Carleson Operators in Harmonic Analysis

• Carleson operators are maximally modulated singular integrals that characterize convergence phenomena in Fourier analysis and underpin modern time-frequency methods.
• Generalizations include polynomial, oscillatory, and weighted variants that use specialized modulation symmetries and curvature to achieve L^p boundedness.
• Recent developments extend Carleson theory to abstract metric spaces, bilinear and anisotropic settings, revealing new insights into oscillatory behavior and endpoint estimates.

Carleson operators are maximally modulated singular integral operators whose prototype is the classical operator

$$Cf(x)=\sup_{N\in\mathbb R}\left|\int_{\mathbb R} f(x-t)e^{iNt}\,\frac{dt}{t}\right|,$$

a central object in harmonic analysis because its boundedness is equivalent to almost-everywhere convergence phenomena for Fourier expansions and partial Fourier integrals (Ramos, 2020). In contemporary usage, the term also encompasses polynomial, oscillatory, variational, anisotropic, multilinear, and abstract metric-space analogues, all of which preserve the characteristic feature of taking a supremum over a modulation family while retaining a singular kernel. The subject now spans classical time-frequency analysis, oscillatory integral theory, weighted inequalities, vector-valued extensions, and non-Euclidean generalizations (Lie, 2011).

1. Classical operator and the convergence problem

The classical Carleson operator can be written either as the maximally modulated Hilbert transform

$$\mathcal C f(x)=\sup_{\xi\in\mathbb R}\bigl|H(M_\xi f)(x)\bigr|, \qquad M_\xi f(x)=e^{2\pi i \xi x}f(x),$$

or in the oscillatory singular-integral form displayed above (Plinio et al., 2013). In one dimension, this operator governs maximal partial Fourier integrals and is the analytic core of Carleson’s theorem on almost-everywhere convergence of Fourier series.

A detailed modern metric-space presentation states the classical theorem in quantitative Egorov form: for a $2\pi$-periodic uniformly continuous $f$ with $|f(x)|\le 1$, and every $0<\epsilon<1$, there is a Borel set $E\subset[0,2\pi]$ with $|E|\le\epsilon$ and an integer $N_0$ such that

$|f(x)-S_Nf(x)|\le \epsilon$

for all $$\mathcal C f(x)=\sup_{\xi\in\mathbb R}\bigl|H(M_\xi f)(x)\bigr|, \qquad M_\xi f(x)=e^{2\pi i \xi x}f(x),$$0 and all $$\mathcal C f(x)=\sup_{\xi\in\mathbb R}\bigl|H(M_\xi f)(x)\bigr|, \qquad M_\xi f(x)=e^{2\pi i \xi x}f(x),$$1 (Becker et al., 2024). That work also makes explicit the reduction from Fourier partial sums to a real-line Carleson maximal operator built from the localized kernel

$$\mathcal C f(x)=\sup_{\xi\in\mathbb R}\bigl|H(M_\xi f)(x)\bigr|, \qquad M_\xi f(x)=e^{2\pi i \xi x}f(x),$$2

thereby exhibiting the classical theorem as a special case of a broader Carleson framework on doubling metric measure spaces (Becker et al., 2024).

