Abstract
In this paper we obtain almost sharp decay estimates for $L^2$ operator norm of strongly singular oscillatory integral operators in $\mathbb{R}^{n+1}$ for $n \geq 2$; we prove some necessary condition for $L^2$ estimates. Also, we prove that the operators are bounded on $L^p$ for some $p \neq 2$ and the range of $p$ depends on the hypersingularity of the operators.
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