Research Article Weighted Inequalities for Potential Operators with Lipschitz and BMO Norms

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1 Hindawi Publishing Corporation Advances in Difference quations Volume 0, Article ID , pages doi:0.55/0/ Research Article Weighted Inequalities for Potential Operators with Lipschitz and MO Norms Yuxia Tong and Jiantao Gu College of Science, Hebei United University, Tangshan , China Correspondence should be addressed to Yuxia Tong, Received January 0; Accepted 7 March 0 Academic ditor: Jin Liang Copyright q 0 Y. Tong and J. Gu. 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. Some Lipschitz norm and MO norm inequalities for potential operator to the versions of differential forms are obtained, and some properties of a new kind of A λ3 r λ,λ, Ω weight are derived.. Introduction In many situations, the process to study solutions of PDs involves estimating the various norms of the operators. Hence, we are motivated to establish some Lipschitz norm inequalities and MO norm inequalities for potential operator to the versions of differential forms. We keep using the traditional notation. Let Ω be a connected open subset of R n,lete,e,...,e n be the standard unit basis of R n,andlet l l R n be the linear space of l-covectors, spanned by the exterior products e I e i e i e il, corresponding to all ordered l-tuples I i,i,...,i l, i <i < < i l n, l 0,,...,n.WeletR R. The Grassman algebra l is a graded algebra with respect to the exterior products. For α α I e I and β β I e I, the inner product in is given by α, β α I β I with summation over all l-tuples I i,i,...,i l and all integers l 0,,...,n. We define the Hodge star operator : by the rule e e e n and α β β α α, β for all α, β. The norm of α is given by the formula α α, α α α 0 R. The Hodge star is an isometric isomorphism on with : l n l and l n l : l l. alls are denoted by, andρ is the ball with the same center as and with diam ρ ρ diam. We do not distinguish balls from cubes throughout this paper. The n-dimensional Lebesgue measure of a set R n is denoted by.

2 AdvancesinDifference quations We call w x a weight if w L loc Rn and that is, w>0. For 0 <p< andaweight w x, we denote the weighted L p -norm of a measurable function f over by /p fp,,w α f x p w α,. where α is a real number. Differential forms are important generalizations of real functions and distributions; note that a 0-form is the usual function in R n.adifferential l-form ω on Ω is a Schwartz distribution on Ω with values in l R n. We use D Ω, l to denote the space of all differential l-forms ω x I ω I x I ω i i,...,i l x i i il.wewrite L p Ω, l for the l-forms with ω I L p Ω, R for all ordered l-tuples I. Thus,L p Ω, l is a anach space with norm ω p,ω Ω /p ω x p Ω ( ωi x p/ /p.. For ω D Ω, l,thevector-valueddifferential form ω ω/ x,..., ω/ x n consists of differential forms ω/ x i D Ω, l, where the partial differentiations are applied to the coefficients of ω. Asusual,W,p Ω, l is used to denote the Sobolev space of l-forms, which equals L p Ω, l L p Ω, l with norm ω W,p Ω, l ω W,p Ω, l diam Ω ω p,ω ω p,ω..3 The notations W,p,p Ω, R and W loc loc Ω, l are self-explanatory. For 0 <p< andaweight w x, the weighted norm of ω W,p Ω, l over Ω is denoted by ω W,p Ω, l,w α ω W,p Ω, l,w α diam Ω ω p,ω,w α ω p,ω,w α,.4 where α is a real number. We denote the exterior derivative by d : D Ω, l D Ω, l for l 0,,...,n. Its formal adjoint operator d : D Ω, l D Ω, l is given by d nl d on D Ω, l,l 0,,...,n. Let u L loc Ω, l, l 0,,...,n.Wewriteu loc Lip k Ω, l, 0 k if u loc Lipk,Ω sup Q n k /n u uq,q <, σq Ω.5 for some σ. Further, we write Lip k Ω, l for those forms whose coefficients are in the usual Lipschitz space with exponent k and write u Lipk,Ω for this norm. Similarly, for u L loc Ω, l, l 0,,...,n,wewriteu MO Ω, l if u,ω sup Q u uq,q <, σq Ω.6 for some σ. When u is a 0-form,.6 reduces to the classical definition of MO Ω.