Classical boundedness is now understood in several complementary forms. Carleson’s original theorem gives the $$\mathcal C f(x)=\sup_{\xi\in\mathbb R}\bigl|H(M_\xi f)(x)\bigr|, \qquad M_\xi f(x)=e^{2\pi i \xi x}f(x),$$3 result, while later developments yield $$\mathcal C f(x)=\sup_{\xi\in\mathbb R}\bigl|H(M_\xi f)(x)\bigr|, \qquad M_\xi f(x)=e^{2\pi i \xi x}f(x),$$4-boundedness for every $$\mathcal C f(x)=\sup_{\xi\in\mathbb R}\bigl|H(M_\xi f)(x)\bigr|, \qquad M_\xi f(x)=e^{2\pi i \xi x}f(x),$$5 (Ramos, 2020). In the weighted setting, Di Plinio and Lerner proved that the classical operator satisfies the same linear $$\mathcal C f(x)=\sup_{\xi\in\mathbb R}\bigl|H(M_\xi f)(x)\bigr|, \qquad M_\xi f(x)=e^{2\pi i \xi x}f(x),$$6-dependence as the Hilbert transform for $$\mathcal C f(x)=\sup_{\xi\in\mathbb R}\bigl|H(M_\xi f)(x)\bigr|, \qquad M_\xi f(x)=e^{2\pi i \xi x}f(x),$$7: $$\mathcal C f(x)=\sup_{\xi\in\mathbb R}\bigl|H(M_\xi f)(x)\bigr|, \qquad M_\xi f(x)=e^{2\pi i \xi x}f(x),$$8 and also obtained the $$\mathcal C f(x)=\sup_{\xi\in\mathbb R}\bigl|H(M_\xi f)(x)\bigr|, \qquad M_\xi f(x)=e^{2\pi i \xi x}f(x),$$9-bound

$2\pi$0

showing that weighted Carleson theory reflects the near-$2\pi$1 endpoint behavior of Fourier convergence (Plinio et al., 2013).

2. Polynomial and oscillatory generalizations

A fundamental nonlinear extension is the polynomial Carleson operator

$2\pi$2

whose one-dimensional $2\pi$3-boundedness for $2\pi$4 was proved by Victor Lie, resolving the one-dimensional case of Stein’s conjecture (Ramos, 2020). Lie’s formulation emphasizes the generalized modulation symmetries

$2\pi$5

and recasts the proof in a higher-order wave-packet framework adapted to polynomial phases rather than linear frequencies (Lie, 2011).

That higher-order framework replaces ordinary time-frequency rectangles by curved polynomial tiles. In Lie’s one-dimensional construction, a tile is a $2\pi$6-tuple

$2\pi$7

encoding a localized neighborhood of a polynomial graph over $2\pi$8. The operator is decomposed as

$2\pi$9

and the proof proceeds through trees, forests, density and geometric factors, and a mass decomposition that yields BMO control of counting functions and eliminates exceptional sets from the analysis (Lie, 2011). One consequence highlighted in that paper is a direct strong $f$0 argument for generalized Carleson operators, without passing through interpolation from weak-$f$1 (Lie, 2011).

A more restricted polynomial family fixes the nonlinear phase and maximizes only over a linear modulation: $f$2 Ramos proved that $f$3 is bounded on $f$4, $f$5, with bounds depending only on $f$6 and $f$7, and did so by reducing the problem to the classical linear Carleson operator together with a quadratic analysis, rather than using the full polynomial Carleson machinery (Ramos, 2020). The result is conceptually narrower than the full theorem of Lie and Zorin-Kranich, but it isolates a mechanism by which linear Carleson control forces boundedness for a nonlinear modulation family (Ramos, 2020).

Oscillatory Carleson operators with nonlinear monomial phase form another major branch: $f$8 For $f$9, $|f(x)|\le 1$0, Ramos proved $|f(x)|\le 1$1-boundedness for $|f(x)|\le 1$2 together with the uniform estimate

$|f(x)|\le 1$3

thereby controlling the constants in a neighborhood of the delicate quadratic regime $|f(x)|\le 1$4 (Ramos, 2020). Earlier proofs treated $|f(x)|\le 1$5 and $|f(x)|\le 1$6 by sharply different methods and lost control as $|f(x)|\le 1$7; the novelty is the uniformity in $|f(x)|\le 1$8 (Ramos, 2020). The same paper notes that the analogous odd-phase family with $|f(x)|\le 1$9 admits the same local uniformity near $0<\epsilon<1$0, while the regimes $0<\epsilon<1$1 and especially $0<\epsilon<1$2 remain more delicate (Ramos, 2020).

