How to show the continuity of the Laplacian of the heat kernel (in t)

Question

Let $K_t(x):=(4\pi t)^{-\frac{d}{2}}e^{-\frac{x^2}{4t}}$ be the heat kernel for $t > 0$. How can I show that the function $t \mapsto (\Delta K_t)*f$ is continuous in $L^p(\mathbb{R}^d)$ (associated with the $L^p$ norm) for $f \in C_c^{\infty}(\mathbb{R}^d)$ ?

I have proven the the continuity of this function $t \mapsto(\Delta K_t)*f(x)$ for every $x$ but clearly this is not sufficient. I have that $\lvert| \Delta K_t |\rvert_{1}\leq \frac{d}{2t}+ \frac{1}{2}(4 \pi t)^{-d/2}(\frac{\pi}{t})^{1/2}$, but I can't find the correct bound function to use this.

Answer 1

Let $t_n, t>0$ such that $t_n \to t$. Then we have

Lemma $\| \Delta K_{t_n} - \Delta K_{t}\|_\infty \xrightarrow{n \to +\infty} 0$.

By Young's inequality, $$ \begin{align} \| (\Delta K_{t_n})*f - (\Delta K_t)*f \|_{\infty} &= \| (\Delta K_{t_n} - \Delta K_t)*f \|_{\infty} \\ &\le \| \Delta K_{t_n} - \Delta K_t \|_\infty \|f\|_{L^1}. \end{align} $$

The claim then follows from above Lemma.