Abstract. Let n

BOUNDS ON MOMENTS OF WEIGHTED SUMS OF FINITE RIESZ PRODUCTS

ALINE BONAMI, RAFA L LATA LA, PIOTR NAYAR, AND TOMASZ TKOCZ

Abstract. Let n

^j

be a lacunary sequence of integers, such that n

j+1

/n

j

≥ r. We are interested in linear combinations of the sequence of finite Riesz products Q

N

j=1

(1 + cos(n

j

t)). We prove that, whenever the Riesz products are normalized in L

^p

norm (p ≥ 1) and when r is large enough, the L

^p

norm of such a linear combination is equivalent to the `

^p

norm of the sequence of coefficients. In other words, one can describe many ways of embedding `

^p

into L

^p

based on Fourier coefficients. This generalizes to vector valued L

^p

spaces.

2010 Mathematics Subject Classification. Primary: 42A55; Secondary: 26D05, 42A05.

Key words. Riesz products, moment estimates, lacunary sequences, trigonometric polynomials.

1. introduction

Let T = R/2πZ be the one dimensional torus and m be the normalized Haar measure on T. Let (n j ) _j≥1 be an increasing sequence of positive integers. Riesz products are defined on T by

(1) R 0 ≡ 1 and R N (t) :=

N

Y

j=1

(1 + cos(n j t)) for N = 1, 2, . . . . To simplify the notation we also put

X ₀ ≡ 1 and X _j (t) := 1 + cos(n _j t), j = 1, 2, . . . .

It was Frigyes Riesz who first realized the usefulness of these objects treated as probability measures. When n j+1 /n j ≥ 2 for j ≥ 1, the numbers P N

j=1 ε j n j are all nonzero for nonzero vectors (ε _j ) ^N _j=1 ∈ {−1, 0, 1} ^N , due to the fact that for every l, P l

k=1 n _k < n _l+1 . In particular, the zero mode of R _N has Fourier weight 1 and thus R _N are densities of probability measures µ _N . The weak-∗ limit of (µ _N ) is a singular measure which admits a number of remarkable Fourier-analytic properties. The reader is referred for instance to [12] for more information on properties of Riesz products and general trigonometric

This material is partially based upon work supported by the NSF grant DMS-1440140, while the authors were in residence at the MSRI in Berkeley, California, during the fall semester of 2017. P. N. and T. T. were also partially supported by the Simons Foundation. R. L. and P. N. were partially supported by the National Science Centre Poland grant 2015/18/A/ST1/00553 and T. T. by NSF grant DMS-1955175. The research leading to these results is part of a project that has received funding from the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme (grant agreement No 637851).

1

polynomials as well as to the short survey [6] of some applications of Riesz products. We will always assume that n j+1 /n j ≥ 3 for j ≥ 1, so that every integer n can be written at most once as P N

j=1 ε j n j for nonzero vectors (ε j ) ^N _j=1 ∈ {−1, 0, 1} ^N . In this article we shall study the sum P N

k=0 v _k R _k where v _k are vectors in a normed space (E, k · k). By the triangle inequality, we trivially have

(2)

Z

T

N

X

k=0

v _k R _k

dm ≤

N

X

k=0

kv _k k .

We are interested in the reverse inequality and in L ^p inequalities. Our interest in this kind of inequalities comes back to a question of Wojciechowski, who asked for the validity of the reverse bound up to some universal constant (personal communication). He first studied this problem in the scalar case and in the following probabilistic context. Suppose we replace the functions X ₁ , X ₂ , . . . appearing in the definition of the Riesz products with a sequence of independent random variables ¯ X 1 , ¯ X 2 , . . . (defined on some probability space (Ω, P)), each having the same distribution as 1 + cos(Y ), where Y is uniform on [0, 2π].

We then take ¯ R N = Q N

k=1 X ¯ k and of course ¯ R 0 ≡ 1. Note that the functions X _j defined on the probability space (T, m) have the same distribution as the random variables ¯ X _j . Even though the X j are not independent, we shall see that they behave, in many ways, like independent random variables. Capturing this phenomenon in a quantitative way is one of the main difficulties in our investigation.

In [11], Wojciechowski showed the existence of universal constants c and C as well as real numbers a ₁ , a ₂ , . . . such that for every n, | P k

i=0 a _i | ≤ C for all k ≤ n and E| P n

i=0 a _i R ¯ _i | ≥ cn. This result was used in [4, 5] in the study of Fourier multipliers on the homogeneous Sobolev space ˙ W ₁ ¹ (R ^d ).

The reverse of (2) for ¯ R _k was proved by the second named author in [7] for general ran- dom variables. More generally, for any sequence ¯ X ₁ , ¯ X ₂ , . . . of i.i.d. non-negative random variables with mean one and such that P( ¯ X 1 = 1) < 1, we have

(3) E

N

X

k=0

v k R ¯ k

≥ c X ¯

1

N

X

k=0

kv _k k ,

for any vectors v _i in an arbitrary normed space (E, k · k), with a constant c X ¯

1

depending only on the distribution of ¯ X 1 (see Theorem 4 in [7]; see also Theorem 3 therein for non identically distributed sequences ( ¯ X _i )). This clearly implies Wojciechowski’s result with a i = (−1) ⁱ (here E = R). According to a theorem of Y. Meyer (see [8]), under a stronger divergence of the sequence of modes, namely when P ∞

k=1 n

k

n

k+1

< ∞, for any real numbers a _i , we have

Z

T

    

N

X

k=0

a _k R _k     

≥ c _S E     

N

X

k=0

a _k R ¯ _k     

2

for a positive constant c _S which depends only on the n _k . In [7], this principle was combined with (3) to show the reverse of (2) in the real case and under the above restrictive condition on the modes n _i .

Later the results of [7] have been generalized by Damek et al. in [2], where it was shown that for any p > 0 and under the same assumptions on the i.i.d. sequence ( ¯ X i ), we have

(4) 1

C _{p, ¯ X}

₁

N

X

k=0

kv _k k ^p E R ¯ ^p _k ≤ E

N

X

k=0

v _k R ¯ _k

p

≤ C _{p, ¯ X}

1

N

X

k=0

kv _k k ^p E R ¯ ^p _k N ≥ 1, with a constant C _{p, ¯ X}

₁

depending only on p and the distribution of ¯ X ₁ .

The aim of this article is to prove the following theorem.

Theorem 1. For every p ≥ 1 there are positive constants d _p , c _p , C _p depending only on p, such that for any integers n j satisfying n j+1 /n j ≥ d _p , j = 1, 2, . . . and for any vectors v 0 , v 1 , . . . in a normed space (E, k · k), we have

(5) c _p

N

X

k=0

kv _k k ^p Z

T

R ^p _k dm ≤ Z

T

N

X

k=0

v _k R _k

p

dm ≤ C _p

N

X

k=0

kv _k k ^p Z

T

R ^p _k dm, for any N ≥ 1, where R _k are defined via (1).