3. Weighted, variational, and vector-valued forms

Weighted Carleson theory has developed into a precise extension of sharp Calderón–Zygmund inequalities. Di Plinio and Lerner treated the classical Carleson operator $0<\epsilon<1$3, its lacunary version $0<\epsilon<1$4, and the Walsh-Carleson operator $0<\epsilon<1$5 using local mean oscillation and sparse domination rather than the original time-frequency decomposition (Plinio et al., 2013). Their abstract framework applies to maximally modulated Calderón–Zygmund operators

$0<\epsilon<1$6

under weak-$0<\epsilon<1$7 input near $0<\epsilon<1$8 (Plinio et al., 2013).

For the classical operator they proved the weak estimate

$0<\epsilon<1$9

while for the lacunary operator

$E\subset[0,2\pi]$0

and for the Walsh-Carleson operator

$E\subset[0,2\pi]$1

(Plinio et al., 2013). The resulting $E\subset[0,2\pi]$2-bounds exhibit a hierarchy: a doubly logarithmic loss for $E\subset[0,2\pi]$3, a single logarithm for $E\subset[0,2\pi]$4, and no logarithmic loss for $E\subset[0,2\pi]$5, mirroring the corresponding endpoint theories (Plinio et al., 2013).

A different refinement replaces maximal control by variation control. For Banach-space-valued functions $E\subset[0,2\pi]$6, partial Fourier integrals are

$E\subset[0,2\pi]$7

and the $E\subset[0,2\pi]$8-variational Carleson operator is

$E\subset[0,2\pi]$9

Hytönen, Lorist, and Veraar proved that if $|E|\le\epsilon$0 is an $|E|\le\epsilon$1-intermediate UMD space, then

$|E|\le\epsilon$2

for all $|E|\le\epsilon$3 and $|E|\le\epsilon$4 (Amenta et al., 2020). This strengthens vector-valued Carleson–Hunt theory by controlling the oscillation of the truncation path $|E|\le\epsilon$5, not merely its supremum (Amenta et al., 2020).

Their method combines time-frequency analysis with outer Lebesgue spaces on the time-frequency-scale domain $|E|\le\epsilon$6, using a wave-packet embedding

$|E|\le\epsilon$7

and a truncated embedding $|E|\le\epsilon$8 adapted to the linearized variation increments (Amenta et al., 2020). The result recovers the maximal vector-valued Carleson theorem for intermediate UMD spaces as the limiting case $|E|\le\epsilon$9, but its actual content is stricter: finite $N_0$0-variation yields almost-everywhere convergence with quantitative oscillation control (Amenta et al., 2020).

4. Bilinear, anisotropic, and geometric variants

Carleson-type maximal modulation also appears in bilinear settings. The non-resonant bilinear Hilbert–Carleson operators are

$N_0$1

with $N_0$2 interpreted as $N_0$3 or $N_0$4 for noninteger $N_0$5 (Benea et al., 2021). These operators interpolate between the bilinear Hilbert transform, recovered at $N_0$6, and nonlinear-phase Carleson operators, recovered when one input is frozen to $N_0$7 (Benea et al., 2021).

For every non-resonant exponent

$N_0$8

Guo, Roos, and Yung proved that $N_0$9 maps $|f(x)-S_Nf(x)|\le \epsilon$0 to $|f(x)-S_Nf(x)|\le \epsilon$1 whenever

$|f(x)-S_Nf(x)|\le \epsilon$2

and they established decay for high-oscillation pieces: $|f(x)-S_Nf(x)|\le \epsilon$3 for some $|f(x)-S_Nf(x)|\le \epsilon$4 (Benea et al., 2021). The excluded exponents $|f(x)-S_Nf(x)|\le \epsilon$5 are resonant because additional modulation symmetries destroy the scale-decay mechanism (Benea et al., 2021). The paper describes the resulting operators as genuinely hybrid, combining zero-curvature features inherited from bilinear Hilbert transform modulation symmetry with nonzero-curvature oscillatory cancellation (Benea et al., 2021).

