Maximal estimates for strong arithmetic means of Fourier series

Bobby Wilson

doi:10.1515/ans-2023-0180

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Maximal estimates for strong arithmetic means of Fourier series

Published/Copyright: April 11, 2025

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From the journal Advanced Nonlinear Studies Volume 25 Issue 2

Abstract

It is well-understood that partial sums of Fourier series for L ¹ functions don’t necessarily converge to the original function in the pointwise-almost everywhere sense. Estimates for averages of partial sum operators demonstrate that the divergence can be quantified. We establish a bilinear maximal estimate for averages of partial sums projectors.

Keywords: Fourier series; L 1 pointwise convergence; maximal average operators

2020 Mathematics Subject Classification: 42A20; 42A24

1 Introduction

We begin by considering the one-dimensional torus T ≔ R / Z and the question of convergence of Fourier series for functions in L 1 ( T ) . For n ∈ N , we define the partial sums operator

S n f ( θ ) = ∑ m = − n n f ̂ ( m ) e 2 π i m θ .

In Ref. [1], we proved that for any f ∈ L 1 ( T ) and λ > 0, there exists E ⊂ T such that the Lebesgue measure of E is less than 1 λ and

lim sup N → ∞ ∫ T \ E 1 N ∑ n = 1 N | S n f ( x ) | 2 d x ≤ C λ ‖ f ‖ 1 2

for some universal C > 0 independent of f and λ. In this article, we address the question of whether one can prove the maximal estimate bound. In other words, we aim to determine whether one can establish the following estimate:

∫ T \ E lim sup N → ∞ 1 N ∑ n = 1 N | S n f ( x ) | 2 d x ≤ C λ ‖ f ‖ 1 2 .

A useful exercise to tests the feasibility of the estimate is to consider two cases: Given λ > 0 consider

f(x) ≤ λ‖f‖₁ for all x ∈ T
f = χ _I f for some interval I ⊂ T such that |I| ≤ λ ⁻¹ and
‖ f ‖ 1 λ ≤ ∫ I f | I | ≤ 2 ‖ f ‖ 1 λ .

For case (1), we let K _N be a uniformly integrable family of kernels and P _N be the typical Littlewood-Paley projection (Littlewood-Paley projections are defined in Section 2). Then one would hope that you could estimate the maximal operator with an infinite sum of frequency localized operators (in the same vein as [2]) and perform the following estimate:

∫ T ∑ k ∈ Z + 1 2 k ∑ | n | ∼ 2 k | S n P 2 k f | 2 = ∑ k ∈ Z + ∫ T 1 2 k ∑ | n | ∼ 2 k | S n P 2 k f | 2 ≲ ∑ k ∈ Z + ∫ T K 2 k ∗ | P 2 k f | 2 ≲ ∑ k ∈ Z + ‖ K 2 k ‖ 1 ‖ P 2 k f ‖ 2 2 ≲ ‖ f ‖ 2 2 ≤ λ ‖ f ‖ 1 2 .

For case (2), we attempt a similar estimate:

∫ T \ 2 ⋅ I ∑ k ∈ Z + 1 2 k ∑ | n | ∼ 2 k | S n P 2 k f | 2 ≲ ∫ T \ 2 ⋅ I ∑ k ∈ Z + K 2 k ∗ | P 2 k f | 2 ≲ ∫ T \ 2 ⋅ I | x − c I | − 2 d x ∑ k ∈ Z + 1 2 k ‖ P 2 k f ‖ 2 2 ≲ | I | − 1 ∑ k ∈ Z + ‖ 1 2 k / 2 P 2 k f ‖ 2 2 ≲ E ω | I | − 1 ∑ k ∈ Z + r k ω 2 k / 2 P 2 k f 2 2

where r k ω represents the kth Rademacher function. Our job would be easy if we could say that

E ω ∑ k ∈ Z + r k ω 2 k P ̃ 2 k f ∞ ≲ ‖ f ‖ 1 = ‖ χ I f ‖ 1

where P ̃ N ≔ P N 2 . However, without more careful analysis, the best estimate that we can obtain is

E ω ∑ k ∈ Z + r k ω 2 k P ̃ 2 k f I ∞ ≲ sup N > 0 ( log ( N ) ) 1 / 2 ‖ V N f ‖ 1 .

It seems that the summation operator produces too many errors that add up over very high scales. However, since the area of integration is separated from the support of χ _I f and the Marcinkiewicz averaging operator acts like an approximate identity in some sense, we should attain the estimate that we desire for relatively high values of N. Therefore, finding a delicate balance between summation and supremum is vital. When this is done, we attain the desired theorem.

