Poincare Inequalities in Punctured Domains

Abstract

The classic Poincare inequality bounds the $L^q$-norm of a function $f$ in a bounded domain $Ω\subset \R^n$ in terms of some $L^p$-norm of its gradient in $Ω$. We generalize this in two ways: In the first generalization we remove a set $Γ$ from $Ω$ and concentrate our attention on $Λ= Ω\setminus Γ$. This new domain might not even be connected and hence no Poincare 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 $Γ$ is arbitrarily small. A Poincare inequality does hold, however, if one makes the additional assumption that $f$ has a finite $L^p$ gradient norm on the whole of $Ω$, not just on $Λ$. The important point is that the Poincare inequality thus obtained bounds the $L^q$-norm of $f$ in terms of the $L^p$ gradient norm on $Λ$ (not $Ω$) plus an additional term that goes to zero as the volume of $Γ$ goes to zero. This error term depends on $Γ$ only through its volume. Apart from this additive error term, the constant in the inequality remains that of the `nice' domain $Ω$. 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 $\|(\nabla +i A)f\|_p $ over all $f$ with a given $\|f\|_q$ 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.