It is well known that given a function $f \in L^p(B\_R)$ such that $|\{x \in B\_R: f(x) = 0\}|>0$, the following Poincaré inequality holds:
$$ \int\_{B\_R} \left(\frac{|f|}{R}\right)^p \ dx \leq c \int\_{B\_R} |\nabla f|^p \ dx \, .$$


My question is: does something like this hold on annular regions? More specifically, given $0<r<t<\infty$, and let $f$ be such that $|\{x \in B\_t \setminus B\_r: f(x) = 0\}| > 0$, then does the following inequality hold?
$$ \int\_{B\_t\setminus B\_r} \left(\frac{|f|}{t-r}\right)^p \ dx \leq c \int\_{B\_t \setminus B\_r} |\nabla f|^p \ dx \, .$$


The only reference for inequalities of Poincare type on punctured domains I could find was Lieb–Seiringer–Yngvason (Ann. Math 2003) [arXiv link](https://arxiv.org/abs/math/0205088).


I suspect the Poincaré inequality on punctured domains in the way it is asked above might be false. If it is false, then I would like to understand is what sort of functions admit the second inequality?



Poincaré inequality on annular regions
No. Consider the positive part of a coordinate function on a large but thin annulus.