In words, the normalized sequence (R k /kR k k _L

_p

_(T) ) is ` p -stable on its span. The lower bound in the case p = 1 answers the original question of Wojciechowski. Let us also note that for p > 1, both the upper and the lower bounds are non-trivial (as opposed to the case p = 1 where the upper bound is easy – see (2)). The values of the constants d p , c p

and C _p that can be obtained from our proofs are far from optimal. In particular, we have lim _p→1

⁺

d p = ∞ and lim _p→1

⁺

c p = 0, which is inconsistent with the case p = 1. Due to these blow-ups as p → 1 ⁺ , our proof in the case p = 1 is different from the proof for p > 1.

It is based on transferring the independent case of [7] using Riesz products. We restate the result for p = 1 with numerical values of the constants. (For explicit bounds on the constants for p > 1, see Remark 25.)

Theorem 2. There exists a constant c 1 > 3.1 · 10 ⁻⁸ such that for any positive integers n j

satisfying n _j+1 /n _j ≥ 3 and for any vectors v ₀ , v ₁ , . . . in a normed space (E, k · k), we have Z

T

N

X

j=0

v j R j

dm ≥ c 1 N

X

j=0

kv _j k for R k defined in (1).

Theorem 1 was proved in [2] in the real case (E = R), with a constant depending on p and the sequence (n _j ), under the condition P ∞

k=1 n

_k

n

k+1

< ∞ mentioned earlier (again by combining the independent case with the decoupling inequality of Meyer). It is easy to see that the same proof implies that it is also valid for vector-valued coefficients under the weaker condition P ∞

k=1

 n

k

n

k+1

 2

< ∞, which is known as Schneider’condition [10]. We

3

do it in the next section for completeness. When E = R and p/2 is an integer, then the condition n k+1 /n k ≥ p + 1 is sufficient.

In general, Theorem 1 cannot be transferred from the independent case by using some generalization of Schneider’s condition: L ^p norms of R _k and ¯ R _k are not equivalent, as we see in the next section. So the core of the proof deals directly with Riesz products on the torus. Many new difficulties appear when compared with the proof for independent frequencies.

We conclude with questions: Is the best constant d _p in Theorem 1 an increasing function?

Can it be chosen so that it does not depend on p?

The article is organized as follows. First we present those results that may be obtained as consequences of the i.i.d case. This concerns the case when Schneider’s Condition P ∞

k=1

 n

k

n

k+1

 2

< ∞ is fulfilled as well as Theorem 2 concerning L ¹ norms. The rest of the paper is devoted to the general case. In Section 4 we give preparatory results. The main section is Section 5, which is devoted to the proof of the lower estimate for p > 1. Finally, in Section 6 we give a proof of the upper bound for p > 1.

Acknowledgements. We would like to thank F. Nazarov for stimulating correspon- dence which encouraged us to continue working on this project. We are also indebted to P. Ohrysko for a helpful discussion, and to anonymous referees for very helpful reports significantly improving the paper.

2. The theorem under Schneider’s Condition

The aim of this section is to prove Theorem 1 under Schneider’s Condition, that is, we have the following result.

Proposition 3. Assume that for each j ≥ 1 one has n j+1 /n j ≥ 3 and that, moreover, P  _n

j

n

j+1

 2

< ∞. Then the conclusion of Theorem 1 holds: for every p ≥ 1 there are positive constants c p , C p depending only on p and the sequence (n j ), such that for any vectors v ₀ , v ₁ , . . . in a normed space (E, k · k), the inequalities (5) hold. Moreover, if P  _n

j

n

j+1

 2

≤ 4/(9π ² ), then constants c p , C p do not depend on the sequence (n j ).

To prove this, we proceed as in [7] making a use of Schneider’s condition. First introduce some notation. For an arbitrarily large integer N , let us denote by Λ _N the set of integers that may be written as P N

j=1 ε j n j , with ε j ∈ {−1, 0, 1}, for all j ≤ N . The condition n _j+1 /n _j ≥ 3 ensures that the mapping T = T _N from Λ _N to Z ^N given by T ( P N

j=1 ε _j n _j ) = (ε j ) ^N _j=1 is injective. For a trigonometric polynomial P (x) = P

n∈Λ

N

a n e ^inx on T with values in E, we define e P (y) = P

n∈Λ

N

a n e ^{iT (n)·y} , which is a trigonometric polynomial on T ^N with values in E. The next proposition is a variant of results one can find in Meyer’s book [9], Chapter VIII.

4

Proposition 4. Under the previous assumptions and notations, there exists a constant C which depends only on the sequence (n j ) such that for all E−valued trigonometric polyno- mials P with frequencies in Λ _N and all p ∈ [1, ∞],

(6) C ⁻¹ k ˜ P k _L

^p

_(T

^N

_,E) ≤ kP k _L

p

(T,E) ≤ Ck ˜ P k _L

^p

_(T

^N

_,E) . Moreover, if P  _n

j

n

j+1

 2

≤ 4/(9π ² ), then one may take C = 2.

Proposition 4 together with (4) easily implies Proposition 3 (observe that f R _k has the same distribution as ¯ R _k ). We present here its simple and complete proof that is inspired by [9], Chapter VIII.

To establish (6), we first consider p = ∞ and E = R and iterate the following simple lemma.

Lemma 5. Let P ₁ , P ₂ and P ₃ be trigonometric polynomials of degree at most d. For an integer M > d, we let

P (x) = P ₁ (x) + P ₂ (x)e ^{iM x} + P ₃ (x)e ^{−iM x} , Q(x, y) = P ₁ (x) + P ₂ (x)e ^{iM y} + P ₃ (x)e ^{−iM y} . Then

sup

x∈T

|P (x)| ≥



1 − π ² d ² 2M ²

 sup

x,y∈T

|Q(x, y)|.

