I'm trying to find some references dealing with regularity and properties of the heat/Gaussian kernel $$ G_t\left(x,y\right) = \frac{1}{\sqrt{2\pi t}}\, e^{-\left.\left(x-y\right)^2\right/2t}, \hspace{2ex} \text{for } \;x,y \in \mathbb{R} $$ In particular, bounds on quantities like $\left\lvert\,\dfrac{\partial\hspace{0.1ex} G_t}{\partial \hspace{0.1ex}x}\,\left(\hspace{0.1ex}x,\hspace{0.2ex}y\hspace{0.1ex}\right)\,\right\rvert$, $\;\big\lvert\,G_t\hspace{0.1ex}\left(\hspace{0.1ex}x,\hspace{0.2ex}z\hspace{0.1ex}\right)-G_t\hspace{0.1ex}\left(\hspace{0.1ex}y,\hspace{0.2ex}z\hspace{0.1ex}\right)\hspace{0.05ex}\big\rvert$, $\;\left\lvert\,\dfrac{\partial\hspace{0.1ex} G_t}{\partial \hspace{0.1ex}x}\,\left(\hspace{0.1ex}x,\hspace{0.2ex}z\hspace{0.1ex}\right) - \dfrac{\partial\hspace{0.1ex} G_t}{\partial \hspace{0.1ex}y}\,\left(\hspace{0.1ex}y,\hspace{0.2ex}z\hspace{0.1ex}\right)\,\right\rvert$, etc.
Do you know of any books, papers, or websites in which the author discusses and/or establishes various bounds on quantities like these?
The answer is long. You should study the book of L C Evans' PDE http://www.amazon.com/Partial-Differential-Equations-Graduate-Mathematics/dp/0821849743 or took a graduate student level of the course on PDE.