A Critical-Scale Extension of Zhizhiashvili’s Theorem for Rectangular Fourier Series

Abstract.

We address a long-standing endpoint problem arising from Zhizhiashvili’s logarithmic modulus theorem for multiple Fourier series. We prove an endpoint Dini criterion for almost-everywhere Pringsheim convergence of ordinary rectangular partial sums. In Zhizhiashvili’s theorem the logarithmic modulus is assumed with exponent strictly above the critical value; here this strict power margin is replaced by a summable endpoint Dini condition. As a consequence, one obtains double-logarithmic endpoint classes lying outside the range of the classical theorem. The proof reduces the endpoint smoothness assumption to the Kaczmarz–Kojima product-logarithmic coefficient criterion by weighted translation-difference estimates.

1. Introduction

Almost-everywhere convergence of Fourier series has a fundamentally different nature in one and in several variables. In one dimension, the Carleson–Hunt theorem gives almost-everywhere convergence of the symmetric partial sums for every function in $L^{p}(\mathbb{T})$, $p>1$ [1, 3]. In several variables, the geometry of the partial sums becomes part of the problem. The present note concerns ordinary symmetric rectangular sums and convergence in the sense of Pringsheim.

Throughout, $\mathbb{T}=\mathbb{R}/2\pi\mathbb{Z}$, and $dx$ denotes the normalized Haar measure on $\mathbb{T}^{d}$; thus, on the fundamental cube,

$$dx=(2\pi)^{-d}\,dx_{1}\cdots dx_{d}.$$

For $f\in L^{1}(\mathbb{T}^{d})$ put

$$\widehat{f}(k)=\int_{\mathbb{T}^{d}}f(x)e^{-ik\cdot x}\,dx,\qquad k=(k_{1},\ldots,k_{d})\in\mathbb{Z}^{d}.$$

The ordinary rectangular partial sums are

$$S_{\mathbf{n}}f(x)=\sum_{\left\lvert k_{1}\right\rvert\leq n_{1}}\cdots\sum_{\left\lvert k_{d}\right\rvert\leq n_{d}}\widehat{f}(k)e^{ik\cdot x},\qquad\mathbf{n}=(n_{1},\ldots,n_{d})\in\mathbb{N}_{0}^{d}.$$ (1.1)

We say that $S_{\mathbf{n}}f$ converges to $f$ in the Pringsheim sense if

$$S_{\mathbf{n}}f(x)\longrightarrow f(x)\qquad\text{as }\min_{1\leq j\leq d}n_{j}\to\infty.$$

This is a genuinely multi-parameter question. Fefferman’s divergence theorem shows that the higher-dimensional theory contains divergence phenomena absent from the one-dimensional theory [2]. It is therefore natural to ask which quantitative smoothness assumptions recover almost-everywhere convergence at the borderline of the logarithmic scale.

For $h=(h_{1},\ldots,h_{d})\in\mathbb{T}^{d}$, we write

$$\left\lvert h\right\rvert=(h_{1}^{2}+\cdots+h_{d}^{2})^{1/2},$$

where $h_{j}\in[-\pi,\pi)$ is the representative near the origin. For $f\in L^{2}(\mathbb{T}^{d})$ let

$$\omega_{2}(f,\delta)=\sup_{\left\lvert h\right\rvert\leq\delta}\left\lVert f(\cdot+h)-f(\cdot)\right\rVert_{L^{2}(\mathbb{T}^{d})},\qquad 0<\delta<1.$$ (1.2)

Zhizhiashvili proved the following fundamental logarithmic criterion: if $d\geq 2$, $f\in L^{2}(\mathbb{T}^{d})$, and

$$\omega_{2}(f,\delta)=O\left(\left(\log\frac{1}{\delta}\right)^{-\beta}\right)\qquad(\delta\downarrow 0)$$ (1.3)

for some $\beta>d/2$, then the rectangular Fourier sums of $f$ converge almost everywhere in the Pringsheim sense [6]. The strict inequality $\beta>d/2$ is the decisive feature of this theorem. It leaves open the critical logarithmic exponent itself.

