Question in Convergence of a integral in the Heat Kernel and Dirac delta function.

Question

In Convergence of a integral - heat Kernel and dirac delta function

Why $$\lim_{t\to 0+}\int_{|x|>\delta}K_t(x)|\varphi(x)-\varphi(0)|\,dx=0?$$

Answer 1

As $|\varphi(x)-\varphi(0)|$ is bounded uniformly, we can replace this by some constant $B$, and it suffices to show that $\lim_{t\to 0^{+}}\int_{|x|>\delta}K_t(x)dx=0$. One way to do this is to do a change of variables to $u=\frac{x}{\sqrt{t}}$. Then, by the change of variables formula, $$ \int_{|x|>\delta}K_t(x)dx=\frac{1}{(4\pi)^{n/2}}\int_{|u|>\delta/\sqrt{t}}e^{-|u|^2/4} du $$ Note that $\delta/\sqrt{t}$ goes to $\infty$, so the RHS goes to zero as $t\to 0^+$ because the integral of a Gaussian converges.