Maximal growth of the Stein-Wainger oscillatory integral

1. Problem Statement

The paper investigates the asymptotic behavior of the Stein-Wainger oscillatory integral as the frequency parameter $\lambda \to \infty$. The integral is defined on the interval $I = [-1, 1]$ as:
$$m(\lambda) := \text{p.v.} \int_{-1}^{1} \frac{e^{i\lambda \psi(t)}}{t} \, dt$$
where $\psi \in C^1(I)$ is a real-valued phase function.

The central question is to determine the maximal possible growth rate of $|m(\lambda)|$ as a function of $\lambda$, specifically in relation to the regularity class of the phase function $\psi$.

• Context: This problem is motivated by eigenfunction restriction estimates on curves and the boundedness of the Hilbert transform along variable curves.
• Previous Work:
  • For polynomials or rational functions, $m(\lambda)$ is bounded ($O(1)$).
  • For $C^\infty$ functions, it was an open question (posed by Wang–Zhang) whether $m(\lambda)$ remains bounded. Nagel–Wainger showed that for certain flat smooth functions, the associated Hilbert transform is unbounded, implying $m(\lambda)$ can grow.
  • The paper aims to establish a precise hierarchy of growth rates for various regularity classes, ranging from $C^\alpha$ to Gevrey classes, refined Gevrey classes, and classes beyond Gevrey.

2. Methodology

The authors employ a combination of harmonic analysis techniques and precise estimates on the behavior of flat functions.

1. Reduction to Flat Phases:

   • The integral is rewritten as $m(\lambda) = \int_0^1 \frac{e^{i\lambda \psi(t)} - e^{i\lambda \psi(-t)}}{t} dt$.
   • Let $\phi(t) = \psi(t) - \psi(-t)$. If $\phi$ has a finite order of vanishing at $t=0$, standard oscillatory integral estimates (van der Corput) show $m(\lambda) = O(1)$.
   • The problem reduces to the case where $\phi$ is flat at $0$(i.e.,$\phi^{(n)}(0) = 0$ for all $n$).

2. Flat-Point Estimates (Bang's Lemma vs. Taylor-Legendre):

   • To bound $m(\lambda)$, the authors must estimate how fast a flat function $\phi(t)$ can grow near $0$ given its derivative bounds (regularity class).
   • Bang's Lemma: Used for Denjoy-Carleman classes defined by log-convex sequences. It provides a sharp estimate on the growth of a function near a flat point based on the sequence of derivative bounds.
   • Taylor-Legendre Method: Used for classes larger than Gevrey (e.g., intermediate regularity classes). This involves minimizing the Taylor remainder bound $K^{n+1} M_n |t|^n / n!$ to find the optimal decay rate.

3. Dyadic Decomposition:

   • The integral is split into dyadic intervals near the origin. The growth of $m(\lambda)$ is controlled by summing contributions where the phase is small versus where it oscillates rapidly.
   • The "tail function" $T_M(N) = \sum_{j \ge N} 1/\mu_j$ (where $\mu_j$ are quotients of the defining sequence $M_n$) plays a crucial role in determining the growth rate.

4. Sharpness Constructions:

   • The authors explicitly construct phase functions $\psi$ within specific regularity classes that achieve the derived upper bounds. These constructions often involve "dyadic plateaus" or specific exponential forms tailored to the derivative bounds of the class.

3. Key Contributions and Results

The paper establishes a precise hierarchy of maximal growth rates for $|m(\lambda)|$ corresponding to different regularity classes of $\psi$.

A. $C^\alpha$ Regularity (Proposition 1.1)

• Class: $\psi \in C^\alpha(I)$ for integer $\alpha \ge 1$.
• Result: $|m(\lambda)| = O(\log \lambda)$.
• Sharpness: There exists $\psi \in C^\alpha$ such that $|m(\lambda)| \approx \log \lambda$.

