Equivalence of the optimal fractional Poincaré constants

Let $\Omega\subset {\mathbb R}^n$ be an open (bounded) domain and let $1<p<\infty$. For $r>z\geq 0$, let $C_{r,z}$ denote the optimal fractional Poincaré constant.

Equivalence of the optimal fractional Poincaré constants. For $t>s\geq 0$, the optimal Poincaré constant is

$$C_{t,s}=C_{r,z}^{\frac{t-s}{r-z}}.$$

The conjecture proposes that all optimal fractional Poincaré constants are determined by any one such constant through interpolation of the fractional orders. The surrounding discussion motivates this from interpolation results for fractional Sobolev spaces, but does not establish the assertion.