Spherical maximal functions and Hardy spaces for Fourier integral operators

1. Problem Statement

The paper addresses the $L^p$ boundedness of spherical maximal functions and related operators associated with the wave equation. Specifically, it investigates the boundedness of the maximal operator:
$$M_{[1,2]}f(x) := \sup_{1 \le t \le 2} \left| \int_{S^{n-1}} f(x - ty) \, d\sigma(y) \right|$$
where $n \ge 2$.

Context and Motivation:

• Classical Results: Stein (1976) and Bourgain (1986) established that the full spherical maximal function (supremum over $t \in (0, \infty)$) is bounded on $L^p(\mathbb{R}^n)$ for $p > \frac{n}{n-1}$ (for $n \ge 3$) and $p > 2$ (for $n=2$).
• Limitations: For the restricted maximal function $M_{[1,2]}$, previous work (e.g., Sogge, Stein, Mockenhoupt, Seeger) established bounds involving Sobolev spaces $W^{s,p}$. However, the optimal regularity index $s$ required for these bounds was not fully determined, particularly for the range $1 \le p \le 2$ and the high-frequency range $p \ge \frac{2(n+1)}{n-1}$.
• Gap: The authors note that for general $p$, the optimal condition on the Sobolev exponent $s$ for inequalities of the form $\|M_{[1,2]}f\|_{L^p} \lesssim \|f\|_{W^{s,p}}$ was unknown. Furthermore, standard Sobolev spaces may not be the sharpest setting for these operators due to the specific geometric nature of Fourier Integral Operators (FIOs).

2. Methodology

The authors employ the theory of Hardy spaces for Fourier Integral Operators ($H^p_{FIO}$), introduced by Smith (1998) and extended by Hassell, Portal, and Rozendaal (2020).

Key Technical Tools:

1. Hardy Spaces for FIOs ($H^p_{FIO}$): These spaces are invariant under FIOs and satisfy sharp Sobolev embeddings:
   $$W^{s(p), p} \subset H^p_{FIO} \subset W^{-s(p), p}$$
   where $s(p) = \frac{n-1}{2}|\frac{1}{2} - \frac{1}{p}|$. This allows the authors to work in a space that is "larger" than $L^p$ for $p \neq 2$ but tailored to the singularities of wave propagation.
2. Decoupling Inequalities: The proofs rely heavily on $\ell^p$ decoupling inequalities (Bourgain-Demeter) for hypersurfaces with non-vanishing Gaussian curvature to establish local smoothing estimates.
3. Microlocal Analysis: To prove sharpness, the authors use microlocalization techniques. They construct specific "Knapp-type" examples (focusing wave packets) and analyze the phase space behavior of the operators to show that the derived bounds cannot be improved.
4. Variable Coefficients and Manifolds: The methodology is extended from constant coefficient operators on $\mathbb{R}^n$ to variable coefficients and compact Riemannian manifolds by utilizing local coordinate charts and the invariance of $H^p_{FIO}$ under coordinate changes.

3. Key Contributions and Results

A. Main Theorems on $\mathbb{R}^n$

The paper establishes sharp $L^p$ bounds for spherical maximal functions and wave propagators when the initial data lies in the Sobolev space over the FIO Hardy space, denoted $H^{s,p}_{FIO}(\mathbb{R}^n)$.

Definitions:
Let $s(p) = \frac{n-1}{2}|\frac{1}{2} - \frac{1}{p}|$ and define $d(p)$ as:
$$d(p) = \begin{cases} s(p) - \frac{1}{p} & \text{if } \frac{2(n+1)}{n-1} \le p \le \infty \\ 0 & \text{if } 2 \le p < \frac{2(n+1)}{n-1} \\ s(p) & \text{if } 1 \le p \le 2 \end{cases}$$

