I'm considering specifically functions $\xi:\partial B(0,1) \to \partial B(0,1)$ in $\mathbb{R}^2$ and $\mathbb{R}^3$ satisfying $\int_{\partial B(0,1)} \xi(y) dS(y) = 0$. I would like to know first of all if there is a Poincare inequality of the form:

$\int_{\partial B(0,1)} |\nabla^{-1}\xi(y)|^2dS(y) \leq C_{B} \int_{\partial B(0,1)}|\xi|^2 dS(y).$

Secondly (and more importantly) I'd like to know if there are any resources I can be directed to regarding the optimal constant in such a case.