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.



Mixed norm estimate for the heat equation
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)*.


