Examples

For $f,g\in L_1(\R^d)$ we have: $$ \Big|\int P_tf(x)g(x)\,dx\Big|\leq\norm f_1\norm g_1\norm{p_t}_\infty $$ and thus: $\norm{P_t:L_1(\R^d)\rar L_\infty(\R^d)}\leq(4\pi t)^{-d/2}$ - $P_t$ is said to be ultracontractive.

For all $x\in \R^d$, we have \[ |P_tf(x)| = \Big| \int f(x-y)p_t(y)\, dy \Big|\le \|f(x-\cdot)\|_1 \|p_t\|_{\infty} = \|f\|_1 \|p_t\|_{\infty}. \] Thus, we conclude by Hölder's inequality: \[ \Big| \int P_tf(x)g(x)\, dx \Big|\le \|P_tf\|_{\infty} \|g\|_1\le \|f\|_1\|p_t\|_{\infty}\|g\|_1. \] The second statement follows because $\|p_t\|_{\infty} = (4\pi t)^{-d/2}$.