Some remarks on the Smith-Pinio-McGehee proof of Littlewood’s conjecture F.L.Nazarov

Abstract The main goal of this paper is to show flexibility of the technique used by McGehee,

Pinio and Smith for obtaining a lower bound for L^{1}-norm of trigonometric sums. Using the same method one can generalize the estimate onto the case of non-integer frequencies, obtain a new proof of the Bochkarev lemma on martingals and its analogue for trigonometric sums. Thus, one can see that the technique is not as tightly connected with exact orthogonality and ”unidirectiveness” of spectrum as was thought before.

§0. Introduction. This article has been written not for the concrete results it contains. Its main task is to

show ”flexibility” of the Smith-Pinio-McGehee technique [2], which most people consider to be very tightly connected with exact orthogonality and ”unidirectiveness” of spectrum. Besides, he reader who knows the subject well will also note an essential simplification in the proof of the so called ”difficult” Bochkarev lemma on martingals. Main cause of this simplification is that we use some automate ( the function xe^{−x} ) instead of a straightforward handiwork to split a function into levels and to choose maximal terms ( It is this automatization that makes the idea of [2] so useful and attractive; in the case of Bochkarev’s lemma it can be seen very clear and the reader may compare our proof with original one due to Bochkarev step by step to enjoy how neatly the function xe^{−x} works. I will only display in the article the original proof of the ”simple” Bochkarev lemma for which the comparison is almost trivial but may be of some interest for the reader who has never dealt with the subject before ).

In conclusion of this introduction I’d like to thank A.N.Podkorytov and S.V.Bochkarev who have acquainted me with this topic.

§1. Smith-Pinio-McGehee inequality for ”well-separated” exponents.

It was conjectured by Littlewood that ∫_{−1/2}^{1/2}|∑_{j=1}^{N} e^{2πim_{j}t}| dt ≥ c log N, where m_{1} <

m_{2} < ⋯ ∈ ℤ are arbitrary different integers. It was proved independently by Koniagin [1] and Smith, Pinio and McGehee [2]. In [2] essentially more general result was obtained:

∫_{−1/2}^{1/2}|∑_{j=1}^{N} a_{j}e^{2πim_{j}t}| dt ≥ c ∑_{j=1}^{N} \frac{|a_{j}|}{j} (a_{j} ∈ ℂ)

We are going to show that the same inequality is true for arbitrary exponential polyno-

mial p(t) = ∑_{j=1}^{N} a_{j}e^{2πiλ_{j}t} with real exponents λ_{j} ∈ ℝ satisfying λ_{j+1} − λ_{j} ≥ 1 if we replace

the integral over [−1/2, 1/2] by that over I = [−Δ/2, Δ/2] with any Δ > 1, the constant c in the right hand part depending on Δ only ( unfortunately I don’t know whether it is possible to take Δ = 1 ). Our proof will differ from that in [2] very slightly. For convenience we denote δ = Δ − 1 and suppose indices of the exponents λ_{j} to change not from 1 to N but from J_{δ} + 1 to J_{δ} + N where J_{δ} is some large positive integer depending on δ only ( its precise value will be chosen later ).

Let ψ be a C^{∞}-smooth function supported at the interval (−δ/2, δ/2) and such that

∫_{ℝ} ψ = 1, η(t) = \{ \begin{matrix}cos πt, & t ∈ (−\frac{1}{2}, \frac{1}{2}) \\ 0, & t ∉ (−\frac{1}{2}, \frac{1}{2}) \end{matrix} , φ = \frac{π}{2} φ ∗ ψ. Then supp φ ⊂ I,

φ̂(λ) = \frac{cos πλ}{1 − 4λ^{2}} ψ̂(λ)

If J_{δ} is large enough then ∑_{\begin{matrix}j=J_{δ}+1 \\ j≠j_{0} \end{matrix}}^{J_{δ}+N} \frac{|φ̂(λ_{j} − λ_{j_{0}})|}{j} ≤ \frac{1 − η}{j_{0}} for each j_{0} > J_{δ} ( η > 0

depends on δ only ).

Indeed, |ψ̂(λ)| ≤ 1 − 3η when |λ| > 1 and therefore |φ̂(λ)| ≤ \frac{1 − 3η}{4λ^{2} − 1} when |λ| > 1.

So, ∑_{j≠j_{0}} |φ̂(λ_{j} − λ_{j_{0}})| ≤ (1 − 3η) ∑_{j≠j_{0}} \frac{1}{4(j − j_{0})^{2} − 1} = 1 − 3η.

Beside it, since φ is smooth, |φ(λ)| < \frac{1}{|λ|^{3}} , provided that λ is greater than some

positive constant A > 0. Write

∑_{j≠j_{0}} \frac{|φ̂(λ_{j} − λ_{j_{0}})|}{j} = ∑_{j<(1−η)j_{0}} + ∑_{j≥(1−η)j_{0}} = Σ_{1} + Σ_{2}.

The sum Σ_{1} consists of at most j_{0} terms. Each of them doesn’t exceed \frac{1}{(ηj_{0})^{3}} modulo,

provided that |λ_{j} − λ_{j_{0}}| ≥ |j − j_{0}| ≥ ηj_{0} ≥ ηJ_{δ} > A ( This is a condition on J_{δ} ). Thus

Σ_{1} ≤ j_{0} \frac{1}{(ηj_{0})^{3}} ≤ \frac{1}{η^{3}J_{δ}j_{0}} ≤ \frac{η}{j_{0}}

for each J_{δ} > max{Aη^{−1}, η^{−4}}.

