The Hormander lemma about oscillatory integral operators states that $T_\lambda f(x)=\int e^{i\lambda S(x,y)}a(x,y)f(y)dy$, while the Hessian of $S(x,y)$ is nondegenerate, then $||T_\lambda||_2 \leq C\lambda^{-n/2}$. I wonder if this decay rate is optimal?
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1$\begingroup$ Where is the dependence on $\lambda$ in the integral? $\endgroup$– Jochen WengenrothJan 6, 2016 at 16:08
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1$\begingroup$ I suppose you meant to write $\int e^{i\lambda S}\ldots $. Then the bound is clearly optimal, as you can see by considering the special case of the Fourier transform. $\endgroup$– Christian RemlingJan 6, 2016 at 19:41
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$\begingroup$ What I mean is that for every $S(x,y)$ satisfying the nondegenerate condition, we can choose appropriate $a(x,y)$ and $f(y)$ s.t. $||T_\lambda f||_2 \geq C^{*}λ^{−n/2}$ $\endgroup$– Kingsley XuJan 7, 2016 at 0:17
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Yes it is optimal: take $a=1$, $S=-π y^2$,$f=e^{-π y^2}$ then $$ (T_\lambda f)(x)=\int_{\mathbb R} e^{-πi\lambda y^2} e^{-π y^2} dy=(1+i\lambda)^{-1/2}. $$