=== Heat kernel (Heat operator) In Euclidean space $\mathbb{R}^n$, the heat kernel $k(t,x)$ is $$ k(t,x) = (4\pi t)^{n/2} e^{-x^2/4t}; $$ and the Heat operator $e^{t\Delta}$ is defined by $$ e^{t\Delta} u_0 = k*u_0 = (4\pi t)^{n/2} \int e^{-(x-y)^2/4t} u_0(y) dy, $$ which is the classical solution to the heat equation \begin{equation} \begin{cases} u_t = \Delta u,\hspace{6pt} t>0,\, x\in \mathbb{R}^n, ~\\ u(0) = u_0. \end{cases} \end{equation} **Lemma 1 (Young's inequality)** Let $f,g$ be measurable functions, and let $p,q,r$ satisfy $1\leq p,q,r\leq\infty$ and that $\f{1}{p} + \f{1}{1} = \f{1}{r} + 1$, then $$ \norm{ f*g }_{r} \leq \norm{ f }_{p} \norm{ g }_{q},$$ where $\norm{\cdot}_r$ denotes the $L^r$-norm, similarly for the other two norms. Direct computation shows that $$\norm{ k(t,x) }_q = c_1 t^{-n/2} t^{n/(2q)},$$ for some constant $c_1>0$ independent of $t$. By Young's inequality, we obtain the following claim. **Claim 4:**We have the estimates: $$\norm{ e^{t\Delta} u }_r \lesssim t^{\f{n}{2}(\f{1}{r}- \f{1}{p})} \norm{u}_p,\hspace{6pt} 1\leq p\leq r < \infty.$$ Similarly, $$\norm{\grad(e^{t\Delta} u)}_r \lesssim t^{\f{1}{2}(\f{n}{r}- \f{n}{p}-1)} \norm{u}_p,\hspace{6pt} 1<p\leq r < \infty. $$