From the other side

Σ_{2} ≤ \frac{1}{(1 − η)j_{0}} ∑_{j≠j_{0}} |φ̂(λ_{j} − λ_{j_{0}})| ≤ \frac{1 − 3η}{1 − η} \frac{1}{j_{0}} < \frac{1 − 2η}{j_{0}} .

Adding these inequalities we get the estimate desired.

Without loss of generality we may assume that J_{δ} is a power of 2, J_{δ} = 2^{m(δ)}, say. Let b_{j} be any complex numbers, equal to 1 modulo. For each m ≥ m(δ) put

f_{m}(t) = ∑_{2^{m}<j≤2^{m+1}} \frac{b_{j}e^{−2πiλ_{j}t}}{j} .

So, for each j_{0} > J_{δ} we have

∫_{I} \{ ∑_{m≥m(δ)} f_{m}(t)\}φ(t)e^{2πiλ_{j_{0}} t} dt = \frac{b_{j_{0}}}{j_{0}} + R_{j_{0}},

where |R_{j_{0}}| = | ∑_{j≠j_{0}} b_{j} \frac{φ̂(λ_{j} − λ_{j_{0}})}{j}| ≤ ∑_{j≠j_{0}} \frac{|φ̂(λ_{j} − λ_{j_{0}})|}{j} ≤ \frac{1 − η}{j_{0}}

The well-known Salem inequality implies that ∥f_{m}∥_{2} ≤ Const 2^{−m/2} ( here ∥·∥ denotes

the norm in L^{2}(I) ). There exist functions h_{m}(t) = ∑_{l=0}^{∞} h_{m,l}e^{(2π/Δ)ilt} ( Δ-periodic ) such

that ℜh_{m} = |f_{m}| , ∥h_{m}∥_{2} ≤ 2∥f_{m}∥_{2} ≤ Const 2^{−m/2}. Put H_{m}(t) = ∑_{n≥m} h_{n}(t). Obviously

∥H_{m}∥_{2} ≤ Const 2^{−m/2} and H_{m}(t) = ∑_{l=0}^{∞} H_{m,l}e^{(2π/Δ)ilt}.

Fix small ε > 0 and put

F (t) = ∑_{m≥m(δ)} f_{m}(t)e^{−εH_{m}(t)}.

It is easy to check that ∥F ∥_{∞} ≤ \frac{1}{ε}. Let 2^{m_{0}} < j_{0} ≤ 2^{m_{0}+1} , m_{0} ≥ m(δ). Let’s now calculate the integral

∫_{I} F (t)e^{2πiλ_{j_{0}} t}φ(t) dt = ∫_{I} { ∑_{m≥m(δ)} f_{m}(t)}e^{2πiλ_{j_{0}} t}φ(t) dt+

∫_{I} { ∑_{m≥m(δ)} f_{m}(t)(e^{−εH_{m}(t)} − 1)}e^{2πiλ_{j_{0}} t}φ(t) dt = J_{1} + J_{2}

J_{1} has already been estimated above:

J_{1} = \frac{b_{j_{0}}}{j_{0}} + R_{j_{0}} |R_{j_{0}}| ≤ \frac{1 − η}{j_{0}}

Further, we have

J_{2} = ∫_{I} ∑_{m≤m_{0}−2} ⋯ + ∫_{I} ∑_{m≥m_{0}−1} ⋯ = J′_{2} + J′′_{2}

We shall estimate J′′_{2} in the most trivial way:

|J′′_{2}| ≤ Const ∑_{m≥m_{0}−1} ε∥f_{m}∥_{2}∥H_{m}∥_{2} ≤Const ε ∑_{m≥m_{0}−1} 2^{−m} =

Const 2^{−m_{0}} ≤ \frac{Const ε}{j_{0}} < \frac{η}{3j_{0}} ,

provided that ε is small enough.

The function e^{−εH_{m}(t)} − 1 can be represented as e^{−εH_{m}(t)} − 1 = ∑_{l=0}^{∞} q_{m,l}e^{(2π/Δ)ilt}, all

the coefficients q_{m,l} being less than 2 modulo ( for the function doesn’t exceed 2 modulo at each point ). Therefore the sum in the definition of J′_{2} consists of at most j_{0} terms of the kind

\frac{b_{j}}{j} e^{−2πiλ_{j}t}(∑_{l=0}^{∞} q_{l}e^{(2π/Δ)ilt})e^{2πiλ_{j_{0}} t}φ(t)

where j ≤ \frac{j_{0}}{2} , |q_{l}| ≤ 2.

The integral of each of these terms over I can be estimated from above by

2 ∑_{l=0}^{∞} |φ̂(λ_{j} − λ_{j_{0}} − \frac{2π}{Δ} l)| ≤ \frac{η}{12(λ_{j_{0}} − λ_{j})^{2}} ,

provided that λ_{j_{0}} − λ_{j} is large enough ( we use here the smoothness of φ again ). But

λ_{j_{0}} − λ_{j} ≥ \frac{j_{0}}{2} ≥ \frac{J_{δ}}{2} , and again it is enough to choose J_{δ} not very small. Thus,

|J′_{2}| ≤ j_{0} \frac{η}{12(j_{0}/2)^{2}} = \frac{η}{3j_{0}}

Finally we have:

∫_{I} F (t)e^{2πiλ_{j_{0}} t}φ(t) dt = \frac{b_{j_{0}}}{j_{0}} + R̃_{j_{0}} , |R̃_{j_{0}}| ≤ \frac{1 − η/3}{j_{0}}.