3 Advances in Difference quations 3 ased on the above results, we discuss the weighted Lipschitz and MO norms. For u L loc Ω, l,w α, l 0,,...,n,wewriteu loc Lip k Ω, l,w α, 0 k if u loc Lipk,Ω,w ( n k /n α sup μ Q u u,q,w Q α <,.7 σq Ω for some σ >, where Ω is a bounded domain, the Radon measure μ is defined by dμ w x α, w is a weight and α is a real number. For convenience, we will write the following simple notation loc Lip k Ω, l for loc Lip k Ω, l,w α. Similarly, for u L loc Ω, l,w α, l 0,,...,n,wewriteu MO Ω, l,w α if u,ω,w α ( sup μ Q u uq,q,w α <,.8 σq Ω for some σ>, where the Radon measure μ is defined by dμ w x α, w is a weight, and α is a real number. Again, we use MO Ω, l to replace MO Ω, l,w α whenever it is clear that the integral is weighted. From, ifω is a differential form defined in a bounded, convex domain M, then there is a decomposition ω d Tω T dω,.9 where T is called a homotopy operator. Furthermore, we can define the k-form ω M D M, k by ω M M ω ( y dy, k 0, ω M d Tω, k,,...,n,.0 M for all ω L p M, k, p<. For any differential k-form ω x, we define the potential operator P by Pω x I K ( x, y ω I ( y dy I,. where the kernel K x, y is a nonnegative measurable function defined for x / y, andthe summation is over all ordered k-tuples I. It is easy to find that the case k 0 reduces to the usual potential operator. That is, Pf x K ( x, y f ( y dy,. where f x is a function defined on R n. Associated with P, the functional ϕ is defined as ϕ sup K ( x, y, x,y, x y Cr.3

4 4 AdvancesinDifference quations where C is some sufficiently small constant and is a ball with radius r. Throughout this paper, we always suppose that ϕ satisfies the following conditions: there exists C ϕ such that ϕ C ϕ ϕ for all balls,.4 and there exists ε>0suchthat ( r ε ϕ μ C ϕ ϕ μ for all balls..5 r On the potential operator P and the functional ϕ,see for details. The nonlinear elliptic partial differential equation d A x, du 0 is called the homogeneous A-harmonic equation or the A-harmonic equation, and the differential equation d A x, du x, du.6 is called the nonhomogeneous A-harmonic equation for differential forms, where A : Ω l R n l R n and : Ω l R n l R n satisfy the conditions A x, ξ a ξ p, A x, ξ,ξ ξ p, x, ξ b ξ p,.7 for almost every x Ω and all ξ l R n.herea, b > 0 are constants and <p< is a fixed exponent associated with.6. A solution to.6 is an element of the Sobolev space W,p loc Ω, l such that A x, du dϕ x, du ϕ 0,.8 Ω for all ϕ W,p loc Ω, l with compact support. When u is a 0-form, that is, u is a function,.6 is equivalent to div A x, u x, u..9 Lots of results have been obtained in recent years about different versions of the A-harmonic equation, see The stimate for Potential Operators with Lipschitz Norm and MO Norm In this section, we give the estimate for potential operators with Lipschitz norm and MO norm applied to differential forms. The following strong type p, p inequality for potential operators appears in 6.

5 Advances in Difference quations 5 Lemma. see 6. Let u D, k, k 0,,...,n, beadifferential form defined in a bounded, convex domain, andletu I be coefficient of u with supp u I for all ordered k-tuples I. Assume that <p< and P is the potential operator with k x, y ϕ ε x y for any ε>0, then there exists a constant C, independent of u,suchthat P u P u p, C diam u p,.. We will establish the following estimate for potential operators. Theorem.. Let u D, k,k 0,,...,n, beadifferential form defined in a bounded, convex domain, andletu I be coefficient of u with supp u I for all ordered k-tuples I. Assume that <p< and P is the potential operator with k x, y ϕ ε x y for any ε>0, then there exists a constant C, independent of ω, suchthat P u, P u loc Lipk, C u p,.. Proof. y the definition of the Lipschitz norm,., andhölder s inequality with /p p /p, wehave P u loc Lipk, ( n k /n P u sup μ P u, σ ( /p n k /n sup μ P u P u p σ p /p p/ p ( n k /n p /p P u sup μ P u p, σ.3 ( n k /n p /p sup μ C diam u p, σ C n k /n p /p /n u p, C u p,, since /p k/n /n > 0and Ω <,whereσ is a constant and σ Ω. y the definition of the MO norm, we have ( P u P u, sup μ P u, σ ( k/n ( n k /n P u sup μ μ P u, σ ( n k /n P u C sup μ P u, σ.4 C P u loc Lipk,. We have completed the proof of Theorem..