Proof. Let (x ₀ , y ₀ ) be a point where |Q| reaches its maximum, which we assume to be nonzero. Without loss of generality we may assume that Q(x 0 , y 0 ) = 1, so that it is also the maximum of its real part. This implies in particular that the derivative in the x variable of its real part vanishes at (x 0 , y 0 ). To conclude it is sufficient to find x 1 ∈ T such that the real part of Q(x 0 , y 0 ) − P (x 1 ) is smaller than ^π _2M

²

^d

2²

. We take x 1 ∈ T to be such that

|x ₁ − x ₀ | ≤ π/M and exp(iM x ₁ ) = exp(iM y 0 ). Then by Taylor’s expansion

<(Q(x ₀ , y 0 ) − P (x 1 )) = <(Q(x 0 , y 0 ) − Q(x 1 , y 0 )) ≤ π ² 2M ² sup

x∈T

|Q ⁰⁰ (x, y 0 )|,

where Q ⁰⁰ stands for the second derivative in the x variable. By Bernstein’s inequality, this supremum is bounded by d ² , which allows to conclude.  Corollary 6. There exists a constant C ∞ which depends only on the sequence (n _j ) such that for all trigonometric polynomials P with frequencies in Λ N ,

(7) C _∞ ⁻¹ sup

y∈T

^N

| ˜ P (y)| ≤ sup

x∈T

|P (x)| ≤ sup

y∈T

^N

| ˜ P (y)|.

Moreover one may take C ∞ = 2 if P  _n

j

n

j+1

 2

≤ 4/(9π ² ).

Proof. Let P (x) = P

n∈Λ

N

a n e ^inx . Here, for convenience, instead of ˜ P , we shall consider Q(y) = ˜ P

n∈Λ

N

a _n e ^{i P}

^j

^ε

^j

ⁿ

^j

^y

^j

, y ∈ T ^N , where ε = T (n). Clearly, sup | ˜ Q| = sup | ˜ P |. The upper bound is obvious because ˜ Q(x, x, . . . , x) = P (x).

5

We use Lemma 5, with M = n _N and d = n ₁ + . . . + n _{N −1} ≤ n _{N −1} 1 + ¹ ₃ + ₃ ¹

2

+ . . .
=

3

2 n N −1 . It implies that sup

x∈T

|P (x)| ≥ c _N sup

x,y

N

∈T

| ˜ Q(x, . . . , x, y _N )|, where

c _N = 1 − 9π ² 8

 n _{N −1} n _N

 2

.

For every fixed y N , ˜ Q(x, . . . , x, y N ) as a function of x is a trigonometric polynomial with frequencies in Λ _{N −1} and therefore we can iterate the above argument to obtain

sup

x∈T

|P (x)| ≥ c _N · . . . · c _N

₀

sup

x,y

_N0

,...,y

N

∈T

| ˜ Q(x, . . . , x, y _N

₀

, . . . , y _N )|.

Observe that Schneider’s condition implies the existence of N ₀ , depending only on the sequence (n j ), such that first, c _k > 0 for every k ≥ N 0 (because necessarily _n ⁿ

^j

j+1

→ 0 as j → ∞), and second, c N · . . . · c _N

₀

≥ ¹ ₂ for N ≥ N 0 . Indeed,

N

Y

k=N

0

c _k ≥ 1 − 9π ² 8

X

k≥N

0

 n _k−1 n _k

 2

since for every real numbers a 1 , . . . , a _l > −1 of the same sign, we have Q l

i=1 (1 + a i ) ≥ 1 + P _l

i=1 a _i . Therefore there is N ₀ depending only on the sequence (n _j ) such that for every polynomial P , we have

(8) sup

x∈T

|P (x)| ≥ 1

2 sup

x,y

_N0

,...,y

N

∈T

| ˜ Q(x, . . . , x, y N

0

, . . . , y N )|.

Now we handle the first M := N 0 − 1 coordinates. Let P _M be the space of trigonometric polynomials on T ^M spanned by {e ⁱ ( ^P

j≤M

ε

j

n

j

y

j

)} _{ε∈{−1,0,1}}

M

. Any two norms on a finite- dimensional space P M are comparable, in particular there exists δ > 0 such that

sup

x∈T

|Q(x, . . . , x)| ≥ δ sup

(y

1

,...,y

_M

)∈T

^M

|Q(y ₁ , . . . , y _M )| for Q ∈ P _M .

The above bound together with (8) implies the lower bound in (7) with C ∞ = 2δ ⁻¹ . To get the last part of the assertion it suffices to observe that if P  _n

j

n

j+1

 2

≤ ⁴

9π

²

then c _k > 0 for all k and

Y

k

c _k ≥ 1 − 9π ² 8

X

k

 n _k−1 n _k

 2

≥ 1 2 .

 Proof of Proposition 4. Let µ be a bounded measure on T and ˜ E N be a set of all functions of the form ˜ P = P

n∈Λ

N

a _n e ^{iT (n)·y} and a _n ∈ R. We may treat ˜ E _N as a subset of the space of continuous functions C(T ^N ). On ˜ E N we define a functional ϕ by the formula

6

ϕ( ˜ P ) = R P dµ. The upper bound in (7) shows that kϕk ≤ kµk _{M (T)} . By the Hahn- Banach theorem we may extend ϕ to C(T ^N ) and thus show that there exists a measure µ ∈ M (T ˜ ^N ) such that k˜ µk _{M (T}

N

) ≤ kµk _{M (T)} , by the Riesz-Markov-Kakutani representation theorem (kµk _{M (T)} is the total variation of µ). Moreover, b µ(T (n)) = ˜ µ(n) for n ∈ Λ b _N because

b ˜

µ(T (n)) = Z

e ^{−iT (n)·y} d˜ µ(y) = ˜ ϕ(e ^{−iT (n)·y} ) = ϕ(e ^{−iT (n)·y} ) = Z

e ^−inx dµ(x) = µ(n). b In the same way we show that for any measure ˜ µ ∈ M (T ^N ), there exists a measure µ ∈ M (T) such that kµk M (T) ≤ C _∞ k˜ µk _{M (T}

N

) and the previously stated relation holds.

Using these observations for Dirac measures we find for x ∈ T and y ∈ T ^N measures

˜

µ _x ∈ M (T ^N ) and µ _y ∈ M (T) such that k˜ µ _x k ≤ 1, kµ _y k ≤ C _∞ and c µ ˜ _x (T (n)) = e ^−inx , c µ y (n) = e ^{−iT (n)·y} for n ∈ Λ N .

Fix now a trigonometric E−valued polynomial ˜ P = P

n∈Λ

N

a n e ^{iT (n)·y} and p ∈ [1, ∞).

Observe that for any x ∈ T, k ˜ P k _L

p

(T

^N

,E) =

X

n∈Λ

N

a _n e ^inx e ^{iT (n)·y} ∗ ˜ µ _x

L

^p

(T

^N

,E)

≤ k˜ µ _x k _{M (T}

N

)

X

n∈Λ

N

a _n e ^inx e ^{iT (n)·y}

L

^p

(T

^N

,E)

≤

X

n∈Λ

N

a _n e ^inx e ^{iT (n)·y}

L

^p

(T

^N

,E)

. Integrating over x ∈ T and changing the order of integration we get

k ˜ P k ^p _L

_p

(T

^N

,E) ≤ Z

T

^N

Z

T

X

n∈Λ

_N

a _n e ^inx e ^{iT (n)·y}

p

dm(x)dm ^N (y).