The rest doesn’t differ a bit from the classic reasoning for integer exponents:

Putting b_{j} = \overline{sign a}_{j}, we find:

|∫_{I} (∑_{j} a_{j}e^{2πiλ_{j}t})F (t)φ(t) dt| = |∑_{j}(\frac{|a_{j}|}{j} + a_{j}R̃_{j})| ≥ \frac{η}{3} ∑_{j=J_{δ}+1}^{J_{δ}+N} \frac{|a_{j}|}{j},

and since |F (t)φ(t)| ≤ \frac{π}{2ε} we obtain

∥ ∑_{j=J_{δ}+1}^{J_{δ}+N} a_{j}e^{2πiλ_{j}t}∥_{1} ≥ \frac{2εη}{3π} ∑_{j=J_{δ}+1}^{J_{δ}+N} \frac{|a_{j}|}{j}.

If we return to the initial enumeration, we shall get

∥∑_{j=1}^{N} a_{j}e^{2πiλ_{j}t}∥_{1} ≥ \frac{2εη}{3π} ∑_{j=1}^{N} \frac{|a_{j}|}{j + J_{δ}} ≥ \frac{2εη}{3π(J_{δ} + 1)} ∑_{j=1}^{N} \frac{|a_{j}|}{j},

which completes the proof.

§2. The Bochkarev lemma for marthingals

In 1974 S.V.Bochkarev proved the following inequality The “simple” Bochkarev lemma

Let χ_{k} be the Haar functions, i.e. χ_{k} : [0, 1] → ℝ,

χ_{k}(t) = \{ \begin{matrix}0 , & t ∉ [\frac{k − 2^{s} − 1}{2^{s}}, \frac{k − 2^{s}}{2^{s}} ) \\ 2^{s/2}, & t ∈ [\frac{k − 2^{s} − 1}{2^{s}}, \frac{k − 2^{s} − 1/2}{2^{s}}) \\ −2^{s/2}, & t ∈ [\frac{k − 2^{s} − 1/2}{2^{s}}, \frac{k − 2^{s}}{2^{s}} ) \end{matrix} (2^{s} < k ≤ 2^{s+1}; s = 0, 1, 2, …)

Let δ_{s} = ∑_{2^{s−1}<k≤2^{s}} c_{k}χ_{k}. The functions δ_{s} are characterized by the following two

properties: δ_{s} is constant on each diadic interval of length 2^{−s} and its integral over each diadic interval of length 2^{−(s−1)} equals 0. By other words, the sequence of partial sums of the series ∑_{s=0}^{∞} δ_{s} forms a marthingal with respect to the sequence of diadic σ-algebras ( only this marthingal property will be used below; so, the following result remains valid for arbitrary marthingals and belongs rather to probability theory than to analysis ).

Let q_{s} be a sequence of positive real numbers satisfying q_{s+1} ≤ (1 − σ)q_{s} for each s = 1, 2, … with some σ > 0. If ∥δ_{s}∥_{∞} ≤ q_{s} for each s then

∫_{0}^{1} ∑_{s=1}^{∞} \frac{1}{q_{s}} δ_{s}^{2}(x) dx ≤ \frac{C}{σ} ∫_{0}^{1}|∑_{s=1}^{∞} δ_{s}(x)| dx.

There is a generalization of the last inequality also due to Bochkarev: The “difficult” Bochkarev lemma.

Let δ_{s} and q_{s} be the same as above, 0 = s_{0} < s_{1} < s_{2} < ⋯ ∈ ℤ_{+} be an increasing sequence of positive integers. Suppose that instead of ||δ_{s}||_{∞} ≤ q_{s} we know that

∥( ∑_{s_{m−1}<s≤s_{m}} δ_{s}^{2})^{1/2}∥_{∞} ≤ q_{m}.

Then

∫_{0}^{1}\{ ∑_{m=1}^{∞} \frac{1}{q_{m}} ( ∑_{s_{m−1}<s≤s_{m}} δ_{s}^{2}(x))\} dx ≤ \frac{C}{σ} ∫_{0}^{1}|∑_{s=1}^{∞} δ_{s}(x)| dx.

Bochkarev applied these lemmas to construct a L^{1}-function Fourier series of which with respect to an arbitrary uniformly bounded orthogonal system diverges on a set of positive measure and a function of bounded variation with prescribed module of continuity ω

satisfying ∑_{n=1}^{∞} \frac{\sqrt{ω(1/n)}}{n} = ∞, Fourier series of which with respect to an arbitrary uniformly

bounded orthogonal system doesn’t converge absolutely, correspondingly.

Proof of the “simple” Bochkarev lemma.

As promised, we will give two proofs. Proof I ( Bochkarev, 1974 )

Define sets E_{m,s} as follows:

E_{m,s} = \{\frac{q_{s}}{2^{m+1}} < |δ_{s}(x)| ≤ \frac{q_{s}}{2^{m}} \}, m = 0, 1, 2, …; s = 1, 2, 3, ….

