-Weighted Inequalities with Lipschitz and BMO Norms
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Research Article Aλr 3 λ1 , λ2 , Ω-Weighted Inequalities with Lipschitz and BMO Norms Yuxia Tong,1 Juan Li,2 and Jiantao Gu1 1 2
College of Science, Hebei Polytechnic University, Tangshan 063009, China Department of Mathematics, Ningbo University, Ningbo 315211, China
Correspondence should be addressed to Yuxia Tong, [email protected] Received 29 December 2009; Revised 25 March 2010; Accepted 31 March 2010 Academic Editor: Shusen Ding Copyright q 2010 Yuxia Tong 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. We first define a new kind of Aλr 3 λ1 , λ2 , Ω two-weight, then obtain some two-weight integral inequalities with Lipschitz norm and BMO norm for Green’s operator applied to differential forms.
1. Introduction Green’s operator G is often applied to study the solutions of various differential equations and to define Poisson’s equation for differential forms. Green’s operator has been playing an important role in the study of PDEs. In many situations, the process to study solutions of PDEs involves estimating the various norms of the operators. Hence, we are motivated to establish some Lipschitz norm inequalities and BMO norm inequalities for Green’s operator in this paper. In the meanwhile, there have been generally studied about Ar Ω-weighted 1, 2 and Aλr Ω-weighted 3, 4 different inequalities and their properties. Results for more applications of the weight are given in 5, 6. The purpose of this paper is to derive the new weighted inequalities with the Lipschitz norm and BMO norm for Green’s operator applied to differential forms. We will introduce Aλr 3 λ1 , λ2 , Ω-weight, which can be considered as a further extension of the Aλr Ω-weight. We keep using the traditional notation. Let Ω be a connected open subset of Rn , let e1 , e2 , . . . , en be the standard unit basis of n R , and let l l Rn be the linear space of l-covectors, spanned by the exterior products eI ei1 ∧ ei2 ∧ · · · ∧ eil , corresponding to all ordered l-tuples I i1 , i2 , . . . , il , 1 ≤ i1 < i2 < · · · < il ≤ n, l 0, 1, . . . , n. We let R R1 . The Grassman algebra ⊕ l is a graded algebra with I I respect to the exterior products. For α α eI ∈ and β β eI ∈ , the inner product
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Journal of Inequalities and Applications
I I in is given by α, β α β with summation over all l-tuples I i1 , i2 , . . . , il and all integers l 0, 1, . . . , n. We define the Hodge star operator : → by the rule 1 e1 ∧ e2 ∧ · · · ∧ en and α ∧ β β ∧ α α, β1 for all α, β ∈ . The norm of α ∈ is given by the formula 0 R. The Hodge star is an isometric isomorphism on with |α|2 α, α α ∧ α ∈ n−l l → and −1ln−l : l → l . : Balls are denoted by B and ρB is the ball with the same center as B and with diamρB ρ diamB. We do not distinguish balls from cubes throughout this pape
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