In higher-dimensional anisotropic Euclidean settings, Roos studied

$|f(x)-S_Nf(x)|\le \epsilon$6

where $|f(x)-S_Nf(x)|\le \epsilon$7 is a bounded $|f(x)-S_Nf(x)|\le \epsilon$8 multiplier homogeneous of degree $|f(x)-S_Nf(x)|\le \epsilon$9 under anisotropic dilations

$$\mathcal C f(x)=\sup_{\xi\in\mathbb R}\bigl|H(M_\xi f)(x)\bigr|, \qquad M_\xi f(x)=e^{2\pi i \xi x}f(x),$$00

(Roos, 2017). If $$\mathcal C f(x)=\sup_{\xi\in\mathbb R}\bigl|H(M_\xi f)(x)\bigr|, \qquad M_\xi f(x)=e^{2\pi i \xi x}f(x),$$01, then

$$\mathcal C f(x)=\sup_{\xi\in\mathbb R}\bigl|H(M_\xi f)(x)\bigr|, \qquad M_\xi f(x)=e^{2\pi i \xi x}f(x),$$02

recovering the classical weak $$\mathcal C f(x)=\sup_{\xi\in\mathbb R}\bigl|H(M_\xi f)(x)\bigr|, \qquad M_\xi f(x)=e^{2\pi i \xi x}f(x),$$03 Carleson bound when $$\mathcal C f(x)=\sup_{\xi\in\mathbb R}\bigl|H(M_\xi f)(x)\bigr|, \qquad M_\xi f(x)=e^{2\pi i \xi x}f(x),$$04, $$\mathcal C f(x)=\sup_{\xi\in\mathbb R}\bigl|H(M_\xi f)(x)\bigr|, \qquad M_\xi f(x)=e^{2\pi i \xi x}f(x),$$05 (Roos, 2017). The proof extends the Lacey–Thiele tree method to anisotropic tiles, anisotropic wave packets, and an anisotropic cone decomposition (Roos, 2017).

A geometric Radon-transform analogue appears in polynomial Carleson operators along the monomial curve $$\mathcal C f(x)=\sup_{\xi\in\mathbb R}\bigl|H(M_\xi f)(x)\bigr|, \qquad M_\xi f(x)=e^{2\pi i \xi x}f(x),$$06: $$\mathcal C f(x)=\sup_{\xi\in\mathbb R}\bigl|H(M_\xi f)(x)\bigr|, \qquad M_\xi f(x)=e^{2\pi i \xi x}f(x),$$07 Guo and Pierce studied partial linearizations in which the stopping-time function depends on only one ambient variable: $$\mathcal C f(x)=\sup_{\xi\in\mathbb R}\bigl|H(M_\xi f)(x)\bigr|, \qquad M_\xi f(x)=e^{2\pi i \xi x}f(x),$$08 and proved $$\mathcal C f(x)=\sup_{\xi\in\mathbb R}\bigl|H(M_\xi f)(x)\bigr|, \qquad M_\xi f(x)=e^{2\pi i \xi x}f(x),$$09-boundedness for all $$\mathcal C f(x)=\sup_{\xi\in\mathbb R}\bigl|H(M_\xi f)(x)\bigr|, \qquad M_\xi f(x)=e^{2\pi i \xi x}f(x),$$10 when $$\mathcal C f(x)=\sup_{\xi\in\mathbb R}\bigl|H(M_\xi f)(x)\bigr|, \qquad M_\xi f(x)=e^{2\pi i \xi x}f(x),$$11 and $$\mathcal C f(x)=\sup_{\xi\in\mathbb R}\bigl|H(M_\xi f)(x)\bigr|, \qquad M_\xi f(x)=e^{2\pi i \xi x}f(x),$$12, as well as for the special modulation-invariant cases $$\mathcal C f(x)=\sup_{\xi\in\mathbb R}\bigl|H(M_\xi f)(x)\bigr|, \qquad M_\xi f(x)=e^{2\pi i \xi x}f(x),$$13 with $$\mathcal C f(x)=\sup_{\xi\in\mathbb R}\bigl|H(M_\xi f)(x)\bigr|, \qquad M_\xi f(x)=e^{2\pi i \xi x}f(x),$$14 and $$\mathcal C f(x)=\sup_{\xi\in\mathbb R}\bigl|H(M_\xi f)(x)\bigr|, \qquad M_\xi f(x)=e^{2\pi i \xi x}f(x),$$15 with $$\mathcal C f(x)=\sup_{\xi\in\mathbb R}\bigl|H(M_\xi f)(x)\bigr|, \qquad M_\xi f(x)=e^{2\pi i \xi x}f(x),$$16 (Guo et al., 2016). Their analysis combines Stein–Wainger-style $$\mathcal C f(x)=\sup_{\xi\in\mathbb R}\bigl|H(M_\xi f)(x)\bigr|, \qquad M_\xi f(x)=e^{2\pi i \xi x}f(x),$$17 estimates, including phases with fractional monomials, with vector-valued Carleson–Hunt bounds in the symmetric cases (Guo et al., 2016). The same paper observes that $$\mathcal C f(x)=\sup_{\xi\in\mathbb R}\bigl|H(M_\xi f)(x)\bigr|, \qquad M_\xi f(x)=e^{2\pi i \xi x}f(x),$$18 bounds for certain partial curved operators imply the usual Carleson theorem or even the quadratic Carleson theorem, so these are not merely auxiliary models (Guo et al., 2016).