We have:

∫_{0}^{1} ∑_{s=1}^{∞} \frac{1}{q_{s}} δ_{s}^{2}(x) dx = ∑_{m=0}^{∞} ∑_{s=1}^{∞} \frac{1}{q_{s}} ∫_{E_{m,s}} δ_{s}^{2}(x) dx.

For fixed m the sets E_{m,s} with different s may intersect. To avoid this inconvenience put H_{m,s} = E_{m,s} \ ⋃_{r=0}^{s−1} E_{m,r}

Since H_{m,r} can intersect E_{m,s} only when r ≤ s we get

∫_{0}^{1} ∑_{s=1}^{∞} \frac{1}{q_{s}} δ_{s}^{2}(x)dx = ∑_{m=0}^{∞} ∑_{s=1}^{∞} \frac{1}{q_{s}} ∑_{r=1}^{s} ∫_{E_{m,s}∩H_{m,r}} δ_{s}^{2}(x)dx = ∑_{m=0}^{∞} ∑_{r=1}^{∞} ∑_{s=r}^{∞} \frac{1}{q_{s}} ∫_{E_{m,s}∩H_{m,r}} δ_{s}^{2}(x)dx

It follows immediately from the definitions of E_{m,s} and H_{m,r} that

∫_{E_{m,s}∩H_{m,r}} δ_{s}^{2}(x) dx ≤ \frac{q_{s}^{2}}{2^{2m}} mes H_{m,r}

\frac{q_{r}}{2^{m+1}} mes H_{m,r} ≤ ∫_{H_{m,r}} |δ_{r}(x)| dx

Using these inequalities and the obvious estimate ∑_{s=r}^{∞} q_{s} ≤ \frac{1}{σ} q_{r}, we obtain

∫_{0}^{1} ∑_{s=1}^{∞} \frac{1}{q_{s}} δ_{s}^{2}(x) dx ≤ ∑_{m=0}^{∞} \frac{1}{2^{2m}} ∑_{r=1}^{∞} mesH_{m,r} ∑_{s=r}^{∞} q_{s} ≤ \frac{2}{σ} ∑_{m=0}^{∞} \frac{1}{2^{m}} ∑_{r=1}^{∞} ∫_{H_{m,r}} |δ_{r}(x)| dx

Note that the sets E_{m,r} consist of entire diadic intervals the Haar functions of s-th bundle δ_{s} are supported on. So, the sets H_{m,r} possess the same property and therefore

∫_{H_{m,r}} |δ_{r}(x)| dx ≤ ∫_{H_{m,r}} | ∑_{s=1}^{∞} δ_{s}(x)| dx

Using the last two inequalities and taking into account that the sets H_{m,s} are mutually disjoint for fixed m , we get

∫_{0}^{1} ∑_{s=1}^{∞} \frac{1}{q_{s}} δ_{s}^{2}(x) dx ≤ \frac{4}{σ} ∫_{0}^{1} | ∑_{s=1}^{∞} δ_{s}(x)| dx

which completes the proof. Proof II ( compare with [2] )

Fix ε > 0 and put

F = ∑_{s=1}^{∞} \frac{δ_{s}}{q_{s}} exp\{−ε ∑_{r=1}^{s} \frac{|δ_{r}|}{q_{r}}\}.

Note that ∥F ∥_{∞} ≤ \frac{1}{ε} and for each p ∈ ℕ

∫_{0}^{1} F δ_{p} = ∫_{0}^{1} \frac{δ_{p}^{2}}{q_{p}} exp\{−ε ∑_{r=1}^{p} \frac{|δ_{r}|}{q_{r}}\}.

The latter follows from the simple observation that the integral ∫_{0}^{1} \frac{δ_{s}}{q_{s}} exp\{−ε ∑_{r=1}^{s} \frac{|δ_{r}|}{q_{r}}\}δ_{p}

equals 0 if p > s because first two factors are constant on diadic intervals of length 2^{−s} while the integral of δ_{p} over each such an interval equals 0. If, conversely, p > s then the last two factors are constant on diadic intervals of length 2^{s−1} while the integral of δ_{s} over each such an interval equals 0 ( The fact that |δ_{s}| is constant on twice longer intervals than δ_{s} is, of course, nothing but a lucky coincidence but it is too pleasant to disregard it. When we prove the more general “difficult” Bochkarev lemma we shall not use this specific feature of the Haar system ).

Thus

∫_{0}^{1} F ∑_{s=1}^{∞} δ_{s} = ∑_{s=1}^{∞} ∫_{0}^{1} \frac{δ_{s}^{2}}{q_{s}} exp\{−ε ∑_{r=1}^{s} \frac{|δ_{r}|}{q_{r}}\} ≥ ∑_{s=1}^{∞} ∫_{0}^{1} \frac{δ_{s}^{2}}{q_{s}} − ε ∑_{r,s:} ∑_{r≤s} ∫_{0}^{1} \frac{δ_{s}^{2}|δ_{r}|}{q_{s}q_{r}}

( we have used the inequality e^{−εt} ≥ 1 − εt )

