If $P$ and $P'$ are linear partial differential operators with constant complex coefficients on $U = \mathring U \subseteq \Bbb R^m$, we say that $P \sim P'$ if and only if $\dfrac {\tilde P} {\tilde {P '}}$ and $\dfrac {\tilde {P '}} {\tilde P}$ are bounded on $\Bbb R^m$, where $P = \sum_\alpha a_\alpha D^\alpha$, $D^\alpha = (-\Bbb i)^{|\alpha|} \partial _\alpha$, $\tilde P = \sqrt{ \sum _\alpha |P^\alpha|^2 }$ and $P^\alpha (v) = \partial ^\alpha P (v)$ (as usual, $\alpha$ is a multi-index). We say that $P$ and $P'$ "are of equal strength".

If $P \sim P'$ on $U$ and $Q \sim Q'$ on $V$, then is it true that $(P+Q) \sim (P' + Q')$ on $U \times V$? If this is not true in general, are there (necessary and) sufficient conditions under which it is?

By $P + Q$ I understand $P \otimes \text{id} + \text{id} \otimes Q$.

I am investigating the hypoellipticity of an operator and if the above were true, then it would allow me to split my problem into smaller, more tractable blocks.

while P is itself a differential operator. For example what is ${|P^{\alpha}|}^2$ where $P$ is Laplacian and $\alpha=(2,0)$. I really do not understand your question. $\endgroup$ – Ali Taghavi Jun 18 '18 at 13:57