Almost-Everywhere Convergence Weyl Multipliers

• Almost-everywhere convergence Weyl multipliers are sequences that set weighted ℓ² conditions to guarantee the convergence of orthonormal expansions like Fourier, wavelet, and polynomial systems.
• Logarithmic weights (e.g., log n or (log n)²) serve as optimal criteria, derived via maximal function estimates and block constructions, ensuring sharp convergence thresholds across different systems.
• Extensions to multidimensional, lacunary, and spectral settings highlight the versatility of these multipliers in controlling convergence even under rearrangements and complex operator frameworks.

Almost-everywhere convergence Weyl multipliers are central objects in harmonic analysis, capturing the precise weighted ℓ² summability conditions on coefficients of orthogonal series (Fourier, wavelet, Franklin polynomial, or general orthonormal expansions) that guarantee almost-everywhere convergence of the associated function series or transforms. The theory, developed from the classical Menshov–Rademacher and Kaczmarz–Moricz results, has been extended and sharpened for various systems—from trigonometric and Walsh to Jacobi, multi-dimensional trigonometric, and orthogonal polynomial frameworks—culminating in explicit, often logarithmic, optimal Weyl multipliers for numerous orthonormal bases.

1. Formal Definition and General Framework

Given an orthonormal system $\{\varphi_n\}_{n=1}^\infty$ in $L^2(X,\mu)$, a sequence $w(n)\to\infty$ is called an almost-everywhere convergence Weyl multiplier (or $C$-multiplier) for $\{\varphi_n\}$ if every series $\sum_{n=1}^\infty a_n \varphi_n(x)$ with $\sum_{n=1}^\infty |a_n|^2 w(n) < \infty$ converges $\mu$-almost everywhere. If this property holds for every rearrangement of the series (i.e., under all permutations), $w$ is said to be an almost-everywhere unconditional convergence Weyl multiplier ($UC$-multiplier) (Kamont et al., 2021, Karagulyan, 15 Jan 2026, Karagulyan, 2020).

Formally, for sequences $\{a_n\}\subset\mathbb C$ and for all $x$ outside a set of measure zero,

$$\sum_{n=1}^\infty a_n \varphi_n(x) \text{ converges } \quad \text{whenever }\quad \sum_{n=1}^\infty |a_n|^2 w(n)<\infty.$$

The definition extends naturally to multidimensional systems: For function systems $$\mathcal T^d=\{e^{2\pi i\mathbf{n}\cdot \mathbf x} : \mathbf n\in\mathbb Z^d\}$$, similar $w(n)$ can serve as Weyl multipliers for multidimensional orthogonal series (Karagulyan, 15 Jan 2026, Bloshanskii et al., 2017).

2. Optimal Almost-everywhere Convergence Weyl Multipliers: Logarithmic Weights

Determining the optimal sequence $w(n)$ for a given orthonormal system is a deep problem, hinging on harmonic analysis, maximal operator estimates, and probabilistic/block constructions.

Trigonometric, Wavelet, and Franklin Systems

For classical non-overlapping orthogonal systems such as trigonometric functions, non-overlapping wavelet polynomials, and Franklin polynomials, it is established that $\log n$ is both a sufficient and necessary (in a precise asymptotic sense) Weyl multiplier for almost-everywhere convergence:

• Kamont and Karagulyan proved that for any orthonormal system of non-overlapping wavelet polynomials, $\log n$ is a sharp a.e. convergence Weyl multiplier. That is, $\sum |a_n|^2\log n<\infty$ guarantees a.e. convergence, and no slower sequence suffices (Kamont et al., 2021).
• The same is confirmed for non-overlapping Franklin polynomial systems (Karagulyan, 2020).
• For the trigonometric system, a sequence $w(n)\gtrsim\log n$ is necessary to serve as a $C$- or $UC$-multiplier, and $w(n)\lesssim (\log n)^2$ suffices for $UC$-multipliers for any orthonormal system (Menshov–Rademacher theorem) (Kamont et al., 2021, Karagulyan, 15 Jan 2026, Karagulyan, 2020).

Table 1: Sharp Weyl Multipliers for Key Systems

System	Sufficient $w(n)$ for a.e. convergence	Necessary $w(n)$ for a.e. convergence
General Orthonormal System	$(\log n)^2$	$\not\ll (\log n)^2$ (Menshov–Rademacher)
Non-overlapping Trig/Wavelet	$\log n$	$\not\ll \log n$
Trigonometric, $UC$-multiplier	$\log n \lesssim w(n) \lesssim (\log n)^2$	$\log n$ [precisely, see divergence threshold]

Any $o(\log n)$ cannot serve as a Weyl multiplier for these systems (Kamont et al., 2021, Karagulyan, 2020, Karagulyan, 15 Jan 2026).

3. Techniques and Maximal Function Estimates

The sharpness and sufficiency of the logarithmic weights are established via sharp maximal function inequalities, dyadic decomposition, square-function arguments, and “blockwise” construction methods:

• For wavelet-type systems, Kamont–Karagulyan prove the key maximal-inequality: $$\left\|\max_{1\le m\le n} \left|\sum_{j\in G_m} \langle f, \phi_j\rangle \phi_j\right|\right\|_p \lesssim \sqrt{\log (n+1)} \cdot \|f\|_p, \quad 1<p<\infty.$$ Applying this using “non-overlapping” blocks $G_n$ shows that $\log n$ controls the growth in the maximal partial sum, and, via Kolmogorov–Menshov arguments, ensures a.e. convergence for coefficients with $\sum |a_n|^2\log n < \infty$ (Kamont et al., 2021).
• Conversely, constructing sequences that exploit the maximal operator growth, one can guarantee divergence almost everywhere for sequences with $\sum |a_n|^2 w(n) < \infty$ if $w(n) \ll \log n$ (Kamont et al., 2021).
• For the trigonometric system, Karagulyan exhibits, via block constructions and permutation arguments, that for $UC$-multipliers (i.e., after rearrangements), convergence requires

$\sum_{n=1}^\infty \frac{1}{n\,w(n)}<\infty,$

and divergence occurs otherwise (Karagulyan, 2020).

