what's the idea behind Carleman estimate - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-21T05:34:43Z http://mathoverflow.net/feeds/question/109817 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/109817/whats-the-idea-behind-carleman-estimate what's the idea behind Carleman estimate Shanlin Huang 2012-10-16T13:56:55Z 2012-10-16T14:43:52Z <p>A standard Carleman-type estimate is of the form <code>$$\sum_{|\alpha|&lt;m}{\tau^{2(m-|\alpha|-1)}\int{|D^{\alpha}u|^{2}e^{2\tau\phi}}dx}\leq K\int{|Pu|^{2}e^{2\tau\phi}dx},\quad u\in C_{0}^{\infty}$$</code> where $\phi$ is some weight function.This formula turn to be very useful in the study of uniqueness of Cauchy problem,and many mathematicians have considered this(such as Calderon,Hormander,Kenig,Sogge,and Tataru...)</p> <p>For a first look at this inequality,I'm wondering whether the weight fuction makes a essential role,and besides, what's the original idea of it?Are there some very simple but illuminated examples to show the the reasonableness of the Carleman estimates ?</p> <p>Well,one example in my mind is the first order operator $P=D+ix$,then it's easy to see that <code>$P^*=D-ix$</code>,and <code>$$P^*P-I=PP^*+I=-\frac{d^2}{dx^2}+x^2$$</code> which is the so-called harmonic oscillator,then we have <code>$$2\|u\|_{L^2}\leq \|Pu\|_{L^2},\quad u\in C_{0}^{\infty}$$</code> But in this simple example,there is no need to put a weight function,anyhow, from the proof,I guess the decomposition <code>$P=\frac{P+P^*}{2}+\frac{P-P^*}{2}$</code> may be one of the general idea.</p>