Heat semigroup - Knowlpedia

On $L^p(\mathbb R^n)$ , $1\le p\le\infty$ , the heat semigroup is

T_0f=f,\qquad T_tf=G_\nu(t,\cdot)*f\quad(t>0).

The star denotes Euclidean convolution. Completing the square in the Gaussian integral gives $G_\nu(t)*G_\nu(s)=G_\nu(t+s)$ , hence $T_tT_s=T_{t+s}$ .

Contraction and continuity

Mass one and Young's inequality give $\|T_tf\|_p\le\|f\|_p$ . For $1\le p<\infty$ , $T_tf\to f$ in $L^p$ as $t\downarrow0$ , by continuity of translations and concentration of the Gaussian near zero. On bounded uniformly continuous functions the convergence is uniform. It need not be uniform for a general $L^\infty$ function; a jump discontinuity supplies a counterexample.