Abstract
After introducing the definition of -weights, we establish the -weighted decomposition estimates and -weighted Caccioppoli-type estimates for -harmonic tensors. Furthermore, by Whitney covering lemma, we obtain the global results in domain . These results can be used to study the integrability of differential forms and to estimate the integrals for differential forms.
Highlights
Let e1, e2, . . . , en denote the standard orthogonal basis of Rn
Suppose that Λl Λl Rn is the linear space of all l-vectors, spanned by the exterior product eI ei1 ∧ei2 ∧· · ·∧eil corresponding to all ordered l-tuples I i1, i2, . . . , il, 1 ≤ i1 < i2 < · · · < il ≤ n
Throughout this paper, we always assume that Ω is an open subset of Rn
Summary
Let e1, e2, . . . , en denote the standard orthogonal basis of Rn. Suppose that Λl Λl Rn is the linear space of all l-vectors, spanned by the exterior product eI ei1 ∧ei2 ∧· · ·∧eil corresponding to all ordered l-tuples I i1, i2, . . . , il , 1 ≤ i1 < i2 < · · · < il ≤ n. Caccioppoli-type estimates have been widely studied and frequently used in analysis and related fields, including partial differential equations and the theory of elasticity. These inequalities provide upper bounds for the Lp-norm of ∇u if u is a function or du if u is a form with the Lp-norm of the differential form u. We first introduce the following definition of Aβr,λ-weights or the two-weight , and establish the local Aβr,λ-weighted Caccioppoli-type inequality for solutions to the homogeneous A-harmonic equation. 2.3 for any measurable set E ⊂ Rn. The following weak reverse Holder inequality plays an important role in founding the integral estimate of the nonhomogeneous and homogeneous A-harmonic tensor; see 4.
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