# ca.classical analysis and odes – A simple oscillatory integral with a non-smooth phase

Let $$phiin C_c^infty(mathbb{R})$$ be an even function such that $$chi_{(-1/2,1/2)}lephile chi_{(-1,1)}$$, where $$chi_{(a,b)}$$ stands for the indicator function of the interval $$(a,b)$$. For $$lambda>0$$ consider the oscillatory integral
$$I(lambda)=int_mathbb{R} phi(x), exp left(ilambda(x+epsilon|x|^{sqrt{2}})right), dx,$$
with some fixed (very small) positive constant $$epsilon$$.

My question is: what is the asymptotic behavior of this integral as $$lambdarightarrow infty$$? I can show, by essentially doing careful integration by parts, that the upper bound is $$lesssim lambda^{-sqrt{2}}$$, but I wonder whether $$lambda^{-sqrt{2}}$$ is also a lower bound?

Note, that if the exponent $$sqrt{2}$$ is replaced by $$2k$$ for some positive integer $$k$$, then the integral decays like $$lambda^{-M}$$ for any $$M>0$$ due to the non-stationary phase estimate (the derivative of the function $$x+epsilon x^{2k}$$ is $$gtrsim 1$$).

I would appreciate any hints on how to approach this problem.