Theorem 1.

Let λ > 0. Then for any f ∈ L 1 ( T ) , there exists E ⊂ T , with | E | ≤ 1 λ such that

(1) ∫ T \ E lim sup N → ∞ 1 N ∑ n = 1 N | S n f ( x ) | 2 d x ≤ C λ ‖ f ‖ 1 2 .

The following bilinear estimate follows easily from Theorem 1.

Corollary 1.

Let λ > 0. Then for any f 1 , f 2 ∈ L 1 ( T ) , there exists E ⊂ T , with | E | ≤ 1 λ such that

(2) ∫ T \ E lim sup N → ∞ 1 N ∑ n = 1 N S n f 1 ( x ) S n f 2 ( x ) d x ≤ C λ ‖ f 1 ‖ 1 ‖ f 2 ‖ 1 .

Another direct consequence of Theorem 1 is the weak-type inequality

lim sup N → ∞ 1 N ∑ n = 1 N | S n f | 2 > λ 2 ≤ | E | + λ − 2 ∫ T \ E lim sup N → ∞ 1 N ∑ n = 1 N | S n f ( x ) | 2 d x ≤ ‖ f ‖ 1 λ + λ − 2 C λ ‖ f ‖ 1 ‖ f ‖ 1 2 = ( 1 + C ) ‖ f ‖ 1 λ

which leads to the following estimate proven by Marcinkiewicz ([3]; also appears in Ref. [4], Chapter XIII): for any 0 < r ≤ 2, any f ∈ L 1 ( T ) and for almost every θ ∈ T

(3) 1 N ∑ n = 1 N S n f ( θ ) − f ( θ ) r → k → ∞ 0

which easily leads to the conclusion that for almost every θ, there exists a sequence { n ℓ } ⊂ N such that S n ℓ f ( θ ) → f ( θ ) and

lim inf N → ∞ # ( { n ℓ } ∩ [ 0 , N ] ) N = 1 .

The proofs of these corollaries are standard, so we do not include them in this manuscript. Work on other weak-type inequalities can be found in Refs. [5], [6]. The equivalent estimate to (2) can not possible hold for higher dimensions, because it would imply that (3) would hold in the higher dimensional case. However, there is a counter-example due to Saks:

Theorem 2

([4], Chapter XVII, Theorem 2.2). Let there be given a function φ which is positive and increasing on (0, +∞) and of order o u ln + ⁡ u d − 1 as u → ∞. Then there is a positive function f such that φ ( f ) ∈ L 1 T d but

lim m 1 , … , m d → ∞ 1 ∏ i = 1 d m i + 1 ∑ n 1 = 0 m 1 … ∑ n d = 0 m d S n 1 , … , n d f , x 1 , … , x d − f x 1 , … , x d ≠ 0

for all x 1 , … , x d ∈ T d , ( d ≥ 2 ) .

On the other hand, Rodin proved that if a continuous function, Φ: [0, ∞) → [0, ∞), Φ(0) = 0, satisfies lim sup t → + ∞ ln Φ ( t ) t < ∞ , the for any f ∈ L 1 ( T )

lim n → ∞ 1 N ∑ n = 0 N − 1 Φ ( | S n f ( θ ) − f ( θ ) | ) = 0

for almost every θ ∈ T . Karagulyan [7], [8] proved that Rodin’s condition on Φ is sharp while results similar to Rodin’s were proven in Ref. [9].

Recent work on the convergence problem can be found in Refs. [10], [11], [12] and an overview of the major questions in the area can be found in Konyagin’s survey, [13]. Since we do not offer a novel result for this problem, we will not focus on this aspect.

2 Notation and preliminaries

We condense the notation for complex exponential by

e ( x ) ≔ e 2 π i x

and, for f ∈ L 1 ( T ) , denote the Fourier transform by

f ̂ ( n ) = ∫ T f ( θ ) e ( − n θ ) d θ .

Moreover, for n ∈ Z + and f ∈ L 1 ( T ) , we define the one-dimensional partial sum operator, S n : L 1 ( T ) → C ∞ ( T ) , by

S n f ( θ ) ≔ ∑ m = − n n f ̂ ( m ) e ( m θ ) = ( D n ∗ f ) ( θ )

where

D n ( x ) = ∑ m = − n n e ( m x ) = e ( ( n + 1 ) x ) − e ( − n x ) e ( x ) − 1 .

