L’INSTITUTFOURIER DE ANNALES

A L

E S

E D L ’IN IT ST T U

F O U R

ANNALES

DE

L’INSTITUT FOURIER

Loïc GAILLARD & Pascal LEFÈVRE

Lacunary Müntz spaces: isomorphisms and Carleson embeddings Tome 68, n^o5 (2018), p. 2215-2251.

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LACUNARY MÜNTZ SPACES:

ISOMORPHISMS AND CARLESON EMBEDDINGS

by Loïc GAILLARD & Pascal LEFÈVRE

Abstract. — In this paper we prove thatM_Λ^p is almost isometric to `^p in the canonical way when Λ is lacunary with a large ratio. On the other hand, our approach can be used to study also the Carleson measures for Müntz spacesM_Λ^p when Λ is lacunary. We give some necessary and some sufficient conditions ensuring that a Carleson embedding is bounded or compact. In the hilbertian case, the membership to Schatten classes is also studied. When Λ behaves like a geometric sequence the results are sharp, and we get some characterizations.

Résumé. — Dans cet article, nous montrons queM_Λ^pest presque isométrique à `^p, et ce de façon naturelle, lorsque Λ est lacunaire avec une raison grande.

Par ailleurs, notre approche permet aussi d’étudier les mesures de Carleson pour les espaces MüntzM_Λ^p lorsque Λ est lacunaire. Nous donnons des conditions né- cessaires et des conditions suffisantes qui permettent d’assurer qu’un plongement de Carleson est borné ou compact. Dans le cadre hilbertien, nous étudions aussi l’appartenance de ce plongement aux classes de Schatten. Nous obtenons des ca- ractérisations complètes lorsque Λ se comporte comme une suite géométrique.

1. Introduction

Let m be the Lebesgue measure on [0,1]. For p ∈ [1,+∞), L^p(m) = L^p([0,1], m) (sometimes denoted simply by L^p when there is no ambigu- ity) denotes the space of complex-valued measurable functions on [0,1], equipped with the norm kfkp = (R1

0 |f(t)|^pdt)^p1. In the same way, C = C([0,1]) is the space of continuous functions on [0,1] equipped with the usual sup-norm. We shall also consider some positive and finite measures µ on [0,1) (see the remark at the beginning of Section 2), and the as- sociated L^p(µ) space. For a sequence w = (w_n)_n of positive weights, we denote`^p(w) the Banach space of complex sequences (bn)n equipped with

Keywords:Müntz spaces, Carleson embeddings, lacunary sequences, Schatten classes.

2010Mathematics Subject Classification:30B10, 47B10, 47B38.

the norm kbk_`p(w) = (P

n|bn|^pwn)^1p and the vector space c00 consisting on complex sequences with a finite number of non-zero terms. All along the paper, when p∈(1,+∞), we denote as usualp⁰ = _p−1^p its conjugate exponent.

The famous Müntz theorem ([3, p. 172],[7, p. 77]) states that if Λ = (λn)_n∈N is an increasing sequence of non-negative real numbers, then the linear span of the monomials t^λn is dense in L^p (resp. in C) if and only ifP

n>1 1

λ_n = +∞ (resp. and λ0 = 0). We shall assume that the Müntz conditionP

n>1 1

λn <+∞ is fulfilled and we define the Müntz spaceM_Λ^p as the closed linear space spanned by the monomials t^λn, where n ∈ N. We shall moreover assume that Λ satisfies the gap condition: infn λn+1− λ_n

>0. Under this later assumption Clarkson–Erdös theorem holds [7, Thm. 6.2.3]: the functions inM_Λ^p are the functionsf inL^psuch thatf(t) = Pa_nt^λn (pointwise on [0,1)). This gives a class of Banach spacesM_Λ^p (L^p of analytic functions on (0,1).

In full generality, the Müntz spaces are difficult to study, but for some particular sequences Λ, we can find some interesting properties of the spaces M_Λ^p. Let us mention that lately these spaces received an increasing atten- tion from the point of view of their geometry and operators: the monograph of Gurariy–Lusky [7], and various more or less recent papers (see for in- stance [1, 2, 4, 9, 10]).

