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

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.