However for any y ∈ T ^N

X

n∈Λ

N

a _n e ^inx e ^{iT (n)·y} L

p

(T,E)

= kP ∗ µ _y k _L

_p

_(T,E) ≤ kµ _y k _{M (T)} kP k _L

_p

_(T,E) ≤ C ∞ kP k _L

_p

_(T,E) .

This way we show that k ˜ P k _L

^p

_(T

^N

_,E) ≤ C _∞ kP k _L

_p

_(T,E) . The case p = ∞ follows by taking the limit. The upper bound in (6) is shown in an analogous way.  In the rest of this section we discuss the question of generalizing this method to sequences that do not satisfy Schneider’s condition. It was observed in [1] Chapter I that, as a consequence of Plancherel’s formula, the double inequality (6) is valid for p an even integer and E = R as soon as n j+1 /n j ≥ p + 1. It means that the conclusions of Theorem 1 are also valid in this case for scalar functions.

For p/2 an integer, condition n _j+1 /n _j ≥ p + 1 is a natural bound for being able to transfer the result for the independent case to the context of the lacunary sequence n j .

7

This is given by the following lemma. Recall that f R k (y 1 , . . . , y N ) = Q k

j=1 (1 + cos(n j y j )) is a polynomial on T ^N (with the same distribution as the random variable ¯ R _k ).

Lemma 7. Let p > 2 be an even integer and n _k = p ^k . Then lim sup kR _k k _p /k f R _k k _p = ∞.

Proof. This comes from a combinatorial argument. We will use the following fact. For a sequence of positive integers q ₁ , . . . , q _k and a trigonometric polynomial g with nonnegative Fourier coefficients, we have

(9)

Z

T

|g(q ₁ x)g(q ₂ x) · · · g(q _k x)| ² dm(x) ≥ kgk ^2k ₂

and the inequality is strict if and only if there exist two different sequences of integers (m ₁ , · · · , m _k ) and (m ⁰ ₁ , · · · , m ⁰ _k ) such that q ₁ m ₁ + · · · + q _k m _k = q ₁ m ⁰ ₁ + · · · + q _k m ⁰ _k while all Fourier coefficients b g(m j ), b g(m ⁰ _j ) are strictly positive. Indeed, by Plancherel’s formula, the inequality (9) is equivalent to

X

m

X

m

1

,··· ,m

k

: q

1

m

1

+···+q

k

m

k

=m

b g(m 1 ) · · · b g(m k )

! 2

≥ X

m

1

,··· ,m

_k

| b g(m 1 )| ² · · · | b g(m k )| ² . This is a direct consequence of the inequality ( P

J a _j ) ² ≥ P

J a ² _j , while the strict inequality comes from the fact that this last inequality is strict whenever a _j ’s are positive and J has more than one element.

Let us come back to the proof of the lemma and prove that kR _2k k _p /k e R _2k k _p tends to ∞.

If we take q = p/2 and

(10) f (x) = (1 + cos(x)) ^q (1 + cos(px)) ^q , g(x, y) = (1 + cos(x)) ^q (1 + cos(y)) ^q , then R _2k (x) ^p = h

f (px)f (p ³ x)·· · ··f (p ^2k−1 x) i 2

and we can use the previous fact to prove that kR _2k k ^p _p ≥ R

T f (x) ² dm(x)
k

. Moreover, kg R 2k k ^p _p =

 R

T×T g(x, y) ² dm(x)dm(y)

 k

. To prove that kR _2k k ^p _p /kg R _2k k ^p _p tends to ∞, it is sufficient to prove that the L ² norm of f is strictly larger than the norm of g, that is, to prove that, at least for one value of m, the Fourier coefficient of b f (m) is obtained through different writings of m as a sum of two frequencies that belong respectively to the two factors. But, for instance, q = q + 0 = −q + 2q, which

allows to conclude. 

The previous lemma allows us to find such examples for other values of p. Namely Lemma 8. Let q ≥ 4 be an even integer. Except possibly for a discrete set of values of p ∈ (1, ∞), there exists a sequence n j such that n j+1 /n j ≥ q for all j ≥ 1 and kR _k k _p /k f R _k k _p does not remain bounded below or above.

Proof. We consider the two quantities kP k ^p p and k e P k ^p p , where P and ˜ P are the trigono- metric polynomials of degree q + 1 and 2, respectively on T and T ² , defined by

P (x) = (1 + cos(x))(1 + cos(qx)), P (x, y) = (1 + cos(x))(1 + cos(y)). ˜

8

We have seen in the proof of the previous lemma that kP k ^p p and k e P k ^p p differ for p = q.

So they differ except on a discrete set of values (this is because kP k ^p p as a function of p is analytic). Let p be such an exponent and let us construct a sequence n _j that satisfies the conclusions of the lemma. We let n _2j = m _j and n _2j+1 = qm _j , where the sequence m _j increases sufficiently rapidly so that P  _(q+1)m

j

m

j+1

 2

< ∞. The L ^p (T ^2k ) norm of e R 2k is easily seen to be the k-th power of the norm of ˜ P . We use for P the analog of Proposition 3, but with the set Λ _N defined with (ε _j ) ^N _j=1 such that ε _j ∈ {0, ±1, · · · ± (q + 1)}. We deduce that the L ^p (T) norm of R 2k is up to a multiplicative constant comparable with the k-th power of the norm of P . The conclusion that kR _k k _p /k f R _k k _p does not remain bounded below or

above follows at once. 

The last lemma shows that in general Theorem 1 cannot be deduced from the indepen- dent case. We will see that it is nevertheless the case for p = 1, which is not contradictory since the L ¹ norms of R _k and e R _k are all equal to 1.

3. Lower bound for p = 1

Proof of Theorem 2. We assume n _j+1 /n _j ≥ 3. Then the Fourier expansion of a Riesz product Q k

j=1 (1+cos(n _j x)) has 3 ^k distinct terms. For a sequence ψ = (ψ ₁ , ψ ₂ , . . .), consider the Riesz product

P _ψ (x) =

∞

Y

j=1

(1 + cos(n _j x + ψ _j )) . Let

R f _k (ψ, x) = (P _ψ ∗ R _k )(x), where ∗ denotes the convolution on T. Then

(11)

Z

T

N

X

j=0

v _j R f _j (ψ, x)

dm(x) ≤ Z

T

N

X

j=0

v _j R _j (x)

dm(x).

