Question

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.

Answer 1

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).