A related higher-dimensional geometric variant acts along the paraboloid in $$\mathcal C f(x)=\sup_{\xi\in\mathbb R}\bigl|H(M_\xi f)(x)\bigr|, \qquad M_\xi f(x)=e^{2\pi i \xi x}f(x),$$19: $$\mathcal C f(x)=\sup_{\xi\in\mathbb R}\bigl|H(M_\xi f)(x)\bigr|, \qquad M_\xi f(x)=e^{2\pi i \xi x}f(x),$$20 where the phase has no linear term and its quadratic homogeneous part is not a nonzero multiple of $$\mathcal C f(x)=\sup_{\xi\in\mathbb R}\bigl|H(M_\xi f)(x)\bigr|, \qquad M_\xi f(x)=e^{2\pi i \xi x}f(x),$$21 (Pierce et al., 2015). For $$\mathcal C f(x)=\sup_{\xi\in\mathbb R}\bigl|H(M_\xi f)(x)\bigr|, \qquad M_\xi f(x)=e^{2\pi i \xi x}f(x),$$22, Pierce proved

$$\mathcal C f(x)=\sup_{\xi\in\mathbb R}\bigl|H(M_\xi f)(x)\bigr|, \qquad M_\xi f(x)=e^{2\pi i \xi x}f(x),$$23

using a Littlewood–Paley decomposition and square-function comparison with a smoother auxiliary operator, precisely because the usual stopping-time $$\mathcal C f(x)=\sup_{\xi\in\mathbb R}\bigl|H(M_\xi f)(x)\bigr|, \qquad M_\xi f(x)=e^{2\pi i \xi x}f(x),$$24 method fails in the Radon setting (Pierce et al., 2015).

