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Consider the inhomogeneous linear heat equation

$$\partial_tu-\Delta u=F$$

on $\mathbb R^n\times [0,1]$ (say) with zero initial data. Assume $F$ is very nice (say Schwarz), so that we have a nice solution $u$. It is quite standard that

$$\|\nabla^2 u\|_{L_x^2L_t^2}\ll \|F\|_{L_x^2L_t^2}. $$

I'm now asking if there is a similar estimate

$$\|\nabla^2 u\|_{L_x^2L_t^p}\ll_p \|F\|_{L_x^2L_t^p} $$

for $1<p<\infty$.

Remark: It is false for $p=\infty$, because the $L^2\to L^2$ norm of second derivatives of the heat kernel $K_t$ (i.e. the $L^\infty$ norm of its spatial Fourier transform) blows up like $1/t$ as $t\to 0$, which is not integrable in $t$. Also by duality it is also false for $p=1$, so I'm interested in what happens in between.

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Yes, the estimate holds and is a very special case of theorem 2.2 in the paper by N.V. Krylov 'The heat equation in $L_q((0,T),L_p)$-spaces with weights', SIAM J. on Math. Anal., Vol. 32, No. 5 (2001).

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  • $\begingroup$ "Surprisingly enough, to the best of my knowledge, the case $q \neq p$ was never addressed before even for the heat equation in $\mathbb R × \mathbb R^d$ without weights." Indeed very surprising that there are things in PDE that are under the nose but never got look at. $\endgroup$ – Fan Zheng Nov 22 '15 at 22:01

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