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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 $\Phi\in L\sp p(\bigwedge\sp lX)\cap\ker d$ and $\Psi\in {L\sp r(\bigwedge\sp{n-l}X)\cap\ker d}$ 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 $\Phi\wedge \Psi\ge 0$ and $\lim\sb{t\to\infty}t\sp{\frac{1}{n}}\int\sb{H>t}H(x)\,dx=0$, where $H=\vert \Phi\vert ^p+\vert \Psi\vert ^r$. In this paper, the authors prove that, for almost every ball $B(x,\rho)$ in $X$, an admissible pair $(\Phi,\Psi)$ satisfies $$\int\sb B\Phi\wedge\Psi\le C(X)\left(\int\sb{\partial B}\vert \Phi\vert \sp p \,d\Cal{H}\sp{n-1}\right)\sp{\frac{1}{p}}\left(\int\sb{\partial B}\vert \Phi\vert \sp s \,d\Cal{H}\sp{n-1}\right)\sp{\frac{1}{s}},$$ where $s=\frac{p(n-1)}{np-n+1}$ and $\rho\le R\sb X$. As applications they obtain an isoperimetric type inequality and the Hölder continuity property for solutions of Hodge systems.

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