Cfrgtkky

Let $$K(x,t) = frac{1}{sqrt{4pi t}^N} e^{-frac{|x|^2}{4t}},
t>0, xinmathbb{R}^n$$

Show that if $u_0in L^{infty}(mathbb{R}^N)$ then $$u(x,t) =
int_{mathbb{R}^N} u_0(y)K(x-y,t)dy$$ is infinitely differentiable in
$mathbb{R}^Ntimes ]0,infty[$.

Suggestion: fix $0<a<b$ and $R>0$ and verify that given
$ainmathbb{Z}_+^N, kinmathbb{Z}_+$, there exists constants $C>0,
c>0, p>0$ such that $$D_x^aD_t^k K(x-y,t)le Cexp(-c|y|^2),
x,yinmathbb{R}^N, |x|le R, |y|le p, tin [a,b]$$

I've used Leibniz product differentiation rule and ended up with

$$D_t^k K = sum_i^k{kchoose i}prod_{j=1}^{k-i}left(frac{-N}{2}-jright)(4pi t)^{-frac{N}{2}-j}4^jpi^i frac{(-|x|)^{2i}}{4i}e^{-frac{|x|^2}{4t}}$$

which is already pretty absurd and looks like won't help. I, however, don't see another way of taking thederivative. I know that that product can be converted into a combinatorial but only for even $N$. I still have to take $D_x^a$. Any ideas?

You don't really need to keep track of the coefficients, only that you have some control:
$$
D^k_t(K(x,t))=p_k(x,t^{-1}) K(x,t)
$$
where $p_k$ is a polynomial of degree $4k$. Similarly, take care of the $D_x^alpha$
$$
D^alpha_x(D^k_t(K(x-t)) = Q_{k,alpha}(x,t^{-1}) K(x,t)
$$
with $Q_{k,alpha}$ a polynomial of degree $4k+2lvertalpharvert$.

For polynomial $Q_{k,alpha}$, you can bound
begin{align}
lvert Q_{k,alpha}(x-y,t)rvert&leq overbrace{(#text{monomials})}^{=C(k,alpha,N)}times\
&quadtimesunderbrace{(maxlverttext{coefficients}rvert)}_{leq C(k,alpha,N)}times
(suptext{bound on monomial})
end{align}
where we use the obvious notation $C(dots)$ for a universal constant depending only on the those variables listed. Now just bound away each monomial, giving
begin{align}
lvert(x-y)^beta t^{-j}rvert&leq C(R,lvertbetarvert)(1+lvert yrvert^2)^{d/2}cdotmax(1,a^{-j})\
&leq C(R,d,a)(1+lvert yrvert^2)^{d/2}
end{align}
for every degree bound $d$, every $lvert xrvertleq R$, every $yinmathbb{R}^N$ and every $tin[a,b]$. Thus we bound
$$
(1+lvert yrvert^2)^{d/2}lvert K(x-y,t)rvertleq C(d,R,a,N,varepsilon)exp(-(1/(4b)-varepsilon)lvert yrvert^2)
$$
for all $lvert xrvertleq R$, $tin[a,b]$ and $varepsilon>0$. Note we weaken the coefficient of $lvert yrvert^2$ in the exponential to help get the bound in front for all $yinmathbb{R}^N$.