A standard Carleman-type estimate is of the form $$ \sum_{|\alpha|<m}{\tau^{2(m-|\alpha|-1)}\int{|D^{\alpha}u|^{2}e^{2\tau\phi}}dx}\leq K\int{|Pu|^{2}e^{2\tau\phi}dx},\quad u\in C_{0}^{\infty}, $$ where $\phi$ is some weight function. This formula turns to be very useful in the study of uniqueness of Cauchy problems, and many mathematicians have considered this (such as Calderon, Hormander, Kenig, Sogge, and Tataru...)

For a first look at this inequality, I'm wondering whether the weight fuction has an essential role, and furthermore, what's the original idea of it? Are there some very simple but illuminating examples that show the reasonability of the Carleman estimates?

One example in my mind is the first order operator $P=D+ix$, where $D = \frac{1}{i} \frac{d}{dx}$. It's easy to see that $P^*=D-ix$, and $$ P^*P-I=PP^*+I=-\frac{d^2}{dx^2}+x^2, $$ which is the so-called harmonic oscillator. Here we have $$ 2\|u\|_{L^2}\leq \|Pu\|_{L^2},\quad u\in C_{0}^{\infty}. $$ But in this simple example, there is no need to use a weight function. From the proof I guess the decomposition $P=\frac{P+P^*}{2}+\frac{P-P^*}{2}$ may be one of the general ideas.