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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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