But

∑_{r,s:} ∑_{r≤s} ∫_{0}^{1} \frac{δ_{s}^{2}|δ_{r}|}{q_{s}q_{r}} ≤^{(|δ_{s}|≤q_{s})} ∑_{r,s:} ∑_{r≤s} ∫_{0}^{1} \frac{|δ_{s}||δ_{r}|}{q_{r}} ≤ ∑_{r,s:} ∑_{r≤s} ∫_{0}^{1} \frac{1}{2}\sqrt{\frac{q_{s}}{q_{r}}} (\frac{δ_{s}^{2}}{q_{s}} + \frac{δ_{r}^{2}}{q_{r}} )

= \frac{1}{2}\{∑_{r,s:} ∑_{r≤s} \sqrt{\frac{q_{s}}{q_{r}}} ∫_{0}^{1} \frac{δ_{s}^{2}}{q_{s}} + ∑_{r,s:} ∑_{r≤s} \sqrt{\frac{q_{s}}{q_{r}}} ∫_{0}^{1} \frac{δ_{r}^{2}}{q_{r}} \} ≤ \frac{2}{σ} ∑_{s=1}^{∞} ∫_{0}^{1} \frac{δ_{s}^{2}}{q_{s}} ,

because it follows from the condition q_{j+1} ≤ (1 − σ)q_{j} that for each s ∈ ℕ  ∑_{r=1}^{s} \sqrt{\frac{q_{s}}{q_{r}}} ≤

\frac{1}{1 − \sqrt{1 − σ}} ≤ \frac{2}{σ} and for each r ∈ ℕ  ∑_{s=r}^{∞} \sqrt{\frac{q_{s}}{q_{r}}} ≤ \frac{2}{σ} .

Therefore, if we put ε = \frac{σ}{4} , we shall get

∫_{0}^{1} F ∑_{s=1}^{∞} δ_{s} ≥ \frac{1}{2} ∑_{s=1}^{∞} ∫_{0}^{1} \frac{δ_{s}^{2}}{q_{s}}

and taking into account that ∥F ∥_{∞} ≤ \frac{4}{σ}

∑_{s=1}^{∞} ∫_{0}^{1} \frac{δ_{s}^{2}}{q_{s}} ≤ \frac{8}{σ} ∫_{0}^{1}|∑_{s=1}^{∞} δ_{s}|

Let’s now prove the “difficult” Bochkarev lemma. Put f_{m} = ∑_{s_{m−1}<s≤s_{m}} δ_{s}. Note that the functions f_{m} are constant on diadic intervals of

length 2^{s_{m}} and have zero integral over each diadic interval of length 2^{s_{m−1}}. To repeat the proof of the “simple” case word by word we lack only one thing: the inequality |f_{m}| ≤ q_{m}. Now we are going to change functions f_{m} a little to satisfy this condition. Lemma.

There exist functions g_{m} such that

(1)  ∥g∥_{∞} ≤ 2q_{m},  ∥g_{m}∥_{2} ≤ ∥f_{m}∥_{2}
(2)  ∫_{0}^{1} f_{m}g_{m} ≥ \frac{1}{2} ∫_{0}^{1} f_{m}^{2}
(3)  g_{m} is constant on each diadic interval of length 2^{−s_{m}} and has zero

integral over each diadic interval of length 2^{−s_{m−1}}.

Suppose the lemma proved. Fix small ε > 0 and put

F = ∑_{m=1}^{∞} \frac{g_{m}}{q_{m}} exp\{−ε ∑_{r=1}^{m−1} \frac{|g_{r}|}{q_{r}}\}.

Note that ∥F ∥_{∞} ≤ \frac{e^{2ε}}{ε} ≤ \frac{e^{2}}{ε}.

Property (3) implies

∫_{0}^{1} F ∑_{m=1}^{∞} f_{m} = ∑_{m=1}^{∞} ∫_{0}^{1} \frac{g_{m}}{q_{m}} exp\{−ε ∑_{r=1}^{m−1} \frac{|g_{r}|}{q_{r}}\}f_{m}

≥ ∑_{m=1}^{∞} ∫_{0}^{1} \frac{g_{m}f_{m}}{q_{m}} − ε ∑_{r,m:} ∑_{r<m} ∫_{0}^{1} \frac{|g_{m}||f_{m}||g_{r}|}{q_{m}q_{r}}

But ∫_{0}^{1} \frac{g_{m}f_{m}}{q_{m}} ≥ \frac{1}{2} ∫_{0}^{1} \frac{f_{m}^{2}}{q_{m}}, while

∑_{r,m:} ∑_{r<m} ∫_{0}^{1} \frac{|g_{m}| |f_{m}| |g_{r}|}{q_{m}q_{r}} ≤^{(|g_{m}|≤q_{m})}2 ∑_{r,m:} ∑_{r<m} ∫_{0}^{1} \frac{|f_{m}| |g_{r}|}{q_{r}}

≤ ∑_{r,m:} ∑_{r<m} ∫_{0}^{1} \sqrt{\frac{q_{m}}{q_{r}}} (∫_{0}^{1} \frac{f_{m}^{2}}{q_{m}} + ∫_{0}^{1} \frac{g_{r}^{2}}{q_{r}} ) ≤ \frac{4}{σ} ∑_{m=1}^{∞} ∫_{0}^{1} \frac{f_{m}^{2}}{q_{m}}

Putting here ε = \frac{σ}{16}, we get

∫_{0}^{1}\{ ∑_{m=1}^{∞} \frac{1}{q_{m}} ( ∑_{s=s_{m−1}+1}^{s_{m}} δ_{s}^{2}(x))\} dx = ∑_{m=1}^{∞} ∫_{0}^{1} \frac{f_{m}^{2}}{q_{m}} ≤

