For any $\kappa>0$, we consider the Gaussian heat kernel $$ p^\kappa_t (x) := (\kappa \pi t)^{-\frac{d}{2}} e^{-\frac{|x|^2}{\kappa t}}, \quad t>0, x \in {\mathbb R}^d. $$
Let $L^0 := L^0 (\mathbb R^d)$ be the space of real-valued measurable functions on $\mathbb R^d$. Let $L^0_+ := L^0_{+} (\mathbb R^d)$ the subspace of $L^0$ consisting of non-negative functions. Let $L^0_b := L^0_b (\mathbb R^d)$ the subspace of $L^0$ consisting of bounded functions. For $f \in L^0_b \cup L^0_+$, let $$ P_t^\kappa f (x) := \int_{\mathbb R^d} p^\kappa_t (x-y) f (y) \, \mathrm d y, \quad t >0, x \in \mathbb R^d. $$
I would like to verify the following claim, i.e.,
It is well-known that for some constant $c>0$, $$ \|P^\kappa_t \|_{L^p \to L^{p'}} := \sup_{\|f\|_{L^p} \le 1} \|P^\kappa_t f\|_{L^{p'}} \le c t^{-\frac{d(p'-p)}{2pp'}}, \quad t>0, 1 \le p \le p' \le \infty. $$
Let $$ q := \frac{d(p'-p)}{2pp'} \in[0,\infty]. $$
By this answer, the above constant $c$ depends on $\kappa$ and it suffices to upper bound $$ C:=\sup_{t>0} t^q \sup_{\|f\|_{L^p} \le 1} \|P^1_t f\|_{L^{p'}}. $$
There are possibly subtle mistakes that I could not recognize in below attempt. Could you please have a check on it?
We have $P^1_t f = p_t^1 * f$ for $f \in L^p$. Let $r>0$ such that $\frac{1}{r} + \frac{1}{p} = \frac{1}{p'} +1$. By Young's inequality, $\|P^1_t f\|_{L^{p'}} \le \|p_t^1\|_{L^r} \|f\|_{L^p}$. So $$ \sup_{\|f\|_{L^p} \le 1} \|P^1_t f\|_{L^{p'}} \le \|p_t^1\|_{L^r}. $$
We consider the case $r=\infty$. Then $q=\frac{d}{2}$ and $\|p_t^1\|_{L^r} = (\pi t)^{-\frac{d}{2}}$. Then $$ C \le \sup_{t>0} t^q (\pi t)^{-\frac{d}{2}}=\pi^{-\frac{d}{2}}. $$
We consider the case $r\in [1, \infty)$. From this thread, there is a constant $C_r>0$ depending only on $r$ such that $$ \|p_t^1\|_{L^r} = C_r t^{-\frac{d}{2}(1-\frac{1}{r})}, \quad t>0. $$ Notice that $q = \frac{d}{2} (1-\frac{1}{r})$. Then $$ C \le \sup_{t>0} t^q C_r t^{-\frac{d}{2}(1-\frac{1}{r})}=C_r. $$