Timeline for Strichartz estimates for the heat equation

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May 23, 2017 at 20:18	comment	added	Piero D'Ancona		That's what I said. Inhomogeneous Strichartz takes you to $L^1L^2$ which is estimated by $L^\infty L^2$. But of course the constant in this inequality is unbounded as $T\to \infty$ (it is exactly $T$)
May 23, 2017 at 3:43	comment	added	Capublanca		I understand the non homogeneous case, thank you. However i'm wonder if also the bound $\Vert e^{t\Delta}F(t,\cdot)\Vert_{L^sL^p}\lesssim \Vert F \Vert_{L^{\infty}L^2}$ is true, at least for finite time intervals and for $F\in\mathcal{C}(\mathbb{R}^+,L^2)$.
May 22, 2017 at 20:00	comment	added	Piero D'Ancona		You mean the non homogeneous equation $u_t-\Delta u=F$? you certainly can apply non homogeneous Strichartz estimates, so $u$ in $L^pL^q$ is estimated by $F$ in $L^1L^2$. For bounded intervals this is controlled by $L^\infty L^2$.
May 22, 2017 at 13:19	vote	accept	Capublanca
May 22, 2017 at 13:18	comment	added	Capublanca		Thank you very much Piero! I have another doubt: is it also true that $e^{t\Delta}$ maps continuosly $L^{\infty}((0,T),L^2(\mathbb{R}^3))$ into $L^{s}((0,T),L^p(\mathbb{R}^3))$? For the Schrodinger flow it should not be true, but maybe it works in the heat case.
May 22, 2017 at 10:52	history	answered	Piero D'Ancona	CC BY-SA 3.0