Assume that P is a real valued strong elliptic polynomial, then what do we know about the following $$ K(\lambda,x)=\int{\frac{e^{ix\xi}}{\lambda-iP(\xi)}}d\xi $$$$ K(\lambda,x)=\int{\frac{e^{ix\xi}}{\lambda-iP(\xi)}}d\xi,\quad \lambda\in \mathbb{R}\0 $$ The reason I want to know more about it is that I need some properties of the resolvent $(\lambda-P(D))^{-1}$$(\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.
