An Oskolkov--Zhizhiashvili Criterion for Rectangular Fourier Sums

Technical Summary: An Oskolkov–Zhizhiashvili Criterion for Rectangular Fourier Sums

Problem Statement and Context
This paper addresses the almost everywhere (a.e.) pointwise convergence of symmetric rectangular partial sums of multiple trigonometric Fourier series on the $d$-dimensional torus $\mathbb{T}^d$ ($d \ge 2$). Unlike the one-dimensional case, where the Carleson–Hunt theorem guarantees convergence for $f \in L^p$ ($1 < p < \infty$), multidimensional rectangular convergence (in the sense of Pringsheim) requires additional quantitative smoothness assumptions.

The central problem investigated is the sharpness of logarithmic moduli of continuity required to ensure convergence. Previous work by Zhizhiashvili established that for $f \in L^p(\mathbb{T}^d)$, if the modulus of continuity $\omega(\delta, f)_p$ satisfies a condition with a logarithmic power of $d/p$ plus an arbitrary $\varepsilon > 0$ (i.e., $\omega(\delta, f)_p \lesssim (\log(1/\delta))^{-d/p-\varepsilon}$), convergence holds. However, the exact endpoint where $\varepsilon = 0$ remained an open problem, particularly for $1 < p < 2$ (the Zhizhiashvili–Marcinkiewicz problem) and for the endpoint cases $p=1$ and $p=2$.

Furthermore, the paper distinguishes between the $L^p$ setting and the uniform norm setting. While Oskolkov provided a sharp criterion for continuous functions in dimension $d=2$ involving triple logarithms, a corresponding sharp $L^p$ criterion for $d > 2$ remains open. This work aims to provide a sharp $L^p$ sufficient condition at the critical logarithmic scale $d/p$, incorporating a summable secondary weight at the iterated-logarithmic level.

Methodology
The proof employs a combination of approximation theory and maximal function estimates, structured as follows:

1. Approximation via de la Vallée Poussin Means: The function $f$ is approximated by a sequence of trigonometric polynomials $P_k$ constructed using product de la Vallée Poussin kernels $V_{N_k}$. These kernels are chosen such that they reproduce frequencies up to a certain scale and have controlled $L^1$ norms. The difference $f - P_k$ is bounded using the modulus of continuity $\omega(1/N_k, f)_p$.
2. Block Decomposition: The function $f$ is decomposed into a telescopic series of "blocks" $g_k = P_k - P_{k-1}$ (with $g_0 = P_0$). Each block $g_k$ is a trigonometric polynomial with a spectrum contained in a specific dyadic frequency cube $[-2^{r_k}, 2^{r_k}]^d$.
3. Finite Rectangular Maximal Estimates: The core of the argument relies on establishing $L^p$ bounds for the maximal operator $M_r f = \sup_{0 \le n_j \le 2^r} |S_n f|$.
   • For $p=1$, the bound is derived using the $L^1$ norm of the maximal Dirichlet kernel, yielding a factor of $(1+r)^d$.
   • For $p=2$, the bound utilizes a product-logarithmic maximal inequality (related to Sjölin's work) and Parseval's identity, yielding a factor of $(1+r)^{d/2}$.
   • For $1 < p < 2$, the Marcinkiewicz interpolation theorem is applied to interpolate between the $L^1$ and $L^2$ estimates, resulting in a bound of $(1+r)^{d/p}$.
4. Summability Argument: By combining the decay of the approximation error (governed by the assumed modulus of continuity) with the growth of the maximal operator bounds, the authors show that the $L^p$ norms of the maximal functions of the blocks, $\|M g_k\|_p$, are dominated by a summable sequence involving the secondary weight function $\psi$.
5. Convergence: The summability of these maximal norms implies that the series of blocks converges absolutely almost everywhere. This allows the authors to conclude that the rectangular partial sums $S_n f(x)$ converge to $f(x)$ as $\min n_j \to \infty$.

Key Contributions and Results

• Main Theorem (Theorem 1.2): The paper proves a general summable endpoint criterion. For $d \ge 2$ and $1 \le p \le 2$, if $f \in L^p(\mathbb{T}^d)$ satisfies:
  $$\omega(\delta, f)_p \le A(f) (\log(1/\delta))^{-d/p} \psi(\log \log(ee/\delta))$$
  where $\psi$ is a non-increasing function such that $\sum_{k=1}^\infty \psi(k) < \infty$, then the rectangular Fourier series of $f$ converges almost everywhere in the sense of Pringsheim.

• Resolution of the Zhizhiashvili–Marcinkiewicz Problem: The main theorem provides an affirmative answer to Problem 1.1 for $1 < p < 2$. It establishes that the critical logarithmic power $d/p$ is sufficient, provided a summable iterated-logarithmic secondary factor is present. This improves upon previous results that required an arbitrary $\varepsilon > 0$ margin.

• Double-Logarithmic Endpoint Criterion (Corollary 1.3): As a specific case, the paper shows that if $\omega(\delta, f)_p \lesssim (\log(1/\delta))^{-d/p} (\log \log(ee/\delta))^{-1-\eta}$ for some $\eta > 0$, convergence holds. This sharpens the sufficient conditions for the endpoint cases $p=1$ and $p=2$ previously known from Zhizhiashvili's work.

• $L^p$ Oskolkov-Type Consequence (Corollary 1.4): The paper derives a condition involving triple logarithms:
  $$\omega(\delta, f)_p \lesssim (\log(1/\delta))^{-d/p} \log \log(ee/\delta) (\log \log \log(eee/\delta))^{-1-\eta}$$
  The authors clarify that this is a $d$-dimensional $L^p$ sufficient criterion, distinct from Oskolkov's classical uniform-norm theorem for $d=2$. It is not claimed to be the solution to the sharp higher-dimensional Oskolkov problem (which would require removing the summable secondary factor entirely).

Significance and Claims
The paper claims to provide the first $d$-dimensional $L^p$ sufficient criterion for rectangular convergence that operates at the dimensionally correct critical logarithmic power $d/p$ for the full range $1 \le p \le 2$.

• Improvement over Existing Theory: The results improve upon Zhizhiashvili's classical theorems by removing the arbitrary $\varepsilon$ and replacing it with a precise, summable secondary weight. This resolves the specific question of how sharp the logarithmic powers are in the context of the Zhizhiashvili–Marcinkiewicz problem.
• Limitations and Open Questions: The authors modestly note that their method relies on summable secondary factors. They explicitly state that the "bare" endpoint problem—replacing the summable weight with the weaker triple-logarithmic term $\Lambda_3(\delta)$ (as in Oskolkov's $d=2$ result) without the summable factor—remains open (Problem 6.1). Their current approximation-maximal method yields block bounds of order $o(1/\log k)$ for such a condition, which are not necessarily summable.

In summary, the paper establishes a robust, sharp sufficient condition for almost everywhere convergence of rectangular Fourier sums in $L^p$ spaces, bridging the gap between the known $\varepsilon$-results and the elusive sharp endpoint, while clearly delineating the boundary of current techniques regarding the triple-logarithmic Oskolkov phenomenon in higher dimensions.