I have a function $u:\mathbb{R}^n\times [0,T]\to \mathbb{R}$ that solves the heat equation $u_t=\Delta u$, is bounded, and $u(x,0)=g(x)$. I need to show that $$\max|\nabla u(x,t)|\leq \frac{C}{\sqrt{T}}\max|g|,$$ where $C$ is some constant.
My attempt: The fundamental solution to the heat equation tells us that $$u(x,t)=\frac{1}{(4\pi t)^{n/2}}\int_{\mathbb{R}^n} g(y)e^{-\frac{(x-y)^2}{4t}}\,dy.$$ Let $M<\infty$ be such that $|g(x)|\leq M$ on $\mathbb{R}^n$. Differentiating under the integral sign, we have
\begin{align*} |\nabla u(x,t)|&\leq \frac{M}{2t(4\pi t)^{n/2}}\int_{\mathbb{R}^n} |x-y|e^{-\frac{(x-y)^2}{4t}}\,dy. \end{align*} Now, if I'm trying to do some change of variables to make everything work out nicely. I'm struggling with cancelling out the t's and adding in the T. Any hints appreciated.