On the other hand,

R f _k (ψ, x) =

k

Y

j=1

 1 + 1

2 cos(n _j x + ψ _j )

 , which can be verified by comparing Fourier coefficients,

c f

R k (ψ, ·)(n) = c P ψ (n) c R k (n) = (

2 ⁻

P

k j=1

|ε

j

|

e ⁱ

P

k

j=1

ε

j

ψ

j

2 ⁻

P

k j=1

|ε

j

|

if n = P k

j=1 ε j n j ,

0 if n / ∈ Γ _k

=





k

Y

j=1

 1 + 1

2 cos(n j x + ψ j )







∧

(n).

We integrate both sides of (11) against dm(ψ) and exchange integration. On the left hand side we have an i.i.d. sequence (with respect to ψ) 1 + ¹ ₂ cos(n j x + ψ j ) (observe also that

9

the distribution does not depend on x), which satisfies conditions of the main theorem of [7]. So we get the desired lower bound. Specifically, we use Theorem 3 from [7] with the i.i.d. sequence X _i = 1 + ¹ ₂ cos(2πU _i ) with U _i being i.i.d. uniform [0, 1] r.v.s for which we can take therein λ = ₁₀₀ ⁹⁹ , A = ³ ₂ , µ = _π ¹ , k = 2000, hence the bound c 1 > 3.1 · 10 ⁻⁸ (to obtain the bound on λ, we use √

1 + x ≤ 1 + x/2 − x ² /12, x ∈ [−1, 1]).  Such techniques involving Riesz products P _ψ have been already used in [1]. Unfortu- nately the same argument based on transferring the i.i.d. case from [2] does not seem to work for L ^p bounds with p > 1. Indeed, the lower bound involves the quantity

R

T 1 + ¹ ₂ cos(t)
p

dm(t)
k

, which is off by an exponential factor (in k).

4. Auxiliary general results

We give here elementary or standard results, which will be our tools in the main proofs.

The following simple result will lie in the heart of our induction procedure to obtain the bound below. It is basically [2, Lemma 9].

Lemma 9. Let µ be a measure on X and let f, g : X → E be measurable functions. Suppose that for some p > 1 and γ > 0, we have

Z

X

kgk ^p−1 kf kdµ ≤ γ Z

X

kf k ^p dµ.

Then,

Z

X

kf + gk ^p dµ ≥  1 3 ^p − 2pγ

 Z

X

kf k ^p dµ + Z

X

kgk ^p dµ.

Proof. For any real numbers a, b we have |a + b| ^p ≥ |a| ^p − p|a| ^p−1 |b|. If, additionally,

|a| ≤ ¹ ₃ |b|, then |a + b| ≥ |b| − |a| ≥ |a| + ¹ ₃ |b| and thus |a + b| ^p ≥ |a| ^p + ₃ ¹

p

|b| ^p . Taking a = kgk, b = −kf k and using the inequality kf + gk ≥ |kf k − kgk|, we obtain

Z

X

kf + gk ^p dµ = Z

X

kf + gk ^p 1 _{kgk≤

¹

3

kf k} dµ + Z

X

kf + gk ^p 1 _{kgk>

¹

3

kf k} dµ

≥ Z

X

kgk ^p 1 _{kgk≤

¹

3

kf k} dµ + 1 3 ^p

Z

X

kf k ^p 1 _{kgk≤

¹

3

kf k} dµ +

Z

X

kgk ^p 1 _{kgk>

¹

3

kf k} dµ − p Z

X

kgk ^p−1 kf k 1 _{kgk>

¹

3

kf k} dµ

= Z

X

kgk ^p dµ + 1 3 ^p

Z

X

kf k ^p (1 − 1 _{kgk>

¹

3

kf k} )dµ − p Z

X

kgk ^p−1 kf k 1 _{kgk>

¹

3

kf k} dµ.

Note that Z

X

 1

3 ^p kf k ^p + pkgk ^p−1 kf k



1 _{kgk>

¹

3

kf k} dµ ≤  1 3 + p

 Z

X

kgk ^p−1 kf kdµ ≤ 2pγ Z

X

kf k ^p dµ.

Therefore, Z

X

kf + gk ^p dµ ≥ Z

X

kgk ^p dµ + 1 3 ^p

Z

X

kf k ^p dµ − 2pγ Z

X

kf k ^p dµ.

10

 The next lemma gives a comparison between explicit constants that we will need.

Lemma 10. For k, p ≥ 1, (12)

Z

T

| cos(t)| ^2p | sin(t)| ^2kp dm ≤ 1 kp + 1

Z

T

| cos(t)| ^2p dm Z

T

| sin(t)| ^2kp dm.

Proof. We have Z

T

| cos(t)| ^α | sin(t)| ^β dm = 2 π

Z π/2 0

cos ^α (t) sin ^β (t)dt = 1

π B  α + 1 2 , β + 1

2



= Γ( ^α+1 ₂ )Γ( ^β+1 ₂ ) πΓ( ^α+β ₂ + 1) , so the ratio between the left and the right hand sides of (12) is equal to

Γ(p + 1)Γ(kp + 1)

Γ(kp + p + 1) = pΓ(p)Γ(kp + 1) Γ(kp + p + 1) = p

Z 1 0

x ^p−1 (1 − x) ^kp dx

≤ p Z 1

0

x ^p−1 dx Z 1

0

(1 − x) ^kp dx = 1 kp + 1 ,

where we have used the continuous version of Chebyshev’s sum inequality.  Our next lemma concerns exact algebraic factorization for integrals of products of trigonometric polynomials and is also standard. (As a side clarifying remark, since func- tions on T may be treated as 2π-periodic functions on R, in the next 3 lemmas, when we say “a function on T”, we implicitly mean, “a T-periodic function”)

Lemma 11. Suppose that g ₁ , . . . , g _{N −1} are trigonometric polynomials of degree at most d, g N is an arbitrary continuous function on T and n j+1 /n j ≥ d + 1 for j ≥ 1. Then

Z

T N

Y

j=1

g j (n j t)dm =

N

Y

j=1

Z

T

g j (n j t)dm.

Proof. Indeed the left hand side is the sum of products of Fourier coefficients g b _j (l _j ), with P N

j=1 l _j n _j =0, |l _j | ≤ d for j ≤ N − 1. This only occurs when all l _j are zero, which allows to

conclude. 

Even if exact factorization does not hold, one can establish approximate factorization in the presence of a highly oscillating factor. This idea is quantified in the following lemma.

Lemma 12. Suppose that f is a Lipschitz function on T and g is an integrable function on T. Then for any integer n ≥ 1, we have

  

Z

T

f (t)g(nt)dm − Z

T

f dm Z

T

g(nt)dm    ≤ 2π

n Z

T

|f ⁰ (t)|dm Z

T

|g(nt)|dm.