We shall focus on two different questions on Müntz spaces. The first one is related to an old result: Gurariy and Macaev proved in [8] that, inL^p, the normalized sequence ((pλn+ 1)^1pt^λn)n is equivalent to the canonical basis of`<sup>p</sup> if and only if Λ is lacunary (see Theorem 2.3 below). More recently, the monograph [7] introduces the notion of quasi-lacunary sequence (see Definition 2.1 below), and states that M<sub>Λ</sub><sup>p</sup> is still isomorphic to `^p when Λ is quasi-lacunary. On the other hand, some recent papers discuss about Carleson measures for Müntz spaces. In [4], the authors introduced the class of sublinear measures on [0,1), and proved that when Λ is quasi- lacunary, the sublinear measures are Carleson embeddings forM_Λ¹. In [10], the authors extended this result to the case p = 2 but only when the sequence Λ is lacunary.

In this paper, we introduce another method to study the lacunary Müntz spaces: for a weightwand a measureµon [0,1), we define T_Λ,µ^w :`^p(w)→ L^p(µ) by T_Λ,µ^w (b) = P

nbnt^λn for b = (bn) ∈ `^p(w). The operator T_Λ,µ^w depends on w, µ, p and Λ, and when it is bounded we shall denote its norm by

T_Λ,µ^w

_p. We shall see that an estimate of T_Λ,µ^w

_p can improve

Gurariy–Macaev theorem, and allows to generalize former Carleson embed- ding results to lacunary Müntz spacesM_Λ^p for anyp>1.

The paper is organized as follows: in Section 2, we specify the miss- ing notation and some useful lemmas. The main result gives an upper bound for the approximation numbers of T_Λ,µ^w (see Proposition 2.9). In Section 3, we focus on the classical case: we fix the weightw(p) defined by w_n(p) = (pλ_n+ 1)⁻¹ and consider T_Λ^w(p) =T_Λ,m^w(p) : `^p(w(p))→ M_Λ^p, the isomorphism occuring in Gurariy–Macaev theorem. For p > 1, we prove that T_Λ^w(p) is bounded exactly when Λ is quasi-lacunary. On the other hand, when Λ is lacunary with a large ratio, we also get a sharp bound for

T_Λ^w(p)−1

p (see Theorem 3.5 below). Our approach leads to an asymp- totically orthogonal version of Gurariy–Macaev theorem exactly for the super-lacunary sequences. In Section 4, we apply the results of Section 2 for a positive and finite measureµ on [0,1) with the weightwn =λ⁻¹_n , in order to treat the Carleson embedding problem. When Λ is lacunary, we give an estimate of the approximation numbers of the embedding operator i^p_µ:M_Λ^p →L^p(µ). In Section 5, we focus on the compactness ofi^p_µusing the same tools as in Section 4. In the casep= 2, this leads to some control of the Schatten norm of the Carleson embedding and some characterizations when Λ behaves like a geometric sequence.

As usual the notation A .B means that there exists a constant c >0 such thatA6cB. This constant c may depend on Λ (or sometimes only on its ratio of lacunarity), on p . . . . We shall specify this dependence to avoid any ambiguous statement. In the same way, we shall use the notation A≈B orA&B.

Acknowledgments. We wish to thank the referee for the very careful reading of the paper and many suggestions in order to improve its quality.

2. Preliminary results

Let us first give a few words of explanation about our choice of mea- sures on [0,1). The measures involved (if considered on [0,1]) must satisfy µ({1}) = 0. Indeed, we focus either on the Lebesgue measurem(satisfying of course m({1}) = 0) or on measures such that the Carleson embedding M_Λ^p→L^p(µ) is (defined and) bounded, so that testing a sequence of mono- mialsgn(t) =t^λn we must have

µ({1}) = limkgnk^p_Lp(µ).limkgnk^p_Lp(m)= 0.

Moreover, by Clarkson–Erdös theorem, the value at any point of [0,1) of any function inM_Λ^p can be defined without ambiguity.

We shall need several notions of growth for increasing sequences.

Definition 2.1.