The main result of this paper gives a partial endpoint solution to this problem by replacing the missing logarithmic power margin with a Dini summability condition.

Theorem 1.1 (Endpoint Zhizhiashvili–Dini criterion).

Let $d\geq 2$ and $f\in L^{2}(\mathbb{T}^{d})$. Suppose that, for some $\eta\in(0,1)$,

$$\int_{0}^{\eta}\omega_{2}(f,t)^{2}\left(\log\frac{e}{t}\right)^{d-1}\frac{\,dt}{t}<\infty.$$ (1.4)

Then

$$S_{\mathbf{n}}f(x)\longrightarrow f(x)\qquad\text{for almost every }x\in\mathbb{T}^{d}$$

in the Pringsheim sense.

Theorem 1.1 is a significant endpoint advance of Zhizhiashvili’s theorem in the $L^{2}$ scale. The classical hypothesis (1.3) with $\beta>d/2$ immediately implies (1.4). More importantly, Theorem 1.1 also includes moduli lying exactly at the critical power $d/2$, provided that the remaining endpoint contribution is summable. The following consequence records the simplest explicit form of this gain.

Corollary 1.2 (Double-logarithmic endpoint class).

Let $d\geq 2$, $f\in L^{2}(\mathbb{T}^{d})$, and $\gamma>0$. If

$$\omega_{2}(f,\delta)=O\left(\left(\log\frac{e}{\delta}\right)^{-d/2}\left(\log\log\frac{e^{e}}{\delta}\right)^{-1/2-\gamma}\right)\qquad(\delta\downarrow 0),$$ (1.5)

then $S_{\mathbf{n}}f(x)\to f(x)$ for almost every $x\in\mathbb{T}^{d}$ in the Pringsheim sense.

Corollary 1.2 is not a restatement of Zhizhiashvili’s result. It permits the critical logarithmic power $d/2$ and compensates only by a summable iterated logarithm. Thus it applies to moduli which may decay more slowly than every power

$$\left(\log\frac{1}{\delta}\right)^{-d/2-\varepsilon},\qquad\varepsilon>0,$$

and hence are outside the range of (1.3).

The proof is short and structural. We use the Kaczmarz–Kojima theorem, which says that a product-logarithmic square summability condition on the Fourier coefficients implies almost-everywhere rectangular convergence. The new point is that the endpoint Dini condition (1.4) forces exactly this coefficient condition. A weighted lower bound for the oscillatory multipliers $e^{imt}-1$ converts one-coordinate translation estimates into one-coordinate logarithmic coefficient estimates. The arithmetic–geometric mean inequality then converts the resulting family of estimates into the Kaczmarz–Kojima product weight.

2. Preliminaries

All logarithms are natural. Constants denoted by $C$ may change from line to line and may depend on fixed parameters such as $d$ and $\eta$, but not on the frequency variable. We use Plancherel’s theorem and uniqueness of trigonometric Fourier coefficients in their standard forms; see, for instance, [7].

For a coordinate vector $e_{j}$, define

$$\Delta_{j}(t)f(x)=f(x+te_{j})-f(x),\qquad 1\leq j\leq d.$$

We shall use the following classical convergence theorem.

Theorem 2.1 (Kaczmarz–Kojima product-log theorem).

Let $d\geq 2$ and $F\in L^{2}(\mathbb{T}^{d})$. If

$$\sum_{k\in\mathbb{Z}^{d}}\left\lvert\widehat{F}(k)\right\rvert^{2}\prod_{j=1}^{d}\log(\left\lvert k_{j}\right\rvert+2)<\infty,$$ (2.1)

then the rectangular partial sums $S_{\mathbf{n}}F$ converge to $F$ almost everywhere in the Pringsheim sense.

For $d=2$ this is due to Kaczmarz; the higher-dimensional form is due to Kojima [4, 5].

