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Isoperimetric type inequalities for differential forms on manifolds. (English) Zbl 1093.58001

For a smooth oriented Riemannian n-manifold X without boundary, let ΦL p ( l X)kerd and ΨL r ( n-l X)kerd be two closed differential forms, where 1<p,r< satisfy Sobolev’s relation 1 p+1 r=1+1 n. The pair (Φ,Ψ) is called admissible pair if ΦΨ0 and lim t t 1 n H>t H(x)dx=0, where H=|Φ| p +|Ψ| r .

In this paper, the authors prove that, for almost every ball B(x,ρ) in X, an admissible pair (Φ,Ψ) satisfies

B ΦΨC(X) B |Φ| p d n-1 1 p B |Φ| s d n-1 1 s ,

where s=p(n-1) np-n+1 and ρR X . As applications they obtain an isoperimetric type inequality and the Hölder continuity property for solutions of Hodge systems.


MSC:
58A10Differential forms (global analysis)
58A14Hodge theory (global analysis)