Considering the heat equation on the flat torus $\mathbf{T}^d$, we have the maximal regularity estimate \begin{align*} \forall \varphi\in\mathscr{D}(\mathbf{R}\times\mathbf{T}^d),\quad \|\Delta \varphi\|_p \lesssim_p \|\partial_t \varphi - \Delta \varphi\|_p,\qquad (\star) \end{align*} the laplacian acting on the space variable only and the norm refering to $L^p(\mathbf{R}\times\mathbf{T}^d)$ for some $1<p<\infty$. I thought that from this specific formulation on compactly supported test functions, one could recover the local version of this result that is, whenever $\varphi$ is smooth and vanishes at $t=0$, there holds \begin{align*} \|\Delta \varphi\|_{T,p} \lesssim_p \|\partial_t \varphi - \Delta \varphi\|_{T,p}, \qquad (\star\star) \end{align*} where this time the norm refers to $L^p(Q_T)$ with $Q_T:=(0,T)\times\mathbf{T}^d$, $\lesssim_p$ being independent of the final time $T>0$.
However, when trying to localize the first estimate with some test function $\theta\in\mathscr{D}(\mathbf{R})$, there's an extra term that I am not able to handle correctly. I thought I could find some answer in this expository paper which states a similar compact-support like estimate (but in the whole physical space instead of the torus), however on page 6 the Theorem 1.1 is used on the function $v$ which precisely is not compactly supported in time (so I don't see this as a direct consequence of Theorem 1.1). The painful fact is that if one localizes $f:=\partial_t \varphi-\Delta \varphi$, the corresponding solution of the heat equation is not compactly supported. On the other hand, if one localizes $\varphi$ directly then there's an extra term which gives a Dirac mass when the test function is sent to $\mathbf{1}_{[0,T]}$.
So my question is : can $(\star\star)$ be easily recovered from $(\star)$ directly or, for some reason, there's a real difficulty and estimates $(\star\star)$ have to be obtained directly ?