6 6 AdvancesinDifference quations 3. The A λ 3 r λ,λ, Ω Weight In this section, we introduce the A λ 3 r λ,λ, Ω weight appeared in 7. Definition 3.. Let w x,w x be two locally integrable nonnegative functions in R n and assume that 0 <w,w < almost everywhere. We say that w x,w x belongs to the A λ 3 r λ,λ, class, <r< and 0 <λ,λ,λ 3 <,orthat w x,w x is an A λ 3 r λ,λ, weight, write w,w A λ 3 r λ,λ, or w,w A λ 3 r λ,λ when it will not cause any confusion, if ( ( ( sup w λ λ / r λ3 r < 3. w for all balls R n. The following results show that the A λ 3 r λ,λ weights have the properties similar to those of the A r weights. Theorem 3.. If <r<s<, thena λ 3 r λ,λ A λ 3 s λ,λ. Proof. Let w,w A λ 3 r λ,λ.since<r<s<, byhölder s inequality, ( w λ / s λ3 s ( λ / r λ3 r λ3 s r λ/ s r w λ 3 s r ( w λ / r λ3 r λ 3 s ( λ / r λ3 r, λ 3 r w 3. so that ( ( λ / s λ3 s ( ( λ / r λ3 r. 3.3 w w Thus, we find that ( ( ( sup w λ λ / s λ3 s w ( ( ( sup w λ λ / r λ3 r, w 3.4

7 Advances in Difference quations 7 for all balls R n since w,w A λ 3 r λ,λ. Therefore, w,w A λ 3 s λ,λ, and hence A λ 3 r λ,λ A λ 3 s λ,λ. Theorem 3.3. If w,w A λ 3 r λ,λ,λ, λ,λ 3 > 0 and the measures μ, ν are defined by dμ w x, dν w x λ, then λ 3r C r, λ λ λ 3 r,λ,λ 3,w,w μ, 3.5 μ λ λ 3 where is a ball in R n and is a measurable subset of. Proof. y Hölder s inequality, we have w λ /r w λ /r w λ /r ( μ /r r /r w λ / r w λ / r r /r. 3.6 This implies r r μ w λ / r. 3.7 Note that λ, by Hölder s inequality again, we have ( /λ w w λ, 3.8 so that Hence, we obtain w ( /λ w λ μ μ. 3.9 μ λ λ w λ. 3.0 Since w,w A λ 3 r λ,λ,thereexistsaconstantc r, λ,λ,λ 3,w,w such that ( ( ( w λ λ / r λ3 r C r, λ,λ,λ 3,w,w. w 3.

8 8 AdvancesinDifference quations Combining 3.7, 3.0, and 3.,wededucethat λ 3r μ λ μ λ 3 λ ( μ λ 3 λ λ 3 r λ3 w λ r / r w λ λ3 r ( w λ / r w λ 3. C r, λ,λ,λ 3,w,w μ λ 3 λ λ 3 r. Hence, λ 3r C r, λ λ λ 3 r,λ,λ 3,w,w μ. 3.3 μ λ λ 3 The desired result is obtained. If we choose λ λ λ 3 andw w w in Theorem 3.3, we will obtain r r C r, w μ μ, 3.4 which is called the strong doubling property of A r weights; see The Weighted Inequality for Potential Operators In this section, we are devoted to develop some two-weight norm inequalities for potential operator P to the versions of differential forms. We need the following lemmas. Lemma 4. see 9. If w A r Ω, then there exist constants β> and C, independent of w,such that w β, C β /β w,, 4. for all balls R n. Lemma 4.. Let 0 <α<, 0 <β<,ands α β.iff and g are measurable functions on R n,then fg s, f α, g β,, 4. for any R n.

9 Advances in Difference quations 9 Lemma 4.3 see 0. Let ω D, k,k 0,,...,n be a solution of the nonhomogeneous A-harmonic equation in, ρ > and 0 <s, t<, thenthereexists a constant C, independent of ω, such that ω s, C t s /st ω t,σq, 4.3 for all with ρ. Theorem 4.4. Let u D, k,υ, k 0,,,...,n, be a solution of the nonhomogeneous A- harmonic equation.6 in a bounded domain and P is the potential operator with k x, y ϕ ε x y for any ε>0, where the Radon measures μ and υ are defined by dμ w αλ x, dυ w αλ λ 3 /s x. Assume that w λ x A r Ω and w x,w x A λ 3 r λ,λ, Ω for some r>, 0 <λ,λ,λ 3 < with w x ε>0for any x Ω, then there exists a constant C, independent of u,suchthat P u,,w αλ C u αλ,ω,w λ 3 /s, 4.4 where α is a constant with 0 <α<. Proof. Since w λ A r Ω, usinglemma 4., thereexistconstantsβ>andc > 0, such that w λ C β /β λ w, 4.5 β,, for any ball R n. Since /s s /s,bylemma 4.,wehave P u P u,,w αλ P u P u w αλ /s s /s P u P u s w αλ w αλ μ s /s P u P u αλ s,,w. 4.6 Choose t s/ α/β where 0 <α<, β>, then <s<tand αt/ t s β. Since /s /t t s /st,bylemma 4. and 4.5,wehave P u P u s,,w αλ ( P u P u w αλ /s s /s /t α/ βs P u P u t w λ β P u P u t, w λ α/s β, P u P u t, C β α/ βs w λ α/s., 4.7

