Boundedness of Fourier Integral Operators

• Fourier integral operators (FIOs) are oscillatory integral operators with nondegenerate phase functions and precise symbol estimates, ensuring bounded mappings across Lebesgue, Hardy, and Sobolev spaces.
• The boundedness thresholds are derived using methods like T*T and atomic decompositions, which reveal sharp endpoint regularity and optimal scaling conditions in various function spaces.
• Recent research extends these estimates to multilinear, multi-parameter, and complex-phase contexts, enhancing applications in dispersive PDEs and microlocal analysis.

Boundedness estimates for Fourier integral operators (FIOs) constitute a central aspect of microlocal and harmonic analysis, governing the mapping properties of oscillatory integral operators arising in linear and nonlinear partial differential equations, pseudodifferential operator theory, and related fields. These estimates characterize the range of Lebesgue, Hardy, Sobolev, and function spaces on which FIOs act boundedly, and further establish sharp thresholds linked to the regularity and oscillatory structure of amplitude-symbol and phase data. Classical local $L^2$ and $L^p$ theory has been systematically extended to global settings, endpoint space phenomena, multilinear and multi-parameter contexts, and operators with rough symbols or complex phase; moreover, semiclassical $h$-FIOs and generalized SG/Fourier-Lebesgue frameworks play an essential role in contemporary analysis.

1. Model Classes of FIOs and Symbol/Phase Hypotheses

A prototypical FIO on $\mathbb{R}^n$ takes the form

$$F f(x) = \int_{\mathbb{R}^n} e^{i\varphi(x,\xi)} a(x,\xi) \widehat{f}(\xi) \,d\xi,$$

where the phase $\varphi(x, \xi)$ is smooth, real- or complex-valued, and homogeneous of degree one in $\xi$. The symbol $a(x, \xi)$ belongs to a Hörmander class $S^m_{\rho,\delta}$ or its multi-parameter and anisotropic generalizations, with estimates of the form

$$|\partial_x^\alpha \partial_\xi^\beta a(x, \xi)| \leq C_{\alpha\beta} (1 + |\xi|)^{m - \rho|\beta| + \delta|\alpha|}.$$

Key structural assumptions include:

• Strong non-degeneracy (SND): The mixed Hessian matrix $\partial^2_{x_j\xi_k}\varphi(x,\xi)$ is everywhere invertible, i.e., $|\det \partial^2_{x\xi} \varphi(x,\xi)| \geq c > 0$.
• Homogeneity and regularity: The canonical phase class $\Phi^2$ imposes homogeneity and global polynomial bounds on all derivatives.

For semiclassical and SG-type FIOs, additional dependence on a small parameter $h$ (arising in pseudodifferential quantization and microlocal propagation) and global phase (not necessarily homogeneous) are allowed (1302.38461412.8050). In multilinear settings, each input variable's frequency variable admits its own phase component, and symbol classes depend on multi-indices in both position and frequency variables (Rodriguez-Lopez et al., 2019).

2. $L^2$-Boundedness and Calderón–Vaillancourt Paradigm

For FIOs with smooth amplitudes in $S^m_{1,0}$ and globally nondegenerate real phase, the Calderón–Vaillancourt and Asada–Fujiwara techniques establish $L^2$-boundedness whenever $m \le 0$, often requiring only uniform boundedness of finite derivatives of $a(x, \xi)$. In semiclassical quantization ($h\to0$), the $L^2$-operator norm of $F_h$ is uniformly controlled by the supremum of the $h$-dependent weight function $m(x,\xi; h)$ defining the symbol class $I^m$; compactness is further characterized by the vanishing of $m(x,\xi; h)$ at infinity (Chahrazed et al., 2013).

The proof leverages the $T^*T$ method: the composition $F F^*$ (or $F^* F$) is a pseudodifferential operator with full symbol $$|a(x, \theta)|^2 |\det \partial_x \partial_\theta S(x, \theta)|^{-1}$$, reducing boundedness to established results for such operators (1302.38461412.8050). For generalized SG-FIOs, the amplitude's uniform boundedness and regularity of the phase in the SG-symbol framework yield $L^2$-boundedness under only mild decay (Coriasco et al., 2014).

