Let $u(x)=\int_\mathbb{R} \frac{1}{\sqrt{2\pi t}}e^{-\frac{|x-y|^2}{2t}}g(y)\, dy,\quad x\in \mathbb{R}$ be the fundamental solution to $$\begin{cases} u_t-\frac{1}{2}u_{xx}=0, &t\in (0,T],x\in \mathbb{R},\\ u(0,x)=g(x),&x\in \mathbb{R}.\end{cases}$$
I am considering the following problem: suppose $g$ is nice enough, does there exist $p>2$ and constant $C=C(p,g)$, such that $$\|D_x u\|_{L^p(\mathbb{R})}\le C, \; \forall t\in (0,T].$$
Formally, by using the PDE and the identity $2DuDu_t=\frac{d}{dt}(|Du|^2)$, I got $$\|D_x u\|_{L^2(\mathbb{R})}\le C\|D_xg\|_{L^2(\mathbb{R})}.$$
First, let me suggest carefulness when dealing with your equation. Your ``fundamental solution'' is a solution of the problem, and the only such that $u \to 0$ as $|x| \to +\infty$ (this condition is satisfied in L^p. Unless you add this condition, the problem you study is ill posed.
Also, when you bound the norms, you should be careful, since you are forgetting the time variable.
If you are looking for solutions $u \in W^{1,p}$ then your best bet is to think about the evolution semigroup, which can be studied by the eigenvalue decomposition of the $-\Delta$ operator.
In a bounded domain $\Omega$ (an I guess you extend that to unbounded domains) you can expect $$ \| \nabla u (\cdot, t) \|_{L^2 (\Omega)} + \| u (\cdot, t) \|_{L^2 (\Omega)} \le C t^{-\frac 1 2} \| g \|_{L^2 (\Omega)} $$
Cheers