Abstract
The classic Poincaré inequality bounds the $L^q$-norm of a function $f$ in a bounded domain $\Omega \subset \mathrm{R}^n$ in terms of some $L^p$-norm of its gradient in $\Omega$. We generalize this in two ways: In the first generalization we remove a set $\Gamma$ from $\Omega$ and concentrate our attention on $\Lambda = \Omega \setminus \Gamma$. This new domain might not even be connected and hence no Poincaré inequality can generally hold for it, or if it does hold it might have a very bad constant. This is so even if the volume of $\Gamma$ is arbitrarily small. A Poincaré inequality does hold, however, if one makes the additional assumption that $f$ has a finite $L^p$ gradient norm on the whole of $\Omega$, not just on $\Lambda$. The important point is that the Poincaré inequality thus obtained bounds the $L^q$-norm of $f$ in terms of the $L^p$ gradient norm on $\Lambda$ (not $\Omega$) plus an additional term that goes to zero as the volume of $\Gamma$ goes to zero. This error term depends on $\Gamma$ only through its volume. Apart from this additive error term, the constant in the inequality remains that of the ‘nice’ domain $\Omega$. In the second generalization we are given a vector field $A$ and replace $\nabla $ by $\nabla +i A(x)$ (geometrically, a connection on a $U(1)$ bundle). Unlike the $A=0$ case, the infimum of $\Vert (\nabla +i A)f \Vert_p $ over all $f$ with a given $\Vert f \Vert_q$ is in general not zero. This permits an improvement of the inequality by the addition of a term whose sharp value we derive. We describe some open problems that arise from these generalizations.
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