11

Proof. Let I _k = [ _n ^k 2π, ^k+1 _n 2π] for k = 0, 1, . . . , n − 1. Observe that for any k, R

T g(nt)dm =

1

|I

_k

|

R

I

_k

g(nt)dt, hence 

  Z

I

k

f (t) 

g(nt) − Z

T

g(ns)dm(s)  dt

   = 1

|I _k |   

Z

I

k

×I

_k

(f (t) − f (s))g(nt)dtds   

≤ sup

t,s∈I

k

|f (t) − f (s)|

Z

I

k

|g(nt)|dt ≤ Z

I

k

|f ⁰ (u)|du Z

I

k

|g(nt)|dt

= 2π n

Z

I

_k

|f ⁰ (u)|du Z

T

|g(nt)|dm.

Summing the above estimate over 0 ≤ k ≤ n − 1 yields the lemma.  In the context of trigonometric polynomials, in the above lemma we can pass from the bound in terms of f ⁰ to the bound in terms of the original factor f . Namely, we have the following lemma. Its first part is the classical Bernstein inequality for vector valued trigonometric polynomials.

Lemma 13. Suppose that f is a vector-valued trigonometric polynomial of order at most d. Then

(13)

Z

T

kf ⁰ k ^p dm ≤ d ^p Z

T

kf k ^p dm.

Moreover, for any integrable (complex valued) function h on T, we have (14)

    Z

T

kf (t)k ^p h(nt)dm − Z

T

kf k ^p dm Z

T

h(nt)dm    

≤ 2π pd n

Z

T

kf k ^p dm Z

T

|h(nt)|dm.

Proof. Formula (3.11) in [12, Chapter X] gives f ⁰ (t) = P 2d

k=1 b _k f (t + t _k ), where P 2d

k=1 |b _k | = d and t _k = ¹ _d (k − ¹ ₂ )π. Thus kf ⁰ (t)k ≤ P _2d

k=1 |b _k |kf (t + t _k )k, so the triangle inequality for the L p norm gives, kf ⁰ k _p ≤ P 2d

k=1 |b _k |kf k _p = dkf k p and (13) follows.

To show (14), take g = kf k ^p . Then |g ⁰ | ≤ pkf k ^p−1 kf ⁰ k (g is in fact almost everywhere differentiable) and

Z

T

|g ⁰ |dm ≤ p

Z

T

kf k ^p dm

 (p−1)/p Z

T

kf ⁰ k ^p dm

 1/p

≤ pd Z

T

kf k ^p dm,

by H¨ older’s inequality and estimate (13). Thus Lemma 12 yields (14).  Lemma 14. Let f 1 and f 2 be vector-valued trigonometric polynomials of degree at most d.

Then for n ≥ 3d, we have Z

T

kf ₁ + f ₂ cos(nt)k ^p dm ≥ 1 3 ^p

Z

T

kf ₂ k ^p dm.

Proof. This is an easy consequence of the use of de la Vall´ ee Poussin kernel V _d (see, e.g.

[3, 2.13, p. 16]). V _d−1 is a trigonometric polynomial of degree 2d − 1 with Fourier co- efficients between −d and d equal to 1. The L ¹ norm of V _d−1 is bounded by 3/2. If

12

g(t) = 2e ^int V _d−1 (t), then e ^int f ₂ coincides with the convolution of f ₁ + f ₂ cos(nt) with g (this is where we need n ≥ 3d). The result follows from

Z

T

kf ₂ k ^p dm = Z

T

k(f ₁ + f ₂ cos(nt)) ∗ gk ^p dm ≤ k2V _d−1 k ^p _L

1

(T)

Z

T

kf ₁ + f ₂ cos(nt)k ^p dm, where the last estimate is justified by Young’s inequality. 

5. Lower bound for p > 1

This section is devoted to the proof of the left hand side inequality in Theorem 1. Remark first that Lemma 14 applied with f 1 = P N −1

k=0 v _k R _k + v N R N −1 and f 2 = v N R N −1 and a simple inequality kR _{N −1} cos(n _N t)k _p ≤ kR _{N −1} k _p yield

kv _N kkR _N k _p = kv _N kkR _{N −1} + R _{N −1} cos(n _N t)k _p ≤ 2kv _N kkR _{N −1} (t)k _p ≤ 6

N

X

k=0

v _k R _k L

^p

(T,E)

under the condition that n k+1 ≥ 4n _k . But we are far from having the possibility of an induction from this. Our first step will concern this inequality, but for a family of weighted measures on the torus.

Let ϕ _k (t) = ( ^{1−cos t} ₂ ) ^k . For k, l ≥ 1, we say that a function g on T belongs to family of weights F _k,l ^p if it has the form

g(t) :=

l

Y

j=1

h _j (n _j t), where h _j ∈

 1, 1

2 ϕ ^p _k , 1 − 1 2 ϕ ^p _k



for j = 1, . . . , l.

We also set F _k,0 ^p := {1}. With a slight abuse of notation we will say that a measure µ on T belongs to F _k,l ^p if it has the form dµ = gdm for some g ∈ F _k,l ^p .

We will approximate these weights by trigonometric polynomials. We start with the next lemma, which is a rather standard application of Bernstein polynomials. We prove it for completeness.

Lemma 15. Let p > 1 and f _p (t) = (1 − ¹ ₂ t ^p ) ^1/p , t ∈ [0, 1]. For any ε > 0, there exists a polynomial w ε,p of degree at most d4ε ⁻² e such that

f p (t) ≤ w ε,p (t) ≤ (1 + ε)f p (t) for t ∈ [0, 1].

Proof. We have |f _p ⁰ (t)| = ¹ ₂ t ^p−1 (1 − ¹ ₂ t ^p ) ^1/p−1 ≤ 2 ^−1/p ≤ 1, so f _p is 1-Lipschitz. Let S _n,t have the binomial distribution with parameters n and t and define ˜ w _n,p (t) := Ef p ( _n ¹ S _n,t ).

Then ˜ w _n,p is a polynomial of degree at most n and

| ˜ w _n,p (t) − f _p (t)| ≤ E    

f _p  1 n S _n,t



− f _p (t)     ≤ E

    1

n S _n,t − t    

≤ 1

n E|S n,t − nt| ²
1/2

= 1 n

p nt(1 − t) ≤ 1 2 √

n .

13

Define w _ε,p = ˜ w _n,p + ₂ ^{√ 1} _n , where n = d4ε ⁻² e. Observe that f p (t) ≤ w ε,p (t) ≤ f p (t) + 1

√ n ≤ f _p (t) + ε

2 ≤ (1 + ε)f _p (t).

 Let us now approximate the weights by trigonometric polynomials.