3. Endpoint and Hardy Space Boundedness

Sharp endpoint regularity is revealed in the mapping properties of FIOs from Hardy (or local Hardy) spaces $h^1$ to $L^1$, and from $L^\infty$ to BMO (Wang et al., 2024Ye et al., 2024Zhu et al., 2024). The Seeger–Sogge–Stein theorem gives $H^1 \to L^1$ for real nondegenerate phases and symbols in $S^{-(n-1)/2}_{1,0}$, which is optimal and cannot be improved for elliptic canonical relations (Cardona et al., 2021).

Recent refinements establish:

• Forbidden symbol classes: $H^1 \to L^1$ boundedness for $S^{(1-n)/2}_{1,1}$ when $n \ge 2$ (Ye et al., 2024); however, the result fails for $n=1$, even for smooth phases (corresponds to loss at the critical/endpoint scaling, e.g., Guo-Zhu counterexample).
• Sharpness with respect to the symbol order and $\delta$ parameter: For $S^m_{0, \delta}$ ($0 \le \delta < 1$), endpoint boundedness holds if and only if $m \le -n/2 - n\delta/2$ for $H^1 \to L^1$ and $m \le -n/2$ for $L^\infty \to$BMO (Wang et al., 2024).
• Complex phase theory: Weak $(1,1)$-boundedness for FIOs of order $-(n-1)/2$ parametrized by canonical relations with complex phase, provided the local graph condition and Kohn–Nirenberg-type symbol estimates in $S^{-(n-1)/2}_{1/2,1/2}$ (Cardona et al., 2024).

In all endpoint Hardy settings, atomic decomposition and careful dyadic/angular decomposition methods are employed, with localized $L^2$ bounds, integration-by-parts, and precise kernel decay estimates controlling both near and far field contributions (Ye et al., 2024Zhu et al., 2024).

4. $L^p$-Mapping Theorems and Interpolation

For $1 < p < \infty$, the sharp threshold for $L^p$-boundedness of classical FIOs with nondegenerate real phase is governed by the Seeger–Sogge–Stein bound: $m \le - (n-1) |\tfrac{1}{p} - \tfrac{1}{2}|,$ with symbol in $S^m_{1,0}$ (Cardona et al., 2021). Analogous parameters control global $L^p$ estimates for FIOs on the full Euclidean space, under global graph-type nondegeneracy and slowly-growing symbol classes—see the local-to-global transference theorems (Ruzhansky et al., 2015Ruzhansky et al., 2017).

Extensions include:

• Multi-parameter and product FIOs: In the $d$-parameter case, with $\Phi(x, \xi) = \sum_{i=1}^d \Phi_i(x_i, \xi_i)$, the threshold becomes $m \le -(n-d)|1/p - 1/2|$ (with $n=\sum n_i$ and $n_i \geq 2$) (Cheng, 2023).
• Product-structure symbol classes: Even with stronger (multi-parameter) symbol regularity, the optimal exponents remain as in the classical case (Tan et al., 2024).
• Bilinear and multilinear FIOs: Mapping $L^{q_1} \times \cdots \times L^{q_N} \to L^r$ is possible when the total order $m$ obeys multilinear analogues of the $L^p$-threshold (cf. $$m_c = -(n-1)(\sum_{j=1}^N (1/p_j - 1/2) + (1/p_0 - 1/2))$$), with $1/p_0 = \sum_{j=1}^N 1/p_j$ and symbol class $S^m(n,N)$ (Rodriguez-Lopez et al., 2019).
• Global regularity for FIOs on noncompact symmetric spaces: Endpoint $h^1\to L^1$ and $h^1\to h^1$ boundedness follows when the symbol belongs to suitable Harish-Chandra classes, with explicit exponential-in-time growth rates (Bruno et al., 2016).

Interpolation (typically Fefferman–Stein or analytic family method) is used to deduce $L^p$-estimates from the $L^2$ and Hardy/BMO endpoints (Zhu et al., 2024Wang et al., 2024). For FIOs with allowed $\delta \to 1$ in symbol regularity, finer interpolation and fractional integration techniques manage endpoint loss (Ye et al., 2024).

