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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 \(\Phi\in L^ p(\bigwedge^ lX)\cap\ker d\) and \(\Psi\in {L^ r(\bigwedge^{n-l}X)\cap\ker d}\) be two closed differential forms, where \(1<p,r<\infty\) satisfy Sobolev’s relation \(\frac{1}{p}+\frac{1}{r}=1+\frac{1}{n}\). The pair \((\Phi,\Psi)\) is called admissible pair if \(\Phi\wedge \Psi\geq 0\) and \(\lim_{t\to\infty}t^{\frac{1}{n}}\int_{H>t}H(x)\,dx=0\), where \(H=| \Phi| ^p+| \Psi| ^r\).
In this paper, the authors prove that, for almost every ball \(B(x,\rho)\) in \(X\), an admissible pair \((\Phi,\Psi)\) satisfies \[ \int_ B\Phi\wedge\Psi\leq C(X)\left(\int_{\partial B}| \Phi| ^ p \,d\mathcal{H}^{n-1}\right)^{\frac{1}{p}}\left(\int_{\partial B}| \Phi| ^ s \,d\mathcal{H}^{n-1}\right)^{\frac{1}{s}}, \] where \(s=\frac{p(n-1)}{np-n+1}\) and \(\rho\leq R_ X\). As applications they obtain an isoperimetric type inequality and the Hölder continuity property for solutions of Hodge systems.

MSC:
58A10 Differential forms in global analysis
58A14 Hodge theory in global analysis
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