SHARP Lp DECAY ESTIMATES FOR DEGENERATE AND SINGULAR OSCILLATORY INTEGRAL OPERATORS: HOMOGENEOUS POLYNOMIAL PHASES

Abstract

We consider the degenerate and singular oscillatory integral operator with a singular kernel which is not a Calderón–Zygmund kernel and satisfies suitable size and derivative conditions related to a real parameter $\mu$. For any given homogeneous polynomial phases, except monomial phases, of degree $n$, we give the range of $p$ for which the sharp decay rate $-\frac{1-\mu }{n}$ on $L^2$ spaces can be preserved on $L^p$ spaces.