3. Weighted differences and logarithmic coefficients

The next elementary estimate extracts a full logarithmic power from a one-dimensional translation difference.

Lemma 3.1.

Let $d\geq 1$ and $0<\eta<1$. There exists $c=c(d,\eta)>0$ such that, for every integer $m$ with $\left\lvert m\right\rvert\geq 1$,

$$\int_{0}^{\eta}\left\lvert e^{imt}-1\right\rvert^{2}\left(\log\frac{e}{t}\right)^{d-1}\frac{\,dt}{t}\geq c\,\log^{d}(\left\lvert m\right\rvert+2).$$ (3.1)

Proof.

By symmetry it is enough to consider $m=N\geq 1$. For every finite range $1\leq N\leq N_{0}(d,\eta)$ the assertion is absorbed by decreasing the constant, since the left-hand side is positive for each fixed $N\geq 1$. Hence we assume that $N$ is large.

Choose $0<\alpha<\pi/8$. For integers $q\geq 0$ set

$$I_{q}=\bigl\{t>0:\left\lvert Nt-(2q+1)\pi\right\rvert<\alpha\bigr\}.$$

On $I_{q}$ we have $\left\lvert e^{iNt}-1\right\rvert^{2}\geq c_{0}$. If $0\leq q\leq Q$, where $Q=\lfloor c_{1}N\eta\rfloor$ and $c_{1}>0$ is chosen sufficiently small, then $I_{q}\subset(0,\eta)$ for all large $N$. Moreover, on $I_{q}$ one has $t\asymp(q+1)/N$ and $\left\lvert I_{q}\right\rvert\asymp N^{-1}$. Hence

$$\int_{0}^{\eta}\left\lvert e^{iNt}-1\right\rvert^{2}\left(\log\frac{e}{t}\right)^{d-1}\frac{\,dt}{t} \geq C\sum_{q=0}^{Q}\frac{1}{q+1}\left(\log\frac{eN}{C(q+1)}\right)^{d-1}$$

$$\geq C\int_{1}^{c_{2}N}\left(\log\frac{eN}{Cu}\right)^{d-1}\frac{\,du}{u}.$$

The change of variables $v=\log(eN/(Cu))$ shows that the last integral is bounded from below by $C(\log N)^{d}-C_{d,\eta}$. This is at least $c(d,\eta)\log^{d}(N+2)$ for all sufficiently large $N$, and the remaining finite range has already been absorbed into the constant. ∎

The next proposition is the main reduction step. It converts the endpoint Dini control of coordinate translation differences into the product-logarithmic Fourier coefficient condition needed for the Kaczmarz–Kojima convergence theorem.

Proposition 3.2.

Let $d\geq 2$ and $f\in L^{2}(\mathbb{T}^{d})$. Suppose that, for some $\eta\in(0,1)$ and every coordinate $j=1,\ldots,d$,

$$\int_{0}^{\eta}\left\lVert\Delta_{j}(t)f\right\rVert_{2}^{2}\left(\log\frac{e}{t}\right)^{d-1}\frac{\,dt}{t}<\infty.$$ (3.2)

Then

$$\sum_{k\in\mathbb{Z}^{d}}\left\lvert\widehat{f}(k)\right\rvert^{2}\prod_{j=1}^{d}\log(\left\lvert k_{j}\right\rvert+2)<\infty.$$ (3.3)

Proof.

Fix a coordinate $j$. Since $f\in L^{2}(\mathbb{T}^{d})$,

$$\widehat{\Delta_{j}(t)f}(k)=(e^{ik_{j}t}-1)\widehat{f}(k),\qquad k\in\mathbb{Z}^{d},$$

and Plancherel’s theorem gives, for every $t$,

$$\left\lVert\Delta_{j}(t)f\right\rVert_{2}^{2}=\sum_{k\in\mathbb{Z}^{d}}\left\lvert\widehat{f}(k)\right\rvert^{2}\left\lvert e^{ik_{j}t}-1\right\rvert^{2}.$$