Let V _N denote the de la Vallée Poussin kernel with piecewise linear Fourier transform satisfying

V ̂ N ( n ) = 1 | n | ≤ 10 N 0 | n | ≥ 20 N

along with

Θ N f ≔ V N ∗ f .

V _N can be defined to satisfy

V N ( θ ) ∼ N − 1 ⁡ min ( N 2 , ‖ θ ‖ − 2 ) .

Along the same lines, define the Littlewood-Paley projector (For N > 1) by

P N f ≔ ( V N − V N / 100 ) ∗ f

and P ₁ f ≔ V ₁∗f. For N ∈ Z + , define

A ° N f ( θ ) ≔ N − 1 ∑ n = N / 2 N | S n f ( θ ) | 2 A N f ( θ ) ≔ N − 1 ∑ n = 1 N | S n f ( θ ) | 2 .

2.1 Covering lemma

The following geometric covering lemma was proven in Ref. [1].

Lemma 1

(Lemma 8 from [1]). Let G be a finite collection of pairwise disjoint, nonadjacent dyadic intervals. Let G * be the collection of dilated intervals that are each of the form 9 8 ⋅ I ≕ I * for all I ∈ G . If ∪ J ∈ G * J is an interval, then

(4) ⋃ J ∈ G * J ⊂ 4 ⋅ J 0

where J ₀ is the largest interval in G .

The lemma essentially says that one can decompose a collection of nonadjacent intervals into families such that two intervals, I and J, from different families are strongly spaced in the sense that their dilations are disjoint: for any ε ∈ [ 0 , 1 8 ] ,

( 1 + ε ) ⋅ I ∩ ( 1 + ε ) ⋅ J = ∅ .

Moreover, each individual family has a unique interval of maximal length whose dilate contains the rest of the intervals in the family [1]. Also contains a higher-dimensional version of the covering lemma.

2.2 Pointwise operator bounds

Fix N ∈ N . Using

D n ( x ) = e ( ( n + 1 ) x ) e ( x ) − 1 − e ( − n x ) e ( x ) − 1

we can estimate N − 1 ∑ n = 1 N | S n f ( θ ) | 2 by

N − 1 ∑ n = 1 N | S n f ( θ ) | 2 = N − 1 ∑ n = 1 N | D n ∗ f ( θ ) | 2 ≲ N − 1 ∑ n = − 2 N 2 N ∫ | x | > N − 1 e ( n x ) e ( x ) − 1 f ( θ − x ) d x 2 + ∑ n = 1 N N − 1 ∫ | x | ≤ N − 1 D n ( x ) f ( θ − x ) d x 2 .

Now we use Plancherel in x to obtain

N − 1 ∑ n = − 2 N 2 N ∫ | x | > N − 1 e ( n x ) e ( x ) − 1 f ( θ − x ) d x 2 ≲ K N ∗ f 2 ( θ )

where K _N(z) = N ⁻¹min(N ², |z|⁻²) for z ∈ T . Also, since n ≤ N, and ‖D _n(x)‖_∞ ≲ n, Jensen’s inequality implies

∑ n = 1 N N − 1 ∫ | x | ≤ N − 1 D n ( x ) f ( θ − x ) d x 2 ≲ ∑ n = 1 N N − 1 ∫ | x | < N − 1 n f ( θ − x ) 2 d x ≲ ∫ | x | < N − 1 N f ( θ − x ) 2 d x ≲ ∫ T K N ( x ) f ( θ − x ) 2 d x .

Therefore,

(5) N − 1 ∑ n = 1 N | S n f ( θ ) | 2 ≲ K N ∗ f 2 ( θ )

and since S _n f = S _nΘ_N f for n ≤ N, we also have

(6) N − 1 ∑ n = 1 N | S n f ( θ ) | 2 ≲ K N ∗ Θ N f 2 ( θ ) .

The splitting on D _n into its component near zero and its part away from zero may seem overly technical, but they are essential to achieving the pointwise bound with a nice approximate identity, {K _N}.

3 Important estimates

Here we collect all of the requisite estimates to prove the main theorem. We start with the standard Calderón-Zygmund decomposition.

3.1 Calderón-Zygmund decomposition

We perform a Calderón-Zygmund decomposition at height 10λ‖f‖₁ > 0. We label the set of bad intervals by B . Without loss of generality, we can assume that the bad intervals are dyadic intervals that are not pairwise adjacent ( I 1 ̄ ∩ I 2 ̄ = ∅ ) . Now we decompose f according the set of intervals, B by

b ≔ ∑ I ∈ B χ I f g ≔ f − b .