5. Techniques and Structural Proof Ideas

The central techniques employed across this literature include:

• Dyadic & cone decomposition: Decomposition in $|\xi|$ and angular variables, localizing the operator's kernel near specific "directions" to control nonstationary phase and integration-by-parts arguments (Ye et al., 2024Tan et al., 2024).
• Atomic decomposition: Reduction of $H^1$ or $h^1$-control to single atom estimates, with attention to exceptional (non-stationary-phase) sets, and use of volume and $L^2$ bounds or cancellation (Zhu et al., 2024Wang et al., 2024).
• Reduction to $T^*T$ structure: Principal symbol computation via adjoint arguments and identification with pseudodifferential operators; off-diagonal kernel decay leading to Schur/Cotlar estimates and kernel integrability (1302.38461412.8050).

In multi-parameter and multilinear settings, almost-orthogonal decompositions and nested blockwise frequency localization are critical (Cheng, 2023Hong et al., 2015Rodriguez-Lopez et al., 2019). For rough amplitude symbols, boundedness relies on localized control and integration-by-parts decay in each variable (Rodríguez-López et al., 2013).

6. Global, Multilinear, and Complex-Phase Extensions

Recent advances address:

• Global versus local theory: Transference methods and explicit decay estimates for the oscillatory kernel allow extension of local operator-boundedness to global $L^p$-estimates under mild phase and symbol growth constraints (Ruzhansky et al., 2015Ruzhansky et al., 2017).
• Fourier-Lebesgue spaces: For FIOs with complex phase and canonical relations satisfying spatial smooth factorization conditions, mapping $\mathcal{F}L^p \to \mathcal{F}L^p$ holds at orders $m \le -\varkappa |1/p-1/2|$ (sharp in the rank $\varkappa$ of the spatial Hessian) (Cardona et al., 31 Dec 2025).
• Invariant Hardy spaces $\mathcal{H}^p_{FIO}$: Global boundedness of order-zero operators on the full scale of Hardy spaces defined via wave packet and tent space transforms on the cosphere bundle (Hassell et al., 2018).

Multilinear and bilinear FIOs are analyzed with endpoint results (e.g., $H^1\times L^\infty \to L^1$ at $m=-n/2$ for bilinear amplitudes in $BS_{1,0}^{-n/2}$ and nondegenerate phase, improving linear thresholds) (Kato et al., 2023Rodriguez-Lopez et al., 2011).

7. Significance, Applications, and Optimality

Boundedness thresholds for FIOs reflect the fundamental geometric and analytic microlocal properties of wave and oscillatory phenomena:

• Propagation of singularities for hyperbolic equations is controlled by FIO mapping properties; the $L^p$-threshold determines Sobolev exponents in dispersive estimates and microlocal parametrices (Ruzhansky et al., 2015).
• Endpoint sharpness—for both real and complex phase, and in the presence of forbidden symbols—is crucial for regularity theory in dispersive PDE, control of energy flow in nonlinear equations, and explicit counterexamples at or above threshold (Cardona et al., 2021).
• Extensions to non-Euclidean or symmetric spaces retain endpoint structure, with time-growth and exponential normalization reflecting the geometry of the underlying manifold (Bruno et al., 2016).

Future work addresses global theory for multilinear/multi-parameter operators, sharp weighted and endpoint regularity on noncompact manifolds, and decoupling inequalities for FIOs in higher codimensional and degenerate settings.

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References:

• (Chahrazed et al., 2013, Coriasco et al., 2014, Hong et al., 2015, Ruzhansky et al., 2015, Bruno et al., 2016, Ruzhansky et al., 2017, Hassell et al., 2018, Rodriguez-Lopez et al., 2019, Cardona et al., 2021, Kato et al., 2023, Cheng, 2023, Cardona et al., 2024, Ye et al., 2024, Tan et al., 2024, Zhu et al., 2024, Wang et al., 2024, Cardona et al., 31 Dec 2025)