4. Multidimensional, Lacunary, and Mixed Systems

For multivariate Fourier systems, the theory extends under both “rectangular” and “lacunary” summation schemes:

• Bloshanskii–Bloshanskaya–Grafov determine that for multiple Fourier series

$$W(\nu) = \prod_{j=1}^{N-k} \log(|\nu_{\alpha_j}|+2)$$

is a Weyl multiplier in the case where $k$ of the coordinates are restricted to lacunary sequences and $N-k$ are “free”. Each free direction requires a logarithmic factor, mirroring the “worst-case” one-dimensional growth (Bloshanskii et al., 2017).

• In two dimensions, Nikishin showed that $\log^2[\min(|v_1|,|v_2|)+2]$ is optimal (Bloshanskii et al., 2017). For $N\ge3$ and two or more free directions, a product of $\log$ factors is both sufficient and, up to constants, necessary for a.e. convergence.
• Karagulyan’s equivalence principle establishes that, for UC-multipliers, the problem reduces to the one-dimensional case under mild regularity of the weight: $w(n^2)\lesssim w(n)$ (Karagulyan, 15 Jan 2026). This allows direct transfer of convergence results between one- and higher-dimensional trigonometric systems.

5. Weyl Multipliers in Harmonic Analysis Beyond Orthogonal Systems

Weyl multipliers generalize to the spectral multipliers of self-adjoint operators, including inverse Jacobi transforms, Bochner–Riesz means, and Weyl transforms in subelliptic and nilpotent group settings:

• In Jacobi analysis (rank one, symmetric space analogues), maximal disc-multiplier operators are bounded from $L^p$ into $L^p + L^2$ in the range $p_0 < p \leq 2$, with $p_0 = \frac{4\alpha+4}{2\alpha+3}$, and a.e. convergence fails outside this range—precisely paralleling the Euclidean radial disc-multiplier situation (Johansen, 2011).
• On Heisenberg-type groups, Bochner–Riesz means $S_R^\delta f$ converge a.e. for $f\in L^p$ in a nontrivial $(\delta,p)$ trapezoid, and in the twisted Laplacian setting the a.e. convergence index is exactly half the Euclidean Bochner–Riesz threshold (Jeong et al., 2023, Horwich et al., 2019).

6. Further Directions and Open Problems

Open questions persist regarding the precise threshold for $UC$-multipliers for the classical trigonometric system: is $\log n$ or $(\log n)^2$ the universal sharp threshold, or does an intermediate sequence (e.g., $\log n \cdot (\log\log n)^a$) characterize the boundary? For systems without a direct product structure, such as spherical harmonics and special wavelet expansions, finding analogous equivalence principles and explicit optimal Weyl multipliers remains challenging (Karagulyan, 15 Jan 2026).

Methods based on discretization, probabilistic equivalence, non-overlapping block encoding, and maximal operator theory are expected to play a central role in further advances—both in identifying the sharp Weyl multipliers and extending the theory to broader function classes and spectral-analytic settings.

7. Summary Table: Principal Results for a.e. Convergence Weyl Multipliers

Context	Optimal Weyl Multiplier $w(n)$	Key Reference
General Orthonormal Systems	$(\log n)^2$ (sharp)	(Kamont et al., 2021)
Non-overlapping Wavelet/Franklin Systems	$\log n$ (sharp)	(Kamont et al., 2021, Karagulyan, 2020)
Trigonometric System (rearrangements)	$\log n \lesssim w(n) \lesssim (\log n)^2$	(Karagulyan, 2020, Karagulyan, 15 Jan 2026)
Multidimensional Trigonometric	Product of $\log$s in free indices	(Bloshanskii et al., 2017)
Jacobi, Spherical, and Radial Disk Cases	Endpoint $L^p$ bounds with sharp divergence outside	(Johansen, 2011)
Bochner–Riesz on H-type/Twisted Laplacian	Trapezoid in $(\delta,p)$, explicit threshold	(Jeong et al., 2023, Horwich et al., 2019)

All results establish $\log n$ (or variants thereof) as the minimal necessary growth, and, in various structured systems, as the optimal weight for almost-everywhere convergence under square-summability.

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References:

• (Kamont et al., 2021) Kamont, Karagulyan: On wavelet polynomials and Weyl multipliers.
• (Karagulyan, 2020) On Weyl multipliers of non-overlapping Franklin polynomial systems.
• (Karagulyan, 2020) Karagulyan: On Weyl multipliers of the rearranged trigonometric system.
• (Karagulyan, 15 Jan 2026) Karagulyan: On UC-multipliers for multiple trigonometric systems.
• (Bloshanskii et al., 2017) Bloshanskii, Bloshanskaya, Grafov: Sufficient conditions for convergence of multiple Fourier series with $J_k$-lacunary sequence of rectangular partial sums in terms of Weyl multipliers.
• (Johansen, 2011) Clerc: Almost everywhere convergence of the inverse Jacobi transform.
• (Jeong et al., 2023) Jeong, Lee, Ryu: Almost everywhere convergence of Bochner--Riesz means for the twisted Laplacian.
• (Horwich et al., 2019) Müller, Ricci, Stein, Thangavelu: Bochner-Riesz means on Heisenberg-type groups.