Suppose $X$ and $Z$ are open sets in $\mathbb{R}^d$ and $\mathbb{R}^{d+1}$, respectively. Define $T_\lambda f:L^2(Z)\to L^2(X)$ by $$T_\lambda f(x)=\int e^{i\lambda\Phi(x,z)}a(x,z)f(z)dz,$$where the phase function $\Phi\in C^\infty(X\times Z)$ and $a\in C_0^\infty(X\times Z)$. Let $C_\Phi=\{(x,\Phi'_x;z,\Phi'_z)\}$ be the associated canonical relation. If the projection $\pi_L:C_\Phi\to T^*X$ is a submersion with folds, then we have$$ ||T_\lambda f||_2\le C\lambda^{-d/2}(\log \lambda)^{1/2}||f||_2,\ \lambda\gg1,$$ by Theorem 2.1 in the paper Fourier integral operators with fold singularities by Greenleaf and Seeger(http://www.math.wisc.edu/~seeger/papers/crelle.pdf).

My questions: When can we remove the $(\log\lambda)^{1/2}$ factor? Are there any papers on the sufficient conditions(or necessary conditions) for the improvement?

Remark: Under the same assumption above, the $\log$ loss can not be removed. But maybe it can be removed with additional assumptions. I tried hard to read some papers citing G-S, but I haven't seen any improvement on this estimate so far. I would appreciate recommendations of references for the improvement if anyone has them. Any comments are welcome :-)


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