For every interval I ∈ B , let I * ≔ 9 8 ⋅ I . We must account for the fact that the set I* are not necessarily pairwise disjoint, so we will utilize Lemma 1 to deal with this. First we break up ∪ I ∈ B I * into disjoint connected components { E i } i :

⋃ I ∈ B I * = ⋃ i E i = ⋃ i ⋃ I ∈ B i I * .

For each i, there exists a largest I such that I ∈ B i . We will denote this maximal interval by J _i. We also denote the subcollection of intervals contained in E _i by B i . Lemma 1 implies that

E i = ⋃ I ∈ B i I * ⊂ 9 2 ⋅ J i

for each i ∈ {1, …, n _λ}.

We are now prepared to define our exceptional set, E, by

⋃ I ∈ B I ⊂ ⋃ i = 1 n λ E i ⊂ ⋃ i = 1 n λ 10 ⋅ J i ≕ E .

Clearly, |E| ≤ 10∗(10λ)⁻¹ = λ ⁻¹.

3.2 The good part

We aim to remove the redundancy in our averaging operator by approximating with a smooth operator. We make the simple observation that for every θ ∈ T ,

We further observe that

N − 1 ∑ 2 k ≤ N ∑ n = 2 k − 1 2 k | S n P 2 k f ( θ ) | 2 ≤ ∑ 2 k ≤ N 2 − k ∑ n = 2 k − 1 2 k | S n P 2 k f ( θ ) | 2

and

sup N > 0 N − 1 ∑ 2 k ≤ N ∑ n = 2 k − 1 2 k | S n P 2 k f ( θ ) | 2 ≤ ∑ k ∈ Z + 2 − k ∑ n = 2 k − 1 2 k | S n P 2 k f ( θ ) | 2 .

If we let T N be the operator defined by

T N f ( θ ) ≔ N − 1 ∑ 2 k ≤ N 2 k | Θ 2 k − 1 f ( θ ) | 2 ,

then we can split our main operator into two pieces:

lim sup N > 0 A N f ( θ ) ≲ lim sup N > 0 T N f ( θ ) + ∑ k ∈ Z + 2 − k ∑ n = 2 k − 1 2 k | S n P 2 k f ( θ ) | 2 = lim sup N > 0 T N f ( θ ) + ∑ k ∈ Z + A 2 k ° ( P 2 k f ) ( θ ) .

We will only need to use this decomposition for bounded functions, the domain on which we can not conceal in a small subset of the torus. Of course, a reduction to bounded functions opens up a variety of heavy machinery to prove the boundedness of sup N > 0 A N . In fact, one only needs to observe that sup N > 0 A N can be bounded by the Carleson maximal operator to arrive at a satisfactory conclusion. We are opting to take the long path, to demonstrate that our setting does not require the heavy lifting necessary for the Carleson-Hunt theorem, particularly because the averaging operator behaves so much better than the partial sums operator on L ^p for p ∈ (1, ∞).

Lemma 2.

Let λ > 0, and f ∈ L ∞ ( T ) . Suppose that f satisfies

‖ f ‖ ∞ ≤ λ ‖ f ‖ 1 .

Then

∫ T lim sup N > 0 T N f ( θ ) d θ ≤ C λ ‖ f ‖ 1 2

for some C > 0 independent of f and λ.

Proof.

We make the simple observation that the family, { Θ N } N > 0 , forms a radially bounded, approximate identity. Therefore, for any f ∈ L 1 ( T ) , Θ_N f → f pointwise almost everywhere. Which then implies lim sup N > 0 T N f ≤ C | f | 2 pointwise almost everywhere. Therefore,

∫ T lim sup N > 0 T N f ( θ ) d θ ≤ C ∫ T | f ( θ ) | 2 d θ ≤ C λ ‖ f ‖ 1 2 .

□

Of course, ‖g‖_∞ ≲ λ‖f‖₁, so we can apply Lemma 2 to g. Moreover, Young’s inequality, Monotone Convergence theorem, estimate (5), and the Littlewood-Paley inequality implies

(7) ∫ T \ E ∑ k ∈ Z + A 2 k ° ( P 2 k g ) ( θ ) d θ ≲ ∑ k ∈ Z + ∫ T \ E K 2 k ∗ P 2 k g 2 ≲ ∑ k ∈ Z + ‖ P 2 k g ‖ 2 2 ≲ ‖ g ‖ 2 2 ≲ λ ‖ f ‖ 1 ‖ g ‖ 1 ≲ λ ‖ f ‖ 1 2 .