5. Abstract metric-space formulations

Recent work has detached Carleson theory from Euclidean algebra and rebuilt it on doubling metric measure spaces. In that formulation, the ambient singular kernel is a one-sided Calderón–Zygmund kernel $$\mathcal C f(x)=\sup_{\xi\in\mathbb R}\bigl|H(M_\xi f)(x)\bigr|, \qquad M_\xi f(x)=e^{2\pi i \xi x}f(x),$$25 satisfying

$$\mathcal C f(x)=\sup_{\xi\in\mathbb R}\bigl|H(M_\xi f)(x)\bigr|, \qquad M_\xi f(x)=e^{2\pi i \xi x}f(x),$$26

together with Hölder regularity in the second variable (Becker et al., 2024). The generalized Carleson operator is then

$$\mathcal C f(x)=\sup_{\xi\in\mathbb R}\bigl|H(M_\xi f)(x)\bigr|, \qquad M_\xi f(x)=e^{2\pi i \xi x}f(x),$$27

where $$\mathcal C f(x)=\sup_{\xi\in\mathbb R}\bigl|H(M_\xi f)(x)\bigr|, \qquad M_\xi f(x)=e^{2\pi i \xi x}f(x),$$28 is a compatible cancellative collection of phase functions equipped, for each ball $$\mathcal C f(x)=\sup_{\xi\in\mathbb R}\bigl|H(M_\xi f)(x)\bigr|, \qquad M_\xi f(x)=e^{2\pi i \xi x}f(x),$$29, with a metric $$\mathcal C f(x)=\sup_{\xi\in\mathbb R}\bigl|H(M_\xi f)(x)\bigr|, \qquad M_\xi f(x)=e^{2\pi i \xi x}f(x),$$30 measuring relative oscillation (Becker et al., 2024).

The metric-space theorem of Buss, Cladek, and others proves a restricted weak-type bound: if the associated non-tangential maximal truncation $$\mathcal C f(x)=\sup_{\xi\in\mathbb R}\bigl|H(M_\xi f)(x)\bigr|, \qquad M_\xi f(x)=e^{2\pi i \xi x}f(x),$$31 satisfies

$$\mathcal C f(x)=\sup_{\xi\in\mathbb R}\bigl|H(M_\xi f)(x)\bigr|, \qquad M_\xi f(x)=e^{2\pi i \xi x}f(x),$$32

then for $$\mathcal C f(x)=\sup_{\xi\in\mathbb R}\bigl|H(M_\xi f)(x)\bigr|, \qquad M_\xi f(x)=e^{2\pi i \xi x}f(x),$$33, Borel sets $$\mathcal C f(x)=\sup_{\xi\in\mathbb R}\bigl|H(M_\xi f)(x)\bigr|, \qquad M_\xi f(x)=e^{2\pi i \xi x}f(x),$$34, and $$\mathcal C f(x)=\sup_{\xi\in\mathbb R}\bigl|H(M_\xi f)(x)\bigr|, \qquad M_\xi f(x)=e^{2\pi i \xi x}f(x),$$35,

$$\mathcal C f(x)=\sup_{\xi\in\mathbb R}\bigl|H(M_\xi f)(x)\bigr|, \qquad M_\xi f(x)=e^{2\pi i \xi x}f(x),$$36

(Becker et al., 2024). The 2025 sequel sharpens and streamlines the abstract theory, formulating main results as Theorem 1.1 and Theorem 1.2, and states that these results have been computer verified in Lean/mathlib, with the 2024 paper serving as the detailed formalization-oriented blueprint (Becker et al., 7 Aug 2025).

The central structural innovation is the axiomatization of the phase family. A compatible collection $$\mathcal C f(x)=\sup_{\xi\in\mathbb R}\bigl|H(M_\xi f)(x)\bigr|, \qquad M_\xi f(x)=e^{2\pi i \xi x}f(x),$$37 is required to satisfy oscillation control

$$\mathcal C f(x)=\sup_{\xi\in\mathbb R}\bigl|H(M_\xi f)(x)\bigr|, \qquad M_\xi f(x)=e^{2\pi i \xi x}f(x),$$38

monotonicity and doubling-type properties for the local metrics $$\mathcal C f(x)=\sup_{\xi\in\mathbb R}\bigl|H(M_\xi f)(x)\bigr|, \qquad M_\xi f(x)=e^{2\pi i \xi x}f(x),$$39, and a covering condition in phase space (Becker et al., 2024). A cancellative collection also satisfies an oscillatory integral estimate against Lipschitz test functions on each ball, abstracting Euclidean van der Corput or integration-by-parts arguments (Becker et al., 2024). This framework is explicitly designed to include the classical linear phases, polynomial phases, and related non-Euclidean modulation classes (Becker et al., 7 Aug 2025).

