This is a question that I am asking to help myself with a homework question. Assume $f$ has is smooth with compact support ( so that $f$ and $\hat f$ are in $\mathscr{S}$), and $G_t$ is the heat kernel.
A fourier approach seems the best way to go about this(because [see below] I am subtracting a first fourier coefficient) - so I am going to try to prove $L_2$ convergence.
That is for some constant $c$, $||\frac{e^{t \Delta}f}{[\int f] G_t(x)} - c||_2 \to 0$ as $t \to \infty$.
Since $f$ has compact support this happens iff $||e^{x^2/4t} (\int e^{-(x-y)^2/4t}f(y))-c[\int f] ||_2 \to 0$. Now replacing c by $C/(2\pi)^{d/2}$, we would like to take the fourier transform and use plancherel. Thus the above would tend to 0 iff (if the fourier transforms exist which they don't) $||\mathscr{F}[e^{x^2/4t}](\xi)* t^{d/2} e^{-|\xi|^2 t} - \hat f(0) ||_2$. What we are seeing above is that the fourier transform $e^{x^2/4t} (\int e^{-(x-y)^2/4t}f(y))$ can't be written nicely as the convolution written above. I need to do something different. I am not sure how to get around this.
I think it is still reasonable to try to show $L_2$ convergence because this converges (by bounded convergence theorem)in $L_2^{loc}$ since it happens pointwise when $f \in C^{\infty}_c$ before taking the fourier transform.
Do you have any ideas or hints? Thanks.