TITLE: Different ways to prove $L^p$-estimates for the heat equation
QUESTION [5 upvotes]: Let $p \in (1,\infty)$. We are interested in strong $L^p$-solutions to the heat equation in $\mathbb{R}^n$.
$$
\begin{cases}
    \partial_t u = \Delta u + f \\
u(0) = 0.
\end{cases}
$$
It is well-known that for all $f \in L^p((0,\infty;L^p(\mathbb{R}^n))$ there exists a unique function $u \in H^{1,p}((0,\infty);L^p(\mathbb{R}^n)) \cap L^p((0,\infty);H^{2,p}(\mathbb{R}^n))$,
which solves the heat equation and satisfies the estimate
$$ \| \partial_t u \|_{p} + \|\Delta u \|_{p}\le C \|f \|_{p} $$
for some constant $C>0$.
I am interested in different ways to prove this. To start the discussion let me name two different methods.

Using the theory of singular integrals applied to the solution formula given by means of the fundamental solution shows that the solutions operator is $L^p$-bounded.

Fourier transformation in time and space gives $\hat{u} = \frac{{|\xi|}^2}{i \omega + |{\xi}|^2}\hat{f}$. Applying Mikhlin's multiplier theorem gives the desired estimate.


Do you know of any other methods to prove this? If so, feel free to extend the list.

REPLY [3 votes]: I would say that methods 1) and 2) are very close each other but in both cases the proof is a bit harder than the elliptic counterpart since one has to use the Marcinkiewicz multiplier theorem instead of Mikhlin-Hormander.
There are  a couple of similar approaches with some simplifications. One is in the book of N. Krylov: Elliptic and Parabolic equations in Sobolev spaces and relies on estimates of the sharp function of Fefferman-Stein.
The other one I know is based on an interpolation result originally due to Z. Shen, which can be found in the first  of 4 papers by P. Auscher and J.M Martell. In few words if $Tf=D_t u$, where $D_t u-\Delta u=f$, it suffices to bound the $L^p$ means of $Tf$ over cubes with the corresponding $L^2$ means over the double cube, whenever $f$ vanishes on a bigger cube. This gives boundedness of $T$ in $L^p$, $p>2$, having at hands that in $L^2$. For PDE this is quite manageable since the criterion follows from interior estimates of homogenuous problems.