The proof architecture remains recognizably time-frequency-theoretic: Christ-type dyadic cubes, tiles $$\mathcal C f(x)=\sup_{\xi\in\mathbb R}\bigl|H(M_\xi f)(x)\bigr|, \qquad M_\xi f(x)=e^{2\pi i \xi x}f(x),$$40, antichains, trees, forests, density functionals, and $$\mathcal C f(x)=\sup_{\xi\in\mathbb R}\bigl|H(M_\xi f)(x)\bigr|, \qquad M_\xi f(x)=e^{2\pi i \xi x}f(x),$$41-style tile correlation estimates all reappear in metric form (Becker et al., 2024). What changes is that Euclidean frequency intervals are replaced by modulation cells inside $$\mathcal C f(x)=\sup_{\xi\in\mathbb R}\bigl|H(M_\xi f)(x)\bigr|, \qquad M_\xi f(x)=e^{2\pi i \xi x}f(x),$$42, and derivative-based phase separation is replaced by the local metrics $$\mathcal C f(x)=\sup_{\xi\in\mathbb R}\bigl|H(M_\xi f)(x)\bigr|, \qquad M_\xi f(x)=e^{2\pi i \xi x}f(x),$$43 (Becker et al., 7 Aug 2025).

6. Methods, structural themes, and terminological boundaries

Across the subject, three proof paradigms recur. The first is classical time-frequency analysis: tiles, trees, forests, and almost orthogonality in the style of Fefferman and Lacey–Thiele. Lie’s polynomial theorem extends this to higher-order wave packets and curved tiles adapted to polynomial phase graphs (Lie, 2011). The second is oscillatory integral analysis via $$\mathcal C f(x)=\sup_{\xi\in\mathbb R}\bigl|H(M_\xi f)(x)\bigr|, \qquad M_\xi f(x)=e^{2\pi i \xi x}f(x),$$44, stationary phase, and van der Corput estimates, especially in Stein–Wainger-type settings, oscillatory Carleson families, and Radon-transform analogues (Ramos, 2020). The third is domination technology—sparse domination, local mean oscillation, and outer $$\mathcal C f(x)=\sup_{\xi\in\mathbb R}\bigl|H(M_\xi f)(x)\bigr|, \qquad M_\xi f(x)=e^{2\pi i \xi x}f(x),$$45 embeddings—which has been particularly effective in weighted and vector-valued questions (Plinio et al., 2013).

Several papers emphasize that the main technical difficulty is not merely singularity, but the interaction between maximal modulation and underlying symmetry. In quadratic or near-quadratic regimes, modulation symmetry can survive strongly enough to force a time-frequency treatment, even when oscillatory curvature is present (Ramos, 2020). In bilinear Hilbert–Carleson theory, this interaction is explicit: non-resonant exponents permit curvature-driven decay, whereas the resonant cases $$\mathcal C f(x)=\sup_{\xi\in\mathbb R}\bigl|H(M_\xi f)(x)\bigr|, \qquad M_\xi f(x)=e^{2\pi i \xi x}f(x),$$46 and $$\mathcal C f(x)=\sup_{\xi\in\mathbb R}\bigl|H(M_\xi f)(x)\bigr|, \qquad M_\xi f(x)=e^{2\pi i \xi x}f(x),$$47 produce additional modulation invariances that obstruct the available mechanism (Benea et al., 2021). In the monomial-curve and paraboloid settings, curvature from the underlying geometry and oscillation from the phase are inseparable, and partial or smoothed models become necessary (Guo et al., 2016).

