For the operator with phase x^a t^m + y^b t^n (m>n), the sharp L^2 to L^{2m+2} decay rate is lambda^{-(1/a + 1/max{b,n})/(2(m+1))}, and this rate is optimal when n is at most b.
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Some sharp $L^2 \to L^p$ decay estimates for $(2+1)$-dimensional degenerate oscillatory integral operators
For the operator with phase x^a t^m + y^b t^n (m>n), the sharp L^2 to L^{2m+2} decay rate is lambda^{-(1/a + 1/max{b,n})/(2(m+1))}, and this rate is optimal when n is at most b.