\frac{64}{σ} ∫_{0}^{1}| ∑_{m=1}^{∞} f_{m}(x)| dx = \frac{64}{σ} ∫_{0}^{1}|∑_{s=1}^{∞} δ_{s}(x)| dx

because it follows from the condition q_{j+1} ≤ (1 − σ)q_{j} that for each s ∈ ℕ

∑_{r=1}^{s} \sqrt{\frac{q_{s}}{q_{r}}} ≤\frac{1}{1 − \sqrt{1 − σ}} ≤ \frac{2}{σ} and for each r ∈ ℕ  ∑_{s=r}^{∞} \sqrt{\frac{q_{s}}{q_{r}}} ≤ \frac{2}{σ} .

Therefore, putting ε = \frac{σ}{4} , we obtain

∫_{0}^{1} F ∑_{s=1}^{∞} δ_{s} ≥ \frac{1}{2} ∑_{s=1}^{∞} ∫_{0}^{1} \frac{δ_{s}^{2}}{q_{s}}

and taking into account that ∥F ∥_{∞} ≤ \frac{4}{σ},

∑_{s=1}^{∞} ∫_{0}^{1} \frac{δ_{s}^{2}}{q_{s}} ≤ \frac{8}{σ} ∫_{0}^{1}|∑_{s=1}^{∞} δ_{s}|

It remains to construct functions g_{m} with properties desired. It may be done in many ways, all these ways being quite standard. We shall choose the most direct of them: the stop moments technique.

Let f_{m}^{(j)} =^{def} ∑_{s=s_{m−1}+1}^{j} δ_{s} (j = s_{m−1} + 1, …, s_{m})

Put E_{s_{m−1}} = [0, 1], E_{s} = {|f_{m}^{(j)}| ≤ q_{m} s_{m−1} < j ≤ s} (s_{m−1} < s ≤ s_{m}) Denote g_{m} = ∑_{s_{m−1}<s≤s_{m}} χ_{E_{s−1}} δ_{s} ( by other words, we stop to add terms into f_{m} in the

very moment the sum becomes greater than q_{m} modulo ). Obviously, ∥g_{m}∥_{∞} ≤ 2q_{m} and g_{m} is constant on each diadic interval of length 2^{−s_{m}}. Beside it, as E_{s−1} consists of entire diadic intervals of length 2^{−(s−1)} and the integral of δ_{s} over every such interval equals 0, the integral of g_{m} over each diadic interval of length 2^{−s_{m−1}} equals 0. Further, on the set E_{s_{m}−1} we have g_{m} ≡ f_{m} and therefore

∫_{E_{sm−1}} g_{m}^{2} = ∫_{E_{sm−1}} g_{m}f_{m} = ∫_{E_{sm−1}} f_{m}^{2}.

For each s_{m−1} < s < s_{m} on E_{s−1} \ E_{s} we have: f_{m} = g_{m} + ∑_{s<r≤s_{m}} δ_{r}, functions

g_{m} and δ_{r} (s < r ≤ s_{m}) being orthogonal on E_{s−1} \ E_{s} ( E_{s−1} \ E_{s} consists of entire diadic intervals of length 2^{−s}, the function g_{m} = f_{m}^{(s)} is constant on each of them while the function δ_{r} with s < r ≤ s_{m} has zero integral over each of them. The mutual orthogonality of the functions δ_{r} can be obtained in the same way )

Thus,

∫_{E_{s−1}\E_{s}} f_{m}^{2} = ∫_{E_{s−1}\E_{s}} g_{m}^{2} + ∫_{E_{s−1}\E_{s}} ∑_{s<r≤s_{m}} δ_{r}^{2}.

But g_{m}^{2} ≥ q_{m}^{2} on E_{s−1} \ E_{s} while ∑_{s<r≤s_{m}} δ_{r}^{2} ≤ ∑_{s_{m−1}<r≤s_{m}} δ_{r}^{2} ≤ q_{m}^{2} and therefore

∫_{E_{s−1}\E_{s}} f_{m}g_{m} = ∫_{E_{s−1}\E_{s}} g_{m}^{2} ≥ \frac{1}{2} ∫_{E_{s−1}\E_{s}} f_{m}^{2}.

Uniting these estimates, we find

∫_{0}^{1} f_{m}g_{m} ≥ \frac{1}{2} ∫_{0}^{1} f_{m}^{2}.

The lemma is proved.

§3. The Bochkarev lemma for the trigonometric system. We shall consider the “difficult” case immediately:

Proposition.

Let δ_{s}(z) = ∑_{2^{s−1}<|k|≤2^{s}} c_{k}z^{k}; 0 = s_{0} < s_{1} < ⋯ ∈ ℤ_{+}; {q_{m}} be a sequence of posi-

tive numbers satisfying q_{m+1} ≤ (1 − σ)q_{m}. Suppose that ∥( ∑_{s_{m−1}<s≤s_{m}} δ_{s}^{2})^{1/2}∥_{∞} ≤ q_{m}.

Then

∫_{𝕋} \{ ∑_{m=1}^{∞} \frac{1}{q_{m}} ( ∑_{s_{m−1}<s≤s_{m}} |δ_{s}(z)|^{2})\} dμ(z) ≤ \frac{C}{σ} ∫_{𝕋} |∑_{s=1}^{∞} δ_{s}(x)| dμ

( Here μ denotes Lebesgue measure on the unit circumference 𝕋 normalized by μ(𝕋) = 1 ). Proof.

