) θ lim ε → + ε − θ (1 − ε ) − θ = 0 . (4 . γ ( x ) > 0, a.e. Ω, wearrive at ∇ u = 0, a.e. E . This implies that ( u − M ) + − ( m − u ) + vanishes a.e. in B ,and thus ( u − M ) + − ( m − u ) + must be the zero function in B , completing the proof ofTheorem 4.1. Remark 4.1. We remark that the result in Theorem 4.1 is a generalization of a result ue to Moscariello, see [17, Corollary 4.1]. § A weight is a locally integrable function on R n which takes values in (0 , ∞ ) almosteverywhere. For a weight w and a measurable set E , we define w ( E ) = R E w ( x ) dx andthe Lebesgue measure of E by | E | . The weighted Lebesgue spaces with respect to themeasure w ( x ) dx are denoted by L pw with 0 < p < ∞ . Given a weight w , we say that w satisfies the doubling condition if there exists a constant C > Q ,we have w (2 Q ) ≤ Cw ( Q ), where 2 Q denotes the cube with the same center as Q whoseside length is 2 times that of Q . When w satisfies this condition, we denote w ∈ ∆ , forshort.A weight function w is in the Muckenhoupt class A p with 1 < p < ∞ if there exists C > Q (cid:18) − Z Q w ( x ) dx (cid:19) (cid:18) − Z Q w ( x ) − p ′ dx (cid:19) p − ≤ C, (5 . p + p ′ = 1. We define A ∞ = S
Lemma 5.1. If < p < ∞ and w ∈ ∆ , then the operator M w is bounded on L pw (Ω) . Theorem 5.1. The operator M w is bounded on L θ, ∞ ) w (Ω) for ≤ θ < ∞ and w ∈ ∆ .Proof. By Lemma 5.1, since for 1 < p < ∞ and w ∈ ∆ , the operator M w is bounded on L pw (Ω), then (cid:18)Z Ω | M w f ( x ) | p w ( x ) dx (cid:19) p ≤ C (cid:18)Z Ω | f ( x ) | p w ( x ) dx (cid:19) p . This implies k M w f k L θ, ∞ ) w (Ω) = sup
Lemma 5.3. If < p < ∞ and w ∈ A p , then a Calder´on-Zygmund operator T is boundedon L pw (Ω) . heorem 5.2. A Calder´on-Zygmund operator T is bounded on L θ, ∞ ) w (Ω) for ≤ θ < ∞ and w ∈ A ∞ .Proof. By w ∈ A ∞ and Lemma 5.2, one has w ∈ A q for some q ∈ (1 , ∞ ). For 1 < p Acknowledgement This study was funded by NSFC (10971224) and NSF of HebeiProvince (A2011201011). 18 eferences [1] T.Iwaniec, C.Sbordone. On the integrability of the Jacobian under minimal hypotheses, Arch. Ra-tional Mech. Anal., 119, 1992, 129-143.[2] N.Fusco, P.L.Lions, C.Sbordone. Sobolev imbeddings theorems in borderline cases, Proc. Amer.Math. Soc., 124(2), 1996, 561-565.[3] A.Fiorenza. Duality and reflexivity in grand Lebesgue spaces, Collect. Math. 51(2), 2000, 131-148.[4] A.Fiorenza, G.E.Karadzhov. Grand and small Lebesgue spaces and their analogous, J. Anal. Appl.,23(4), 2004, 657-681.[5] L.Greco, T.Iwaniec, C.Sbordone. Inverting the p -harmonic operator, Manuscripta Math., 92, 1997,249-258.[6] T.Iwaniec, G.Martin. Geometric function theorey and nonlinear analysis, Clarendon Press, Oxford,2001.[7] H.Y.Gao, H.M.Wang, G.Z.Gu. Weak monotonicity for the component functions of weak solutions ofBeltrami system, Acta Math. Sci., 29A(3), 2009, 651-655. (In Chinese).[8] M.D.Cheng, D.G.Deng, R.L.Long. Real analysis, Higher Education Press, 1993.[9] J.B.Garnett. Bounded analytic functions (Revised first edition), Springer, 2007.[10] S.Campanato. Sistemi ellittici in forma di divergenza, Quaderni Scuola Norm. Sup. Pisa, 1980.[11] E.Giusti, Metodi diretti nel calcolo delle variazioni, U.M.I., 1994.[12] A.Kufner, O.John, S.Fu˜cik. Function spaces, Noordhoff International Publishing, Leyden, 1977.[13] H.Lebesgue. Oeuvres scientifiques, Vol I-V, edited by Francois Chˆ a telet et Gustave Choquet. Instituede Math` e matiques de l’Universit` e , Geneva, 1973.[14] J.J.Manfredi. Weakly monotone functions, J. Geom. Anal., 3, 1994, 393-402.[15] P.Koskela, J.J.Manfredi, E.Villamor. Regularity theory and traces of A -harmonic functions, TransAmer. Math. Soc., 348(2), 1996, 755-766. 16] H.Y.Gao. Some properties of weakly quasiregular mappings, Acta Math. Sin., 5(1), 2002, 191-196.(In Chinese)[17] G.Moscariello. On the integrability of finite energy solutions for p -harmonic equations, NoDEA, 11,2004, 393-406.[18] J.Garcia-Cuerva, J.L.Rubio de Francia. Weighted norm inequalities and related topics, North Hol-land Mathematics Studies, Vol 116, North Holland, Amsterdam, 1985.[19] M.Q.Zhou. Lectures on Harmonic Analysis, Beijing University Press, 1999. (In Chinese)[20] J.Duoandikoetxea. Fourier Analysis, American Mathematical Society, Provindence, Rhode Island,29, 2000.[21] L.Grafakos. Classical and Modern Fourier Analysis, Pearson Education, Inc. Upper Saddle River,New Jersey, 2004.-harmonic equations, NoDEA, 11,2004, 393-406.[18] J.Garcia-Cuerva, J.L.Rubio de Francia. Weighted norm inequalities and related topics, North Hol-land Mathematics Studies, Vol 116, North Holland, Amsterdam, 1985.[19] M.Q.Zhou. Lectures on Harmonic Analysis, Beijing University Press, 1999. (In Chinese)[20] J.Duoandikoetxea. Fourier Analysis, American Mathematical Society, Provindence, Rhode Island,29, 2000.[21] L.Grafakos. Classical and Modern Fourier Analysis, Pearson Education, Inc. Upper Saddle River,New Jersey, 2004.