Assume that P is a real valued strong elliptic polynomial, then what do we know about the following $$ K(\lambda,x)=\int{\frac{e^{ix\xi}}{\lambda-iP(\xi)}}d\xi,\quad \lambda\in \mathbb{R}\0 $$ The reason I want to know about it is that I need some properties of the resolvent $(\lambda-iP(D))^{-1}$. It is not hard to show that the resolvent is bounded on $L^p$, a more detailed analysis can obtain $L^p-L^q$ estimates for some {p,q}. However, I want to know if there is a pointwise estimate of its kernel, which will allow me to do more. Any reference is appreciated.
$\begingroup$
$\endgroup$
Add a comment
|
1 Answer
$\begingroup$
$\endgroup$
2
There exist more or less explicit formulas, especially for the case where $P$ is a homogeneous polynomial. See
F. John, Plane Waves and Spherical Means, Interscience, New York, 1955 (there were later reprints of this classical book, in 1981 and 2004).
answered Jan 30, 2013 at 10:36
-
$\begingroup$ Dear Anatoly Kochubei, I couldn't get it through the internet, neither in my school library. Would you show me what the detailed reslult is? Thanks very much. $\endgroup$ Commented Jan 31, 2013 at 2:29
-
1