Let $\left(e^{t \Delta}\right)_{t \geqslant 0}$ be the Neumann heat semigroup in $\Omega$, and let $\lambda_{1}>0$ denote the first nonzero eigenvalue of $-\Delta$ in $\Omega$ under Neumann boundary conditions. Then there exist constants $C_{1}, \ldots, C_{4}$ depending on $\Omega$ only which have the following properties. (i) If $1 \leqslant q \leqslant p \leqslant \infty$ then ...
I don't know the definition of Neumann heat semigroup and first nonzero eigenvalue of $\Delta$?
If you know anything about them I thank you to tell me.
This is lemma 1.3., from here.