Therefore, we have the desired bound for the good part of f: for any E ⊂ T ,

(8) ∫ T \ E sup N ≥ 1 A N g ( θ ) d θ ≲ ∫ T \ E ∑ k ∈ Z + A 2 k ° ( P 2 k g ) ( θ ) d θ + ∫ T lim sup N > 0 T N g ( θ ) d θ ≲ λ ‖ f ‖ 1 2 .

3.3 The bad part

Recall that for x ≠ 0,

(9) | D n ( x ) | ≤ e ( ( n + 1 ) x ) e ( x ) − 1 + e ( − n x ) e ( x ) − 1 ,

so we define

E n , N f ( θ ) ≔ ∫ | x | > 1 N e ( n x ) e ( x ) − 1 f ( θ − x ) d x B N f ( θ ) ≔ ∫ | x | ≤ 1 N N f ( θ − x ) d x .

Note that estimate (9) is very poor when | x | ≪ 1 n due to the fact that |e(x) − 1|⁻¹ ≳ |x|⁻¹. Therefore, we aim only to use estimate (9) only when | x | > 1 N , thereby mitigating the losses from this decomposition. Then for n ≤ N

| S n f ( θ ) | 2 ≲ | E n + 1 , N f ( θ ) | 2 + | E − n , N f ( θ ) | 2 + B N | f | 2 ( θ )

and therefore,

1 N ∑ n = 1 N | S n f ( θ ) | 2 ≲ B N | f | 2 ( θ ) + N − 1 ∑ n = − 2 N 2 N | E n , N f ( θ ) | 2 .

Expanding the bad part and the Dirichlet kernels, we obtain

(10) 1 N ∑ n = 1 N | S n b | 2 = 1 N ∑ n = 1 N | S n Θ N b | 2 ≤ ∑ i B N ∑ I ∈ B i Θ N f I 2 + 1 N ∑ n = − 2 N 2 N E n , N ∑ I ∈ B i Θ N f I 2 + ∑ i ≠ j B N ∑ I ∈ B i Θ N f I ∑ I ∈ B j Θ N f I ̄ + ∑ i ≠ j 1 N ∑ n = − 2 N 2 N E n , N ∑ I ∈ B i Θ N f I E n , N ∑ I ∈ B j Θ N f I ̄ .

Recall that by inequality (6)

∑ i B N ∑ I ∈ B i Θ N f I 2 + 1 N ∑ n = − 2 N 2 N E n ∑ I ∈ B i Θ N f I 2 ≲ ∑ i K N ∗ ∑ I ∈ B i Θ N f I 2 .

It is by this reasoning that we arrive at the following lemma:

Lemma 3

(11) ∫ T \ E lim sup N → ∞ K N ∗ ∑ I ∈ B i Θ N f I 2 ( θ ) d θ ≲ λ ∑ I ∈ B i f I 1 ‖ f ‖ 1 .

Proof.

First we make the simple observation that

∫ T \ E lim sup N → ∞ K N ∗ ∑ I ∈ B i Θ N f I 2 ( θ ) d θ ≲ ∫ T \ E sup N ≥ | J i | − 1 K N ∗ ∑ I ∈ B i Θ N f I 2 ( θ ) d θ .

Since we are outside the support of the function that we are mollifying, the low frequencies should be dominant, because the averaging coefficient is smaller. We see this with the following argument. First, we localize:

∫ T \ E sup N ≥ | J i | − 1 K N ∗ ∑ I ∈ B i Θ N f I 2 ( θ ) d θ ≤ ∫ T \ E sup N ≥ | J i | − 1 K N ∗ χ 5 ⋅ J i ∑ I ∈ B i Θ N f I 2 ( θ ) d θ + ∫ T \ E sup N ≥ | J i | − 1 K N ∗ χ T \ 5 ⋅ J i ∑ I ∈ B i Θ N f I 2 ( θ ) d θ .

Now, for fixed θ, we can use ‖ ∑ I ∈ B i Θ N f I ‖ ∞ ≤ N ‖ ∑ I ∈ B i f I ‖ 1 to achieve

∫ T \ E sup N ≥ | J i | − 1 K N ∗ χ 5 ⋅ J i ∑ I ∈ B i Θ N f I 2 ( θ ) d θ = ∑ I ∈ B i f I 1 ∫ T \ E sup N ≥ | J i | − 1 ∫ T K N ( x ) χ 5 ⋅ J i ( θ − x ) N ∑ I ∈ B i Θ N f I ( θ − x ) d x d θ .