As in the marthingal case, the proof consists of 2 parts: the construction of the auxiliary functions g_{m} and the application of the Smith technique itself. As above, put f_{m} = ∑_{s_{m−1}<s≤s_{m}} δ_{s}. Note that the spectrum of f_{m} is contained in the “symmetrical inter-

val” I(2^{s_{m−1}}, 2^{s_{m}}) ( for convenience we denote by I(a, b) the union [−b, −a) ∪ (a, b] ) Construction of the auxiliary functions g_{m}. We need to construct functions g_{m} such that

(1)  ∥g∥_{∞} ≤ Const q_{m},  ∥g_{m}∥_{2} ≤ ∥f_{m}∥_{2}
(2)  ℜ ∫_{0}^{1} f_{m}g_{m} ≥ const ∫_{0}^{1} |f_{m}|^{2}
(3)  spec g_{m} ⊂ I(2^{s_{m−1}}, 2^{s_{m}})

Split the function f_{m} into two terms: f_{m} = (δ_{s_{m−1}+1} + δ_{s_{m}}) + ∑_{s=s_{m−1}+2}^{s_{m}−1} δ_{s} =

f′_{m} + f′′_{m}. The functions f′_{m} and f′′_{m} are orthogonal and ∫_{𝕋} \overline{f}′_{m}f_{m} + ∫_{𝕋} \overline{f}′′_{m}f_{m} = ∥f_{m}∥_{2}^{2}.

Two cases are possible:

1) ℜ ∫_{𝕋} \overline{f}′_{m}f_{m} ≥ \frac{1}{2}∥f_{m}∥_{2}^{2}.

In this case we may take g_{m} = \overline{f}′_{m}

2) ℜ ∫_{𝕋} \overline{f}′′_{m}f_{m} ≥ \frac{1}{2}∥f_{m}∥_{2}^{2}.

Here the construction is a bit more complicated. Put h_{m} = \overline{f}′′_{m} exp\{−η \frac{|f′′_{m}|}{q_{m}}\} ( η >

0 is a small positive number to be chosen later ). We have

∥h_{m} − \overline{f}′′_{m}∥_{2} ≤ ∥\overline{f}′′_{m}(exp\{−η \frac{|f′′_{m}|}{q_{m}}\} − 1)∥_{2} ≤ η∥\overline{f}′′_{m} \frac{|f′′_{m}|}{q_{m}}∥_{2} = \frac{η}{q_{m}} ∥f′′_{m}∥_{4}^{2}

According to the Littlewood-Paley inequality,

∥f′′_{m}∥_{p}^{p} ≤ C_{p}∥f′′_{m}∥_{2}^{2}∥( ∑_{s=s_{m−1}+2}^{s_{m}−1} |δ_{s}|^{2})^{1/2}∥_{∞}^{p−2}

for each p > 2.

In particular ∥f′′_{m}∥_{4}^{4} ≤ C_{4}∥f′′_{m}∥_{2}^{2}q_{m}^{2} and therefore ∥f′′_{m}∥_{4}^{2} ≤ Const q_{m}∥f′′_{m}∥_{2}. Thus, if η is small enough

∥h_{m} − \overline{f}′′_{m}∥_{2} ≤ \frac{1}{24}∥f′′_{m}∥_{2} ≤ \frac{1}{24}∥f_{m}∥_{2}

while, obviously, ∥h_{m}∥_{∞} ≤ \frac{1}{η} q_{m}. The function h_{m} is almost that we need. It remains only

to correct its spectrum.

Recall that the Valle-Poussin kernel K_{j} is the difference of two Fejer kernels K_{j} = 2Φ_{2j} − Φ_{j}. The convolution with the kernel K_{j} preserves the Fourier coefficients with indices less than or equal to j modulo and kills those with indices greater than 2j modulo. Beside it, the norm of the corresponding convolution operator in each space L^{p}(𝕋) (1 ≤ p ≤ ∞) doesn’t exceed 3.

Let Q_{m} be a convolution operator with the kernel K_{2^{sm−1}} − K_{2^{s_{m−1}}} . Put g_{m} =

Q_{m}h_{m}. It’s easy to check that spec g_{m} ⊂ I(2^{s_{m−1}}, 2^{s_{m}}) , ∥g_{m}∥_{∞} ≤ 6∥h_{m}∥_{∞} ≤ \frac{6}{η} q_{m} and

∥g_{m}∥_{2} ≤ ∥h_{m}∥_{2} ≤ ∥f_{m}∥_{2}.

It remains only to note that

\begin{matrix}ℜ ∫_{𝕋} g_{m}f_{m} = ℜ ∫_{𝕋} (Q_{m}h_{m})f_{m} = ℜ ∫_{𝕋} (Q_{m}\overline{f}′′_{m})f_{m} + ℜ ∫_{𝕋} Q_{m}(h_{m} − \overline{f}′′_{m})f_{m} ≥ \\ ℜ ∫_{𝕋} \overline{f}′′_{m}f_{m} − 6∥h_{m} − \overline{f}′′_{m}∥_{2}∥f_{m}∥_{2} ≥ \frac{1}{2}∥f_{m}∥_{2}^{2} − \frac{1}{4}∥f_{m}∥_{2}^{2} = \frac{1}{4}∥f_{m}∥_{2}^{2} , \end{matrix}

for Q_{m}\overline{f}′′_{m} = \overline{f}′′_{m}.