The available results also leave a clear map of unresolved regimes. Uniform bounds for oscillatory Carleson operators are established only locally near $$\mathcal C f(x)=\sup_{\xi\in\mathbb R}\bigl|H(M_\xi f)(x)\bigr|, \qquad M_\xi f(x)=e^{2\pi i \xi x}f(x),$$48, not globally in $$\mathcal C f(x)=\sup_{\xi\in\mathbb R}\bigl|H(M_\xi f)(x)\bigr|, \qquad M_\xi f(x)=e^{2\pi i \xi x}f(x),$$49 (Ramos, 2020). The resonant bilinear Hilbert–Carleson cases $$\mathcal C f(x)=\sup_{\xi\in\mathbb R}\bigl|H(M_\xi f)(x)\bigr|, \qquad M_\xi f(x)=e^{2\pi i \xi x}f(x),$$50 remain open (Benea et al., 2021). Several partial operators along monomial curves, including $$\mathcal C f(x)=\sup_{\xi\in\mathbb R}\bigl|H(M_\xi f)(x)\bigr|, \qquad M_\xi f(x)=e^{2\pi i \xi x}f(x),$$51, $$\mathcal C f(x)=\sup_{\xi\in\mathbb R}\bigl|H(M_\xi f)(x)\bigr|, \qquad M_\xi f(x)=e^{2\pi i \xi x}f(x),$$52, and $$\mathcal C f(x)=\sup_{\xi\in\mathbb R}\bigl|H(M_\xi f)(x)\bigr|, \qquad M_\xi f(x)=e^{2\pi i \xi x}f(x),$$53, are not covered by full $$\mathcal C f(x)=\sup_{\xi\in\mathbb R}\bigl|H(M_\xi f)(x)\bigr|, \qquad M_\xi f(x)=e^{2\pi i \xi x}f(x),$$54 theory (Guo et al., 2016). In the variational vector-valued setting, no endpoint statements are proved at $$\mathcal C f(x)=\sup_{\xi\in\mathbb R}\bigl|H(M_\xi f)(x)\bigr|, \qquad M_\xi f(x)=e^{2\pi i \xi x}f(x),$$55 or $$\mathcal C f(x)=\sup_{\xi\in\mathbb R}\bigl|H(M_\xi f)(x)\bigr|, \qquad M_\xi f(x)=e^{2\pi i \xi x}f(x),$$56, and it remains open whether intermediate UMD can be replaced by general UMD (Amenta et al., 2020). Abstract metric-space theorems presently yield restricted weak type in the lower range $$\mathcal C f(x)=\sup_{\xi\in\mathbb R}\bigl|H(M_\xi f)(x)\bigr|, \qquad M_\xi f(x)=e^{2\pi i \xi x}f(x),$$57, with strong $$\mathcal C f(x)=\sup_{\xi\in\mathbb R}\bigl|H(M_\xi f)(x)\bigr|, \qquad M_\xi f(x)=e^{2\pi i \xi x}f(x),$$58 bounds for $$\mathcal C f(x)=\sup_{\xi\in\mathbb R}\bigl|H(M_\xi f)(x)\bigr|, \qquad M_\xi f(x)=e^{2\pi i \xi x}f(x),$$59 obtained by interpolation, while broader endpoint and full-range extensions remain natural targets (Becker et al., 2024).

The term “Carleson” also appears outside Fourier-analytic operator theory. In elliptic PDE, “Carleson perturbations” refers to Carleson-measure-controlled perturbations of elliptic coefficients, not to maximally modulated singular integrals (Feneuil et al., 2020). That usage is historically connected through Carleson measure estimates but designates a different theory. Within harmonic analysis proper, however, Carleson operators remain the prototypical maximally modulated singular integrals, and their modern generalizations continue to organize a large part of time-frequency and oscillatory analysis.