---
title: Almost-Everywhere Convergence Weyl Multipliers
url: https://www.emergentmind.com/topics/almost-everywhere-convergence-weyl-multipliers
type: topic
---

# Almost-Everywhere Convergence Weyl Multipliers

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) [2104.03124], [2601.10360], [2004.01003].

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 [2601.10360], [1704.04673].

## 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 [2104.03124].
- The same is confirmed for non-overlapping Franklin polynomial systems [2005.04017].
- 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) [2104.03124], [2601.10360], [2004.01003].

#### 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 [2104.03124], [2005.04017], [2601.10360].

## 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 $\sum |a_n|^2\log n < \infty$ [2104.03124].

- 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 $w(n) \ll \log n$ [2104.03124].

- 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 [2004.01003].

## 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 [1704.04673].

- In two dimensions, Nikishin showed that $\log^2[\min(|v_1|,|v_2|)+2]$ is optimal [1704.04673]. 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: $w(n^2)\lesssim w(n)$ [2601.10360]. 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 [1101.0478].
- 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 [2303.02679], [1908.04049].

## 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 [2601.10360].

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)                                | [2104.03124]                   |
| Non-overlapping Wavelet/Franklin Systems | $\log n$ (sharp)                                    | [2104.03124], [2005.04017]     |
| Trigonometric System (rearrangements)    | $\log n \lesssim w(n) \lesssim (\log n)^2$           | [2004.01003], [2601.10360]     |
| Multidimensional Trigonometric           | Product of $\log$s in free indices                   | [1704.04673]                   |
| Jacobi, Spherical, and Radial Disk Cases | Endpoint $L^p$ bounds with sharp divergence outside  | [1101.0478]                    |
| Bochner–Riesz on H-type/Twisted Laplacian| Trapezoid in $(\delta,p)$, explicit threshold        | [2303.02679], [1908.04049]     |

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.

---

**References:**
- [2104.03124] Kamont, Karagulyan: On wavelet polynomials and Weyl multipliers.
- [2005.04017] On Weyl multipliers of non-overlapping Franklin polynomial systems.
- [2004.01003] Karagulyan: On Weyl multipliers of the rearranged trigonometric system.
- [2601.10360] Karagulyan: On UC-multipliers for multiple trigonometric systems.
- [1704.04673] 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.
- [1101.0478] Clerc: Almost everywhere convergence of the inverse Jacobi transform.
- [2303.02679] Jeong, Lee, Ryu: Almost everywhere convergence of Bochner--Riesz means for the twisted Laplacian.
- [1908.04049] Müller, Ricci, Stein, Thangavelu: Bochner-Riesz means on Heisenberg-type groups.

Source: https://www.emergentmind.com/topics/almost-everywhere-convergence-weyl-multipliers