Therefore,

For every N ≥ |J _i|⁻¹ and θ ∈ T \ 10 ⋅ J i ,

sup x ∈ 5 ⋅ J i N | K N ( θ − x ) | = sup x ∈ 5 ⋅ J i N N − 1 min ( N 2 , | θ − x | − 2 ) = sup x ∈ 5 ⋅ J i | θ − x | − 2 .

Therefore, we get

∑ I ∈ B i f I 1 ∫ T \ 10 ⋅ J i sup N ≥ | J i | − 1 sup x ∈ 5 ⋅ J i N | K N ( θ − x ) | Θ N ∑ I ∈ B i f I 1 d θ ≲ ∑ I ∈ B i f I 1 2 ∫ T \ 10 ⋅ J i sup x ∈ 5 ⋅ J i | θ − x | − 2 ≲ | J i | − 1 ∑ I ∈ B i ‖ f I ‖ 1 2 ≲ λ ‖ f ‖ 1 ∑ I ∈ B i ‖ f I ‖ 1 .

Summarizing, we have shown that

(12) ∫ T \ E sup N ≥ | J i | − 1 K N ∗ χ 5 ⋅ J i ∑ I ∈ B i Θ N f I 2 ( θ ) d θ ≲ λ ‖ f ‖ 1 ∑ I ∈ B i ‖ f I ‖ 1 .

Now, for the second term we switch the roles of the convolution operator with K _N and the operator, Θ_N.

∫ T \ E sup N ≥ | J i | − 1 K N ∗ χ T \ 5 ⋅ J i Θ N ∑ I ∈ B i f I 2 = ∫ T \ E sup N ≥ | J i | − 1 ∫ T K N ( θ − x ) χ T \ 5 ⋅ J i ( x ) Θ N ∑ I ∈ B i f I ( x ) 2 d x d θ = ∫ T \ E sup N ≥ | J i | − 1 ∫ T K N ( θ − x ) χ T \ 5 ⋅ J i ( x ) ∫ T V N ( x − y ) ∑ I ∈ B i f I ( y ) d y 2 d x d θ .

We will now exploit the relative smallness of operators of the form T f = χ I c Θ N ( χ I f ) . First

χ T \ 5 ⋅ J i ( x ) ∫ T V N ( x − y ) ∑ I ∈ B i f I ( y ) d y ≤ χ T \ 9 2 ⋅ J i ( x ) ∑ I ∈ B i ∫ T V N ( x − y ) f I ( y ) d y ≲ χ T \ 5 ⋅ J i ( x ) ∑ I ∈ B i ∫ T χ I ( y ) V N ( x − y ) f I ( y ) d y ≲ ∑ I ∈ B i ‖ f I ‖ 1 χ T \ 5 ⋅ J i ( x ) sup y ∈ I χ I ( y ) V N ( x − y ) .

Now

χ T \ 5 ⋅ J i ( x ) sup y ∈ I χ I ( y ) V N ( x − y ) ≤ sup x ∈ T \ 5 ⋅ J i sup y ∈ 9 2 ⋅ J i V N ( x − y ) ≤ sup x ∈ T \ 5 ⋅ J i sup y ∈ 9 2 ⋅ J i N − 1 | x − y | − 2 ≲ N − 1 | J i | − 2 .

Since N ≥ |J _i|⁻¹, we know that

χ T \ 5 ⋅ J i ( x ) sup y ∈ I χ I ( y ) V N ( x − y ) ≤ | J i | − 1 .

Finally, we can say that

sup x ∈ T χ T \ 5 ⋅ J i ( x ) ∫ T V N ( x − y ) ∑ I ∈ B i f I ( y ) d y ≲ | J i | − 1 ∑ I ∈ B i ‖ f I ‖ 1 ≲ λ ‖ f ‖ 1 .

With this estimate we can proceed as follows

∫ T \ E sup N ≥ | J i | − 1 ∫ T K N ( θ − x ) χ T \ 5 ⋅ J i ( x ) ∫ T V N ( x − y ) ∑ I ∈ B i f I ( y ) d y 2 d x d θ ≲ λ ‖ f ‖ 1 ∫ T \ E sup N ≥ | J i | − 1 ∫ T K N ( θ − x ) χ T \ 5 ⋅ J i ( x ) ∫ T V N ( x − y ) ∑ I ∈ B i f I ( y ) d y d x d θ .