The construction of auxiliary functions g_{m} is now complete.

To proceed further we need the following simple Lemma.

Let g(z) be a trigonometric polynomial spectred at [−4M, 4M] (M ∈ ℕ). Then there exists a nonnegative trigonometric polynomial p(z) spectred at [−M, M] and such that

(1)  const |g(z)| ≤ p(z) ≤ ∥g∥_{∞} for each z ∈ 𝕋
(2)  ∥p∥_{2} ≤ Const ∥g∥_{2}

( all the constants are absolute ) Proof.

Split the unit circumference 𝕋 into M equal arcs L_{i} and put φ(z) = max_{L_{i}} |g| for

z ∈ L_{i}. It is easy to derive from the S.N.Bernstein inequality ∥\frac{d}{dz} g∥_{2} ≤ 4M∥g∥_{2} that ∥φ∥_{2} ≤ Const ∥g∥_{2}. Beside it, obviously ∥φ∥_{∞} ≤ ∥g∥_{∞}.

Put p = φ ∗ Φ_{M} ( Φ_{M} denotes here the Fejer kernel ). It is clear that ∥p∥_{∞} ≤ ∥φ∥_{∞} ≤ ∥g∥_{∞} , ∥p∥_{2} ≤ ∥φ∥_{2} ≤ Const ∥g∥_{2}. The inequality p(z) ≥ const |g(z)| follows immediately from the observation that for each z ∈ L_{i}

p(z) = ∫_{𝕋} φ(ζ)Φ_{M}(zζ^{−1}) dμ(ζ) ≥ ∫_{L_{i}} φ(ζ)Φ_{M}(zζ^{−1})dμ(ζ) = max_{L_{i}} ∫_{zL_{i}^{−1}} Φ_{M}dμ ≥ const |g(z)|

( the letter results from the evident estimate max_{L_{i}} |g| ≥ |g(z)| and the fact that a large part

of the Fejer kernel Φ_{M} of order M is concentrated on the arc zL_{i}^{−1} of length \frac{2π}{M} containing the point 1.)

Lemma is proved. Applying the lemma to each of the trigonometric polynomials g_{m} we get a sequence

of nonnegative trigonometric polynomials {p_{m}} such that

(1)  ∥p_{m}∥_{∞} ≤ Const q_{m}
(2)  ∥p_{m}∥_{2} ≤ Const ∥g_{m}∥_{2}
(3)  p_{m} ≥ const |g_{m}|
(4)  spec p_{m} ⊂ [−2^{s_{m}−2}, 2^{s_{m}−2}]

Fix small ε > 0 and put

F = ∑_{m=1}^{∞} \frac{g_{m}}{q_{m}} ∏_{r=1}^{m−1}(1 − \frac{εp_{r}}{q_{r}} ).

If ε is small enough, the norm ∥\frac{εp_{r}}{q_{r}} ∥_{∞} < 1 and thereby

∏_{r=1}^{m−1}(1 − \frac{εp_{r}}{q_{r}} ) ≤ exp\{−ε ∑_{r=1}^{m−1} \frac{p_{r}}{q_{r}} \} ≤ exp\{−const ε ∑_{r=1}^{m−1} \frac{|g_{r}|}{q_{r}}\}

So, we have ∥F ∥_{∞} ≤ \frac{e^{2}}{const ε} again. Note now that

spec ∏_{r=1}^{m−1}(1 − \frac{εp_{r}}{q_{r}} ) ⊂ [− ∑_{r=1}^{m−1} 2^{s_{r}−2}, ∑_{r=1}^{m−1} 2^{s_{r}−2}] ⊂ (−2^{s_{m−1}−1}, 2^{s_{m−1}−1})

We derive immediately from here that the spectrum

spec \{\frac{g_{m}}{q_{m}} ∏_{r=1}^{m−1}(1 − \frac{εp_{r}}{q_{r}} )\} ⊂ I(2^{s_{m−1}} − 2^{s_{m−1}−1}, 2^{s_{m}} + 2^{s_{m−1}−1}) ⊂ I(2^{s_{m−2}}, 2^{s_{m+1}})

and thereby doesn’t intersect the spectrum of f_{l} when |l − m| > 1.

Thus,

∫_{𝕋} F ∑_{m=1}^{∞} f_{m} dμ = ∑_{m=1}^{∞} ∫_{𝕋} \frac{g_{m}}{q_{m}} ∏_{r=1}^{m−1}(1 − \frac{εp_{r}}{q_{r}} )(f_{m−1} + f_{m} + f_{m+1}) dμ.

The rest is clear. Literature cited

[1]. Koniagin S.V. On the Littlewood problem. Izvestia A.N. SSSR, ser. mat. 45 (1981) no.2, pp.243-265

[2]. McGehee O.C., Pinio l., Smith B., Hardy inequality and the L^{1}-norm of exponential sums. Ann. Math. 113 (1981) no.3 pp.613-618

[3]. Bochkarev S.V. A method of averaging in the theory of orthogonal series and some problems in the theory of bases. Proceedings of the Steclov Institute of Math., 1980, Issue 3.