This is a duality argument (the author is really invoking the adjoint of Lemma 5.2(2), rather than Lemma 5.2(2) directly). We can write $$ I = \sup_g |\int_{{\bf R}^d} \partial_i \partial_j (1-\Delta)^{-\delta/2} p_t(\cdot-y)(x) g(x)|$$ where $g$ ranges over elements of $C_c({\bf R}^d)$ of supremum norm one. The expression inside the absolute value can be rearranged (after an "integration by parts") as $$ |\partial_i \partial_j (1-\Delta)^{-\delta/2} P_t g(y)|.$$ By Lemma 5.2 (with $\alpha=2$, $\beta=0$, $k = \delta/2$), this expression is $\lesssim t^{\frac{\delta}{2}-1} \|g\|_{C^0} = t^{\frac{\delta}{2}-1}$, as claimed.

This is a duality argument (the author is really invoking the adjoint of Lemma 5.2(2), rather than Lemma 5.2(2) directly). We can write $$ I = \sup_g \left|\int_{{\bf R}^d} \partial_i \partial_j (1-\Delta)^{-\delta/2} p_t(\cdot-y)(x) g(x)\ dx\right|$$ where $g$ ranges over test functions in $C^\infty_c({\bf R}^d)$ of supremum norm one. The expression inside the absolute value can be rearranged (after an "integration by parts") as $$ |\partial_i \partial_j (1-\Delta)^{-\delta/2} P_t g(y)|.$$ By Lemma 5.2 (with $\alpha=2$, $\beta=0$, $k = \delta/2$), this expression is $\lesssim t^{\frac{\delta}{2}-1} \|g\|_{C^0} = t^{\frac{\delta}{2}-1}$, as claimed.