Now we can proceed with basic Young’s inequality arguments. Let y J i denote the center of the interval J _i. We get the following estimates

Therefore, we have

(13) ∫ T \ E sup N ≥ | J i | − 1 K N ∗ χ T \ 5 ⋅ J i Θ N ∑ I ∈ B i f I 2 ≲ λ ‖ f ‖ 1 ∑ I ∈ B i f I 1 .

Finally, inequalities (12) and (13) provide the full high-mode estimate:

(14) ∫ T \ E sup N ≥ | J i | − 1 A N ∑ I ∈ B i f I ( θ ) d θ ≲ λ ‖ f ‖ 1 ∑ I ∈ B i f I 1 .

□

It is important to see that Lemma 3 is not strong enough to handle the interaction terms in inequality (10). Therefore, we establish the following lemma:

Lemma 4.

When i ≠ j

∫ T \ E lim sup N → ∞ B N ∑ I ∈ B i Θ N f I ∑ I ∈ B j Θ N f I ̄ = 0 .

Proof.

Let I ∈ B i and J ∈ B j . Since I* ∩ J* = ∅,

B N Θ N f I Θ N f J ̄ ( θ ) = ∫ | x | ≤ N − 1 N Θ N f I ( θ − x ) Θ N f J ̄ ( θ − x ) d x .

For N ≫ max(|I|⁻¹, |J|⁻¹) if θ ∈ T \ E and |x| ≤ N ⁻¹, we can say that |Θ_N f _I(θ − x)| ≲ N ⁻¹|I|⁻¹‖f _I‖₁ and |Θ_N f _J(θ − x)| ≲ N ⁻¹|J|⁻¹‖f _J‖₁. Therefore, for θ ∈ T \ E

| B N Θ N f I Θ N f J ̄ ( θ ) | ≲ 1 N N N − 1 | I | − 1 ‖ f I ‖ 1 ( N − 1 | J | − 1 ‖ f J ‖ 1 ) → N → ∞ 0 .

□

Lemma 5.

When i ≠ j

∫ T \ E lim sup N → ∞ 1 N ∑ n = − 2 N 2 N E n , N ∑ I ∈ B i Θ N f I E n , N ∑ I ∈ B j Θ N f I ̄ = 0 .

Proof.

Let I ∈ B i and J ∈ B j . Since I* ∩ J* = ∅, we have

dist ( I , J ) ≳ max ( | I | , | J | ) .

Therefore,

∑ n = − 2 N 2 N E n , N Θ N f I ( θ ) E n , N Θ N f J ( θ ) ̄ ≲ ∑ n = − 2 N 2 N ∫ | x | > N − 1 e ( n x ) e ( x ) − 1 Θ N f I ( θ − x ) d x ∫ | y | > N − 1 e ( − n y ) e ( − y ) − 1 Θ N f J ̄ ( θ − y ) d y .

For the first term, we recall that D 2 N ( z ) ≔ ∑ n = − 2 N 2 N e ( n z ) and note that |D _2N(z)| ≲ |z|⁻¹ for |z| ≥ 1/2N. This implies

∑ n = − 2 N 2 N ∫ | x | > N − 1 e ( n x ) e ( x ) − 1 Θ N f I ( θ − x ) d x ∫ | y | > N − 1 e ( − n y ) e ( − y ) − 1 Θ N f J ̄ ( θ − y ) d y = ∫ | x | > N − 1 ∫ | y | > N − 1 1 ( e ( x ) − 1 ) ( e ( − y ) − 1 ) D 2 N ( x − y ) Θ N f I ( θ − x ) Θ N f J ̄ ( θ − y ) d y d x ≲ min ( | I | − 1 , | J | − 1 ) ∫ | x | > N − 1 1 | x | Θ N f I ( θ − x ) d x ∫ | y | > N − 1 1 | y | Θ N f J ̄ ( θ − y ) d y .

If θ ∈ T \ E , θ − x ∈ I*, and θ − y ∈ J*, then | x | ≳ dist ( I , T \ E ) and | y | ≳ dist ( J , T \ E ) . Therefore,

∫ | x | > N − 1 1 | x | χ I * ( θ − x ) Θ N f I ( θ − x ) d x ≲ dist ( I , T \ E ) − 1 ‖ f I ‖ 1 and ∫ | y | > N − 1 1 | y | χ J * ( θ − y ) Θ N f J ̄ ( θ − y ) d y ≲ dist ( J , T \ E ) − 1 ‖ f J ‖ 1 .

