Let us consider the fractional heat semigroup $\left(e^{-t(-\Delta)^\alpha}\right)_{t\ge 0}$ for $\alpha\in (0,1)$ (the fractional power is taken in whole $\mathbb R^d$). Is there any result about the limit of $e^{-t(-\Delta)^\alpha} f$ when $\alpha \to 1^-$ with respect to an appropriate norm, e.g., $L^2(0,T;L^2(D))$ for a fixed domain $D\subset \mathbb R^d$ ? In particular, I am interested to some kind of explicit convergence rates with respect to $\alpha$. Any reference on similar topics would be helpful.