Multiplying by $(\log(e/t))^{d-1}/t$ and applying Tonelli’s theorem, the assumption (3.2) implies

$$\sum_{k\in\mathbb{Z}^{d}}\left\lvert\widehat{f}(k)\right\rvert^{2}\int_{0}^{\eta}\left\lvert e^{ik_{j}t}-1\right\rvert^{2}\left(\log\frac{e}{t}\right)^{d-1}\frac{\,dt}{t}<\infty.$$ (3.4)

Lemma 3.1 yields

$$\sum_{\begin{subarray}{c}k\in\mathbb{Z}^{d}\\ \left\lvert k_{j}\right\rvert\geq 1\end{subarray}}\left\lvert\widehat{f}(k)\right\rvert^{2}\log^{d}(\left\lvert k_{j}\right\rvert+2)<\infty.$$ (3.5)

Because $f\in L^{2}(\mathbb{T}^{d})$, Plancherel gives $\sum_{k}\left\lvert\widehat{f}(k)\right\rvert^{2}<\infty$. Therefore the terms with $k_{j}=0$ may be added to (3.5), and for every $j$,

$$\sum_{k\in\mathbb{Z}^{d}}\left\lvert\widehat{f}(k)\right\rvert^{2}\log^{d}(\left\lvert k_{j}\right\rvert+2)<\infty.$$ (3.6)

For non-negative $a_{1},\ldots,a_{d}$,

$$\prod_{j=1}^{d}a_{j}\leq\frac{1}{d}\sum_{j=1}^{d}a_{j}^{d}.$$

Taking $a_{j}=\log(\left\lvert k_{j}\right\rvert+2)$, multiplying by $\left\lvert\widehat{f}(k)\right\rvert^{2}$, and summing over $k$ gives (3.3) by (3.6). ∎

4. Proofs of the endpoint results

Proof of Theorem 1.1.

For each coordinate $j$ and every $t>0$,

$$\left\lVert\Delta_{j}(t)f\right\rVert_{2}\leq\omega_{2}(f,t).$$

Thus the endpoint Dini assumption (1.4) implies (3.2) for every coordinate. Proposition 3.2 gives the Kaczmarz–Kojima product-log condition (3.3). The convergence conclusion follows from Theorem 2.1. ∎

Proof of Corollary 1.2.

For small $t$, the hypothesis (1.5) gives

$$\omega_{2}(f,t)^{2}\left(\log\frac{e}{t}\right)^{d-1}\frac{1}{t}\leq C\frac{1}{t\log(e/t)\left(\log\log(e^{e}/t)\right)^{1+2\gamma}}.$$

With $u=\log(e/t)$, the right-hand side is integrable near the origin, since the corresponding integral is bounded by a constant multiple of

$$\int^{\infty}\frac{\,du}{u(\log u)^{1+2\gamma}}<\infty.$$

Hence (1.4) holds, and Theorem 1.1 applies. ∎

Corollary 4.1 (Recovery of Zhizhiashvili’s $L^{2}$ theorem).

Let $d\geq 2$ and $f\in L^{2}(\mathbb{T}^{d})$. If (1.3) holds for some $\beta>d/2$, then $S_{\mathbf{n}}f(x)\to f(x)$ almost everywhere in the Pringsheim sense.

Proof.

After the change of variables $u=\log(e/t)$, the integral in (1.4) is dominated by a constant multiple of

$$\int^{\infty}u^{d-1-2\beta}\,du,$$

which is finite precisely when $\beta>d/2$. The assertion follows from Theorem 1.1. ∎

Remark 4.2.

Corollary 4.1 shows that the classical theorem of Zhizhiashvili is contained in the endpoint Dini criterion. Corollary 1.2 shows the strict endpoint gain: at the critical logarithmic exponent, a summable iterated-logarithmic improvement is sufficient. This is the precise sense in which Theorem 1.1 strengthens Zhizhiashvili’s logarithmic modulus criterion.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

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