10 0 Advances in Difference quations From Lemma., wehave P u P u t, C 3 diam u t,. 4.8 Applying Lemma 4.3 the weak reverse Hölder inequality for the solutions of the nonhomogeneous A-harmonic equation, weobtain u t, C 4 m t /mt u m,σ, 4.9 where σ is a constant and σ Ω. Choosing m s/ αλ 3 r s, thenm< <s.using Hölder s inequality with /m / αλ 3 r /s,wehave u m,σ σ ( /m u w αλ λ 3 /s w αλ λ 3 /s m ( u w αλ λ 3 /s σ σ ( λ αλ 3 /s u,σ,w αλ λ 3 /s w w / r,σ λ / r αλ3 r /s. 4.0 Since w,w A λ 3 r λ,λ, Ω, then w λ α/s, ( λ αλ 3 /s w / r,σ ( w λ σ σ w λ / r λ3 r α/s ( ( ( σ λ 3 r w λ λ / r λ3 r α/s σ σ σ σ w 4. C 5 σ αλ 3 r /s α/s C 6 αλ 3 r /s α/s. Since m t /mt αλ 3 r /s α/s s /s β α/ βs 0, combining with , we have P u P u,,w αλ μ s /s C β α/ βs C 3 diam C 4 m t /mt C 6 αλ 3 r /s α/s u,σ,w αλ λ 3 /s C 7 diam u,σ,w αλ λ 3 /s. 4.

11 Advances in Difference quations From the definition of the MO norm, we obtain P u,,w αλ sup P u P u αλ,,w σ sup C 7 diam u,σ,w αλ λ 3 /s σ 4.3 C 8 u,σ,w αλ λ 3 /s, for all balls with σ >σ and σ Ω. We have completed the proof of Theorem 4.4. Acknowledgments The authors are supported by NSF of Hebei Province A and Scientific Research Fund of Zhejiang Provincial ducation Department Y References R. P. Agarwal, S. Ding, and C. Nolder, Inequalities for Differential Forms, Springer, New York, NY, USA, 009. J. M. Martell, Fractional integrals, potential operators and two-weight, weak type norm inequalities on spaces of homogeneous type, Journal of Mathematical Analysis and Applications, vol. 94, no., pp. 3 36, H. Gao, Weighted integral inequalities for conjugate A-harmonic tensors, Journal of Mathematical Analysis and Applications, vol. 8, no., pp , H. Gao, J. Qiao, and Y. Chu, Local regularity and local boundedness results for very weak solutions of obstacle problems, Journal of Inequalities and Applications, vol. 00, Article ID , pages, H. Gao, J. Qiao, Y. Wang, and Y. Chu, Local regularity results for minima of anisotropic functionals and solutions of anisotropic equations, Journal of Inequalities and Applications, vol. 008, Article ID , pages, H. i, Weighted inequalities for potential operators on differential forms, Journal of Inequalities and Applications, vol. 00, Article ID 7365, 3 pages, Y.Tong,J.Li,andJ.Gu, A λ3 r λ,λ, Ω -weighted inequalities with Lipschitz and MO norms, Journal of Inequalities and Applications, vol. 00, Article ID 7365, 3 pages, J. Heinonen, T. Kilpeläinen, and O. Martio, Nonlinear Potential Theory of Degenerate lliptic quations, Oxford Mathematical Monographs, The Clarendon Press Oxford University Press, New York, NY, USA, J.. Garnett, ounded Analytic Functions,vol.96ofPure and Applied Mathematics, Academic Press, New York, NY, USA, C. A. Nolder, Hardy-Littlewood theorems for A-harmonic tensors, Illinois Journal of Mathematics, vol. 43, no. 4, pp , 999.