Theorem 1.1 (Spherical Maximal Functions):
For $p \in [1, \infty)$ and $s > d(p) + \frac{1}{p} - \frac{n-1}{2}$, there exists a constant $C$ such that:
$$\|M_{[1,2]}f\|_{L^p(\mathbb{R}^n)} \le C \|f\|_{H^{s,p}_{FIO}(\mathbb{R}^n)}$$

• Sharpness: This bound is essentially sharp for $p \in [1, 2] \cup [\frac{2(n+1)}{n-1}, \infty)$.
• Extension: The result holds for arbitrary compact hypersurfaces with non-vanishing Gaussian curvature and for complex spherical means of order $\alpha \in \mathbb{C}$ (requiring $s > d(p) + \frac{1}{p} - \frac{n-1}{2} - \text{Re}(\alpha)$).

Theorem 1.2 (Wave Equation Maximal Functions):
For the Euclidean wave equation, for $p \in [1, \infty)$ and $s > d(p) + \frac{1}{p}$:
$$\left\| \sup_{0 \le t \le 1} |e^{it\sqrt{-\Delta}}f| \right\|_{L^p(\mathbb{R}^n)} \le C \|f\|_{H^{s,p}_{FIO}(\mathbb{R}^n)}$$
This implies pointwise convergence $e^{it\sqrt{-\Delta}}f(x) \to f(x)$ almost everywhere as $t \to 0$.

• Sharpness: The bound is essentially sharp for $p \in [1, 2] \cup [\frac{2(n+1)}{n-1}, \infty)$.

B. Results on Compact Manifolds

The authors extend these results to compact Riemannian manifolds $(M, g)$:

• Proposition 3.15: Analogous maximal bounds for the wave propagator $e^{it\sqrt{-\Delta_g}}$ on $M$.
• Proposition 3.16: Maximal bounds for geodesic spheres on $M$.
• Sharpness: These bounds are shown to be essentially sharp for the same ranges of $p$ as in the Euclidean case.

C. Comparison with Classical Sobolev Bounds

The paper highlights a crucial distinction:

• If one restricts the initial data to standard Sobolev spaces $W^{s,p}$, the resulting bounds are strictly weaker than those obtained using $H^{s,p}_{FIO}$ for $p \neq 2$.
• For $p \ge \frac{2(n+1)}{n-1}$, the condition on $s$ derived via $H^p_{FIO}$ improves upon previous known results for $W^{s,p}$, even in cases where the "optimal" Sobolev exponent was previously thought to be established.
• For $p \in [1, 2]$, the use of $H^p_{FIO}$ allows for a sharper analysis of the "focusing" of wave packets that standard $L^p$ estimates miss.

4. Significance and Impact

1. Resolution of Sharpness: The paper provides essentially sharp bounds for spherical maximal functions and wave equation maximal functions for the full range $p \in [1, 2] \cup [\frac{2(n+1)}{n-1}, \infty)$. It clarifies the optimal regularity required for these operators, a question that was previously open for general $p$.
2. Optimization of Function Spaces: It demonstrates that Hardy spaces for FIOs are the natural and optimal setting for studying maximal functions associated with wave propagation and curvature. Using these spaces yields stronger results than standard Sobolev spaces.
3. Geometric Generality: The results are not limited to flat space; they apply to general hypersurfaces with non-vanishing Gaussian curvature, complex spherical means, and compact Riemannian manifolds, unifying the theory across different geometric settings.
4. Pointwise Convergence: The maximal function bounds directly imply improved pointwise convergence results for solutions to the wave equation, ensuring convergence almost everywhere for initial data in $H^{s,p}_{FIO}$ with the optimal $s$.
5. Methodological Advancement: The proof techniques, particularly the combination of decoupling inequalities with FIO Hardy space theory and the construction of specific counterexamples for sharpness, provide a robust framework for future research in harmonic analysis and PDEs.

In summary, this paper significantly advances the understanding of maximal operators in harmonic analysis by identifying the precise function spaces ($H^p_{FIO}$) and regularity indices ($s$) required for sharp $L^p$ estimates, bridging the gap between classical $L^p$ theory and the geometric nature of wave propagation.