B. $C^\infty$ Regularity (Theorem 1.2)

• Class: $\psi \in C^\infty(I)$.
• Result: $|m(\lambda)| = o(\log \lambda)$.
• Sharpness: For any $k \ge 2$, there exists $\psi \in C^\infty$ such that along a sequence $\lambda_n \to \infty$:
  $$|m(\lambda_n)| \approx \frac{\log \lambda_n}{\log^{(k)} \lambda_n}$$
  where $\log^{(k)}$ is the $k$-fold iterated logarithm. This resolves the Wang–Zhang problem by showing the integral is not bounded for arbitrary smooth phases, but grows slower than $\log \lambda$.

C. Gevrey Classes $G^s$ (Theorem 1.3)

• Class: $\psi \in G^s(I)$ with $s > 1$ (intermediate between $C^\infty$ and real-analytic $G^1$).
  • Definition: $|\psi^{(n)}(t)| \le K^{n+1} (n!)^s$.
• Result: $|m(\lambda)| = O(\log \log \lambda)$.
• Sharpness: There exists $\psi \in G^s$ such that $|m(\lambda_n)| \approx \log \log \lambda_n$.
• Note: For $s=1$ (real-analytic), $m(\lambda) = O(1)$ (known result).

D. Refined Gevrey Classes $G^s_k$ (Theorem 1.4)

• Class: Refined classes between $G^s$ ($s>1$) and $G^1$. Defined by derivative bounds involving iterated logarithms:
  $$|\psi^{(n)}(t)| \le K^{n+1} n! (\log^{(k)} n)^s \prod_{j=1}^{k-1} \log^{(j)} n$$
• Result: $|m(\lambda)| = O(\log^{(k+2)} \lambda)$.
• Sharpness: There exists $\psi \in G^s_k$ such that $|m(\lambda_n)| \approx \log^{(k+2)} \lambda_n$.
• Significance: This fills the gap between the Gevrey classes and the analytic class, showing how the growth rate slows down as the regularity approaches analyticity.

E. Classes Beyond Gevrey (Sections 7 & 8)

The paper extends the analysis to classes larger than any Gevrey class, defined by sequences like $M_n = \exp(c n^\alpha)$ or $M_n = \exp^{(k)}(c n^\alpha)$.

• Result: As the class becomes "larger" (less regular), the maximal growth increases, approaching the universal $C^\infty$ bound of $o(\log \lambda)$.
  • For $M_n = \exp(c n^\alpha)$, growth is $O((\log \lambda)^{1 - 1/\alpha})$.
  • For $M_n = \exp^{(k)}(c n^\alpha)$, growth is $O\left(\frac{\log \lambda}{(\log^{(k)} \lambda)^{1/\alpha}}\right)$.

4. Significance

1. Resolution of Open Problems: The paper definitively answers the question posed by Wang–Zhang regarding the boundedness of the Stein-Wainger integral for smooth phases, proving it is unbounded but grows sub-logarithmically.
2. New Proof of Nagel-Wainger: The results provide a new proof of the Nagel–Wainger theorem (1994) regarding the unboundedness of the Hilbert transform along curves with flat phases, by explicitly linking the unboundedness to the growth of $m(\lambda)$.
3. Hierarchy of Regularity: The work establishes a rigorous "spectrum" of growth rates. It demonstrates that the maximal growth of the oscillatory integral is a sensitive indicator of the phase function's regularity, distinguishing between $C^\alpha$, $C^\infty$, Gevrey, and intermediate classes with high precision.
4. Methodological Advancement: The paper successfully applies and refines Bang's Lemma and the Taylor-Legendre method to derive sharp flat-point estimates for a wide variety of Denjoy-Carleman classes, offering a unified framework for analyzing singular oscillatory integrals.

In summary, the paper provides a complete characterization of how the regularity of the phase function dictates the divergence rate of the Stein-Wainger integral, bridging the gap between polynomial/rational boundedness and the logarithmic growth of general smooth functions.