Consider the heat flow $e^{t\Delta}$, $t>0$, on an Euclidean domain (say $\mathbb{R}^3$). I expect, in analogy with the Strichartz estimates for the Schrodinger equation, that the following estimates hold:
$$\Vert e^{t\Delta}f\Vert\_{L^s(\mathbb{R}^+,L^p(\mathbb{R}^3))}\lesssim \Vert f\Vert\_{L^2(\mathbb{R}^3)},$$
where $(s,p)$ is a Strichartz pair. Is this fact actually true? In such a case, there exists a reference where it is stated explicitly?



Strichartz estimates for the heat equation
Sure. Actually you have in addition the maximum principle (i.e. an $L^\infty$--$L^\infty$ estimate) which combined with the others gives a full triangle of indices for the Strichartz type estimates in your question. Even more, the strong decay of the symbol $e^{-t|\xi|^2}$ can be exploited for some gains in regularity. A short account is in the book by Wang, Huo, Hao, Guo "Harmonic analysis method for nonlinear evolution equations" (around p.35) [link](http://www.worldscientific.com/worldscibooks/10.1142/8209)