Lemma 16. Suppose that n _j+1 /n _j ≥ 8 for all j ≥ 1 and let k ≥ 1, l ≥ 0. Then for any g ∈ F _k,l ^p , there exists a trigonometric polynomial h of degree at most C 1 (p)n _l k such that g ≤ h ^p ≤ 2g.

Proof. There exist disjoint I ₁ , I ₂ ⊂ {1, . . . , l} such that g := 2 ^−|I

¹

^| Y

j∈I

1

ϕ ^p _k (n j t) Y

j∈I

2

 1 − 1

2 ϕ ^p _k (n j t)

 .

Let ε j := ^{ln 2} _p 2 ^j−l−1 for j ∈ I 2 and h := 2 ⁻

^|I1|p

Y

j∈I

1

ϕ k (n j t) Y

j∈I

2

w ε

j

,p (ϕ k (n j t)),

where w _ε

_j

_,p are polynomials given by Lemma 15. Then h is a trigonometric polynomial of degree at most

deg(h) ≤ X

j∈I

1

n j k + X

j∈I

2

d4ε ⁻² _j en _j k ≤ 8p ² ln ² 2

l

X

j=1

4 ^l+1−j n j k ≤ 64p ² ln ² 2 n _l k.

Moreover,

g ≤ h ^p ≤ g Y

j∈I

2

(1 + ε _j ) ^p ≤ e ^{p P}

^j∈I2

^ε

^j

g ≤ e ^{ln 2 P}

^lj=1

²

^j−l−1

g ≤ 2g.

 The following lemma will comprise a first step in our main inductive argument.

Lemma 17. Suppose that k ≥ 1, l ≥ 0 and n _j+1 /n _j ≥ C ₃ (p)k for j ≥ 1. Then for any µ ∈ F _k,l ^p and any vectors v 0 , . . . , v _l+1 in a normed space (E, k · k), we have

Z

T

l+1

X

j=0

v _j R _j

p

dµ ≥ c ₃ (p)kv _l+1 k ^p Z

T

R ^p _l+1 dµ.

Proof. We may assume that C 3 (p) ≥ 8. Let g = _dm ^dµ and h be a trigonometric polynomial given by Lemma 16. We have

Z

T

l+1

X

j=0

v _j R _j

p

dµ ≥ 1 2

Z

T

l+1

X

j=0

v _j R _j

p

h ^p dm.

14

Observe that

l+1

X

j=0

v j R j h = f h + v l+1 cos(n l+1 t)R l h, where f is a vector-valued trigonometric polynomial. Moreover,

max{deg(R l h), deg(f h)} ≤ deg(h) +

l

X

j=1

n j ≤ (C ₁ (p) + 2)n l k

and the assertion easily follows by Lemma 14. 

Lemma 18. For any p > 1, there exists a real polynomial w _p such that x ^p−1 ≤ w _p ^p (x) for x ∈ [0, 2] and

λ ₁ (p) :=

R

T w ^p p (X ₁ )dm R

T X ₁ ^p dm
(p−1)/p < 1.

Proof. Let I p = R

T (1 + cos t) ^p dm(t)
1/p

. By Jensen’s inequality, I p ≥ I _p−1 , but 1 + cos t is non-constant, so this inequality is in fact strict. Let δ > 0 be such that I _p = (1 + δ)I _p−1 . Note that δ depends only on p. Now choose ε > 0 (depending on δ) such that

  R

T X ₁ ^p−1 dm  1/p

+ ε

 p

R

T X ₁ ^p dm
(p−1)/p =



I _p−1 ^(p−1)/p + ε  p

I p ^p−1

< 1.

By the Weierstrass approximation theorem, there exists a polynomial w _p such that x ^(p−1)/p ≤ w _p (x) ≤ x ^(p−1)/p + ε for x ∈ [0, 2]. To finish, it is enough to observe that

Z

T

w _p ^p (X 1 )dm

 1/p

≤

Z

T

X ₁ ^p−1 dm

 1/p

+ ε

and then λ 1 (p) < 1 by the choice of ε. 

Remark 19. We emphasize that it is clear from the proof that the polynomial w _p depends only on p (in particular it does not depend on n 1 which defines X 1 ).

We are now in position to give the main ingredients for the induction procedure.

Lemma 20. For p > 1, there exist constants C 5 (p), C 6 (p), C 7 (p) and λ 2 (p) < 1 with the following property. If n _j+1 /n _j ≥ C ₅ (p)k for j ≥ 1, k ≥ 1, l ≥ 0, then for any µ ∈ F _k,l ^p , any N ≥ l + 1 and any vector valued polynomial f of order at most 2n l , we have

(15)

Z

T

kf k ^p ϕ ^p _k (n _l+1 t)dµ ≥ 1 4

Z

T

kf k ^p dµ Z

T

ϕ ^p _k dm

15

and Z

T

kf kR ^p−1 _N ϕ ^p _k (n _l+1 t)dµ

≤ C 6 (p)

k ^(p−1)/p λ 2 (p) ^{N −l−1}

Z

T

kf k ^p dµ

 1/p Z

T

ϕ ^p _k dm

 Z

T

R _N ^p dµ

 (p−1)/p

. (16)

Moreover for any v _l+1 , . . . , v _N we have

Z

T

kf k

N

X

j=l+1

v _j R _j

p−1

ϕ ^p _k (n _l+1 t)dµ

≤ C 7 (p) k ^(p−1)/p

Z

T

kf k ^p dµ

 1/p Z

T

ϕ ^p _k dm







N

X

j=l+1

λ 2 (p) ^j−l−1 kv _j k ^p Z

T

R ^p _j dµ





(p−1)/p

. (17)

Proof. Let g = _dm ^dµ and h be a trigonometric polynomial given by Lemma 16. Notice that hf is a vector-valued trigonometric polynomial with degree at most (C 1 (p) + 2)n l k. Thus by (14) we have for sufficiently large C 5 (p),

Z

T

kf k ^p ϕ ^p _k (n l+1 t)dµ ≥ 1 2

Z

T

kf hk ^p ϕ ^p _k (n l+1 t)dm ≥ 1 4

Z

T

kf hk ^p dm Z

T

ϕ ^p _k dm

≥ 1 4

Z

T

kf k ^p dµ Z

T

ϕ ^p _k dm.

To establish (16), let us define d˜ µ = h ^p (t)ϕ ^p _k (n l+1 t)dm. By H¨ older’s inequality, we have Z

T

kf kR ^p−1 _N ϕ ^p _k (n _l+1 t)dµ ≤ Z

T

kf kR ^p−1 _N d˜ µ

≤

Z

T

kf k ^p R ^p−1 _l+2,N d˜ µ

 1/p Z

T

R ^p _l+1 R ^p−1 _l+2,N d˜ µ

 (p−1)/p

.

