Sharp $L^2$ Estimates for $(2+1)$-dimensional oscillatory integral operators with homogeneous binomial phases

Abstract

We study oscillatory integral operators in $(2+1)$-dimensions with a homogeneous binomial phase \[ \Phi(x,y,t)=x^{k-k_P}t^{k_P}+y^{k-k_Q}t^{k_Q}, \qquad 1\le k_P<k_Q<k. \] For compactly supported smooth amplitudes, we establish sharp \(L^2(\R)\to L^2(\R^2)\) estimates with logarithmic losses occurring only in certain critical cases. The proof is based on scale-dependent Phong--Stein estimates.