Different ways to prove $L^p$-estimates for the heat equation

Question

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.

1. 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. See for example, the excellent monograph "Parabolic $L^p$--$L^q$ estimates by Dietmar A. Salamon https://people.math.ethz.ch/~salamon/PREPRINTS/parabolic.pdf

2. 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.

Answer 1

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.

Answer 2

In the book "Second Order Parabolic Differential Equations" of G. M. Lieberman, such estimates are recovered (see Chapter VII) on a domain $\Omega$ with adequate boundary conditions. He introduces a parabolic version of the Calderón-Zygmund decomposition (second paragraph of Chapter VII) but its use in the remaining part of the proof is a bit difficult to grasp. The important thing is that there is no direct use of the kernel in his proof, so it does not seem to be part of your first item 1. of proofs. The proof uses properties of the maximal function to first establish $\|\nabla_x u\|_p \lesssim \|f\|_p$ and $\|\textrm{D}^2_x u\|_p \lesssim \|f\|_p + \|\nabla_x u\|_p + \|u\|_p$ and eventually recovers the estimate by some interpolation argument.

However the presentation is a bit harsh and solves far more cases than the simple heat equation. If anyone has a reference of Liberman's approach in a more "down-to-earth" way for the heat equation, I'll be pleased to hear from it !