[Paper Review] Optimal regularity of Fourier integral operators with one-sided folds

Andrew Comech|ArXiv.org|Sep 1, 2006

Advanced Harmonic Analysis Research11 references4 citations

TL;DR

This paper establishes the optimal regularity loss for Fourier integral operators (FIOs) associated with canonical relations featuring one-sided Whitney folds. Using $L^2$ estimates for oscillatory integrals and scaling arguments, it proves that the regularity loss is exactly $(4 + 2/k)^{-1}$ derivatives, where $k$ is the type of the non-folding projection, generalizing prior results on two-sided folds and cusp singularities.

ABSTRACT

We obtain optimal continuity in Sobolev spaces for the Fourier integral operators associated to singular canonical relations, when one of the two projections is a Whitney fold. The regularity depends on the type, $k$, of the other projection from the canonical relation ($k=1$ for a Whitney fold). We prove that one loses $(4+\frac{2}{k})^{-1}$ of a derivative in the regularity properties. The proof is based on the $L^2$ estimates for oscillatory integral operators.

Motivation & Objective

• To determine the optimal regularity loss for Fourier integral operators (FIOs) associated with canonical relations that include a one-sided Whitney fold.
• To extend prior results on two-sided Whitney folds (loss of $1/6$ derivative) and cusp singularities (loss of $1/3$ derivative) to the general case of mixed singularity types.
• To characterize the dependence of regularity loss on the type $k$ of the non-folding projection from the canonical relation.
• To prove optimality of the derived regularity loss via model operator constructions and scaling analysis.

Proposed method

• The analysis relies on $L^2$ estimates for oscillatory integral operators, particularly in the framework of Fourier integral operators with singular canonical relations.
• The canonical relation is analyzed via local normal forms, focusing on the case where one projection is a Whitney fold and the other has type $k \geq 1$.
• A key technical tool is the definition of the type $k$ of a map with corank at most 1, based on the order of vanishing of the Jacobian determinant along the kernel direction.
• Model operators are constructed using phase functions with Morin $S_{1_k}$-singularities, such as $S(x,\vartheta) = (x_n^{k+1} + \dots)\vartheta_n + \dots$, to represent the canonical relation.
• Scaling arguments are applied to the oscillatory integral operator $T^{(1,k)}_\lambda$, rescaling $x$ and $\vartheta$ to derive the optimal decay rate of the operator norm.
• The optimality of the regularity loss is proven by showing that the decay rate $\|T^{(1,k)}_\lambda\| \lesssim \lambda^{-n/2 + (4+2/k)^{-1}}$ cannot be improved, using determinant-based Jacobian scaling.

Experimental results

Research questions

• RQ1 What is the optimal regularity loss for Fourier integral operators when one projection in the canonical relation is a Whitney fold and the other is of type $k$?
• RQ2 How does the regularity loss depend on the type $k$ of the non-folding projection?
• RQ3 Can the $L^2$ operator norm decay rate of oscillatory integrals with such singular phase functions be used to derive sharp Sobolev regularity estimates?
• RQ4 Is the derived regularity loss of $(4 + 2/k)^{-1}$ derivatives optimal for such operators?
• RQ5 How do scaling properties of the phase function and the associated Jacobian determinants influence the operator norm decay?

Key findings

• The optimal regularity loss for FIOs with one-sided Whitney fold and a type-$k$ projection is exactly $(4 + 2/k)^{-1}$ derivatives.
• For the case $k=1$ (two-sided Whitney folds), the regularity loss is $1/6$, consistent with Melrose and Taylor's result.
• The decay rate of the $L^2$ operator norm for the model operator $T^{(1,k)}_\lambda$ is $\|T^{(1,k)}_\lambda\| \lesssim \lambda^{-n/2 + (4+2/k)^{-1}}$, which is optimal.
• The optimality is proven via scaling: the norm cannot decay faster than $\lambda^{-n/2 + (4+2/k)^{-1}}$, as shown by analyzing the Jacobian determinant scaling under rescaling.
• The result generalizes previous results on two-sided folds and cusp singularities, providing a unified framework for regularity loss in terms of the type $k$ of the non-folding projection.
• The canonical relation with a Whitney fold and a Morin $S_{1_k}$-singularity is realized via a phase function $S(x,\vartheta)$ with explicit polynomial structure, enabling precise analysis.