On the other hand, if θ ∈ T \ E and either θ − x ∈ T \ I * or θ − y ∈ T \ J * , then |Θ_N f _I(θ − x)| ≲ N ⁻¹|I|⁻¹‖f _I‖₁ or |Θ_N f _J(θ − x)| ≲ N ⁻¹|J|⁻¹‖f _J‖₁, respectively. Therefore,

∫ | x | > N − 1 1 | x | χ T \ I * ( θ − x ) Θ N f I ( θ − x ) d x ≲ log ( N ) N − 1 | I | − 1 ‖ f I ‖ 1 or ∫ | y | > N − 1 1 | y | χ T \ J * ( θ − y ) Θ N f J ̄ ( θ − y ) d y ≲ log ( N ) N − 1 | J | − 1 ‖ f J ‖ 1 .

Thus, for θ ∈ T \ E ⊂ T \ ( 10 ⋅ J i ∪ 10 ⋅ J ℓ )

lim sup N → ∞ 1 N ∑ n = − 2 N 2 N E n , N f I ( θ ) E n , N f J ( θ ) ̄ ≲ lim sup N → ∞ min ( | I | − 1 , | J | − 1 ) N ‖ f I ‖ 1 dist ( I , T \ E ) + log ( N ) ‖ f I ‖ 1 N | I | ⋅ ‖ f J ‖ 1 dist ( J , T \ E ) + log ( N ) ‖ f J ‖ 1 N | J | = 0 .

□

4 Proof of main theorem

Proof of Theorem 1.

Let λ > 0, f ∈ L 1 ( T ) . As done before, we let B represent the collection of bad intervals for f generated by performing a Calderón-Zygmund decomposition of f at height 10λ‖f‖₁. Then

| I | − 1 ∫ I | f | ≤ 20 ‖ f ‖ 1 λ

for all I ∈ B . Of course, without loss of generality, we can assume the bad intervals are nonadjacent. Now we proceed as usual with the definition of E with E _i are the connected components of ∪ I ∈ B I * , and J _i are the maximal intervals in E _i, ∪ I ∈ B i I * = E i , and

E ≔ ⋃ i 10 ⋅ J i .

Of course, |E| ≤ λ ⁻¹. We decompose f in the usual way. Define

b ≔ ∑ I ∈ B f χ I

and g ≔ f − b. We observe that

A N f ( θ ) ≲ A N g ( θ ) + A N b ( θ ) .

By inequality (8),

∫ T \ E lim sup N → ∞ A N g ( θ ) ≲ λ ‖ f ‖ 1 2 .

For the bad part, we recall the decomposition from earlier:

A N b = 1 N ∑ n = 1 N | S n b | 2 = 1 N ∑ n = 1 N | S n Θ N b | 2 ≲ K N ∗ ∑ I ∈ B i Θ N f I 2 + ∑ i ≠ j B N ∑ I ∈ B i Θ N f I ∑ I ∈ B j Θ N f I ̄ + 1 N ∑ n = − 2 N 2 N E n , N ∑ I ∈ B i Θ N f I E n , N ∑ I ∈ B j Θ N f I ̄ .

Now Lemmas 3–5 imply

∫ T \ E lim sup N → ∞ A N b ( θ ) ≲ λ ‖ f ‖ 1 2 .

□

Corresponding author: Bobby Wilson, Department of Mathematics, University of Washington, Seattle, USA, E-mail: blwilson@uw.edu

Funding source: National Science Foundation

Award Identifier / Grant number: DMS 1856124

Award Identifier / Grant number: DMS 2142064

Acknowledgments

This article was written in honor of Professor Robert Fefferman. I was invited to study analysis with Professor Fefferman at the University of Chicago in the summer of 2009. I was given a copy of Stein’s “Singular Integrals and Differentiabilty Properties of Functions”, and I have been captivated by harmonic analysis ever since.

Research ethics: Not applicable.
Informed consent: Not applicable.
Author contributions: The author has accepted responsibility for the entire content of this manuscript and approved its submission.
Use of Large Language Models, AI and Machine Learning Tools: None declared.
Conflict of interest: The author states no conflict of interest.
Research funding: B. W. was supported by NSF grant, DMS 1856124, and NSF CAREER Fellowship, DMS 2142064.
Data availability: Not applicable.

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Received: 2024-08-26

Accepted: 2025-03-10

Published Online: 2025-04-11

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Keywords for this article

Fourier series; L 1 pointwise convergence; maximal average operators

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