We have used the notation, for 1 ≤ l ≤ N,

(18) R _l,N =

N

Y

j=l

X j .

16

Let w _p be given by Lemma 18 and ε = ε _p be a small positive number to be chosen later.

By (14), if C 5 (p) is sufficiently large, we have Z

T

kf k ^p R ^p−1 _l+2,N d˜ µ ≤ Z

T

kf k ^p

N

Y

j=l+2

w _p ^p (X j )d˜ µ

≤ (1 + ε) Z

T

kf k ^p

N −1

Y

j=l+2

w ^p _p (X j )d˜ µ Z

T

w _p ^p (X N )dm ≤ . . .

≤ (1 + ε) ^{N −l−1} Z

T

kf k ^p d˜ µ

N

Y

j=l+2

Z

T

w _p ^p (X _j )dm

≤ (1 + ε) ^{N −l} Z

T

kf hk ^p dm Z

T

ϕ ^p _k dm

N

Y

j=l+2

Z

T

w _p ^p (X j )dm

≤ 2(1 + ε) ^{N −l} λ ₁ (p) ^{N −l−1} Z

T

kf k ^p dµ Z

T

ϕ ^p _k dm

N

Y

j=l+2

Z

T

X _j ^p dm

 (p−1)/p

.

In the same way we show that Z

T

R _l+1 ^p R ^p−1 _l+2,N d˜ µ

≤ 2(1 + ε) ^{N −l} λ 1 (p) ^{N −l−1} Z

T

R ^p _l dµ Z

T

X _l+1 ^p ϕ ^p _k (n l+1 t)dm

N

Y

j=l+2

Z

T

X _j ^p dm

 (p−1)/p

.

The above estimates together with Lemma 10 yield Z

T

kf kR ^p−1 _N ϕ ^p _k (n _l+1 t)dµ ≤ 2

 1

kp + 1

 (p−1)/p

(1 + ε) ^{N −l} λ ₁ (p) ^{N −l−1}

Z

T

kf k ^p dµ

 1/p

×

Z

T

ϕ ^p _k dm





 Z

T

R ^p _l dµ

N

Y

j=l+1

Z

T

X _j ^p dm





(p−1)/p

.

Estimate (14) implies however that for sufficiently large C 5 (p), Z

T

R ^p _N dµ ≥ 1 2 (1−ε)

Z

T

R ^p _{N −1} h ^p dm Z

T

X _N ^p dm ≥ . . . ≥ 1

2 (1−ε) ^{N −l} Z

T

R ^p _l gdm

N

Y

j=l+1

Z

T

X _j ^p dm.

To derive (16) we choose ε = ε _p in such a way that

λ 2 (p) := (1 + ε)(1 − ε) ^(1−p)/p λ 1 (p) < 1.

17

To show (17) we consider two cases. First assume that 1 < p ≤ 2. By (16), we have

Z

T

kf k

N

X

j=l+1

v _j R _j

p−1

ϕ ^p _k (n _l+1 t)dµ

≤ Z

T

kf k

N

X

j=l+1

kv _j R _j k ^p−1 ϕ ^p _k (n _l+1 t)dµ

≤ C 6 (p) k ^(p−1)/p

Z

T

kf k ^p dµ

 1/p Z

ϕ ^p _k dm

 N

X

j=l+1

λ 2 (p) ^j−l−1 kv _j k ^p−1

Z

T

R ^p _j dµ

 (p−1)/p

.

However

N

X

j=l+1

λ ₂ (p) ^j−l−1 kv _j k ^p−1

Z

T

R ^p _j dµ

 (p−1)/p

≤





N

X

j=l+1

λ 2 (p) ^j−l−1





1/p 



N

X

j=l+1

λ 2 (p) ^j−l−1 kv _j k ^p Z

T

R ^p _j dµ





(p−1)/p

≤ (1 − λ ₂ (p)) ^−1/p





N

X

j=l+1

λ ₂ (p) ^j−l−1 kv _j k ^p Z

T

R ^p _j dµ





(p−1)/p

,

which concludes for this case.

Finally, if p > 2, we have by the triangle inequality in L ^p−1 and (16)

Z

T

kf k

N

X

j=l+1

v j R j

p−1

ϕ ^p _k (n _l+1 t)dµ

≤





N

X

j=l+1

kv _j k

Z

T

kf kR ^p−1 _j ϕ ^p _k (n _l+1 t)dµ

 1/(p−1) 



p−1

≤ C ₆ (p) k ^(p−1)/p

Z

T

kf k ^p dµ

 1/p Z

T

ϕ ^p _k dm







N

X

j=l+1

kv _j kλ ₂ (p) (j−l−1)/(p−1)

Z

T

R ^p _j dµ

 1/p 



p−1

.

18

To finish the proof of (17) in this case it is enough to observe that by H¨ older’s inequality

N

X

j=l+1

kv _j kλ ₂ (p) (j−l−1)/(p−1)

Z

T

R ^p _j dµ

 1/p

≤





N

X

j=l+1

λ 2 (p) (j−l−1)/(p−1)

²





(p−1)/p 



N

X

j=l+1

λ 2 (p) ^j−l−1 kv _j k ^p Z

T

R ^p _j dµ





1/p

≤ 

1 − λ ₂ (p) ^1/(p−1)

²

 (1−p)/p





N

X

j=l+1

λ ₂ (p) ^j−l−1 kv _j k ^p Z

T

R ^p _j dµ





1/p

.



Proposition 21. If k ≥ 2, N ≥ l ≥ 0, n j+1 /n j ≥ max{C ₃ (p), C 5 (p), 8}k for j ≥ 1, then for any µ ∈ F _k,l ^p and any vectors v ₀ , v ₁ , . . . , v _N in a normed space (E, k · k) we have

Z

T

N

X

j=0

v j R j

p

dµ ≥ α p

Z

T

l

X

j=0

v j R j

p

dµ +

N

X

j=l+1

(β p − c _p,j−l )kv j k ^p Z

T

R ^p _j dµ,

where

α _p = 1 16 · 3 ^p

Z

ϕ ^p _k dm, β _p = c ₃ (p)

2 α _p , γ _p = (16p3 ^p C ₇ (p))

^p−1p

α p

k and c _p,j = γ _p

j−1

X

i=0

λ ₂ (p) ⁱ .

Proof. We proceed by induction on N − l. If N − l = 0 the assertion is obvious, since α _p ≤ 1. To show the induction step we may assume that l is fixed and we increased N . We consider two cases.

Case 1. α _p R

T

P l

j=0 v _j R _j

p

dµ ≤ γ _p P N +1

j=l+1 λ ₂ (p) ^j−l−1 kv _j k ^p R

T R ^p _j dµ.

19