• A sequenceu= (un)nof positive numbers is calledlacunaryif there existsr > 1 such thatun+1 >run, for everyn∈N. We shall say that such a sequence isr-lacunary and thatris a ratio of lacunarity of this sequence.

• A sequenceuis calledquasi-lacunaryif there is an extraction(n_k)_k such thatsup_k∈N(nk+1−nk)<+∞,and(un_k)k is lacunary.

• A sequenceuis called quasi-geometricif there are two constants r andR such that1< r6 ^uuⁿ⁺¹n 6R <+∞, for everyn∈N. Such a sequence is lacunary.

• A sequenceuis called super-lacunary if ^u_uⁿ⁺¹

n −→+∞.

Remark 2.2. — It is proved in [7, Prop. 7.1.3, p. 94] that a sequence is quasi-lacunary if and only if it is a finite union of lacunary sequences.

The following result is due to Gurariy and Macaev.

Theorem 2.3([7, Cor. 9.3.4, p. 132]). — Letp∈[1,+∞). The following assertions are equivalent

(1) The sequenceΛ is lacunary.

(2) The sequence

t^λn kt^λnk_p

in L^p is equivalent to the canonical basis of`^p.

In particular, whenΛ is lacunary, we have for anyb∈c00

Xbnt^λn _p≈

X |bn|^p pλn+ 1

¹_p

and the underlying constants depend onpandΛonly.

We shall recover and generalize partially this result: for a given sequence of weights (wn)n and a positive finite measure µ on [0,1), we study the boundedness of the operator

T_Λ,µ^w :

`^p(w) −→ L^p(µ)

b 7−→ P

bnt^λn.

Example 2.4. — In the case of the Lebesgue measureµ=mand when the weights are wn(p) = (pλn+ 1)⁻¹ or in a simpler way (if we do not care about the value of the constants) w_n =λ⁻¹_n , Theorem 2.3 states in particular thatT_Λ^w and T_Λ^w(p) are bounded on`<sup>p</sup>(w) or `^p(w(p)), when Λ is lacunary.

Remark 2.5. — In the casep >1, a (rough) sufficient condition ensuring the boundedness ofT_Λ,µ^w is

Z

[0,1)

X

n

w⁻

p0

n pt^p0λn

!_p^p0

dµ <∞.

Indeed, this comes from the estimate T_Λ,µ^w

_p= sup

a∈B_`p a∈c00

sup

g∈BLp0 (µ)

Z

[0,1)

X

n

anw⁻

1

npt^λng(t) dµ

6 sup

g∈B_Lp0(µ)

Z

[0,1)

|g(t)| sup

a∈B`p

a∈c₀₀

X

n

anw⁻

1 p

n t^λn

dµ.

For standard weights, wn ≈λ⁻¹_n and for a quasi-geometric sequence Λ, this condition can be reformulated with the help of Lemma 2.10 below as

Z

[0,1)

1

1−t dµ≈ Z

[0,1)

1

1−t^p0 dµ <∞.

Such a condition will be considered later (see Proposition 5.5 below for instance).

To get a sharper estimate, we introduce the sequence D^w,p_Λ,µ(n)

ndefined forn∈Nandp>1, with a priori values inR+∪ {+∞} by

D^w,p_Λ,µ(n) = Z

[0,1)

w⁻

1 p

n t^λn X

k>0

w⁻

1 p

k t^λk

!p−1

dµ

!¹_p .

Proposition 2.6. — Let p ∈ [1,+∞). Assume that D^w,p_Λ,µ(n)

n is a bounded sequence of real numbers. Then we have for everyb∈`^p(w),

X

n>0

b_nt^λn _Lp(µ)6



 X

n>0

|b_n|^pw_n D_Λ,µ^w,p(n)^p





1 p

.

Proof. — Ifp= 1 the result is obvious. Assume now thatp >1. For any t∈[0,1) andn∈N, we have

b_nt^λn=b_nw

1 pp0

n t^λnp ×w⁻

1 pp0

n t^λnp0 , By Hölder’s inequality, we get

Xbnt^λn

6 X

n

|bn|^pw

1 p0

n t^λn

!_p¹ X

k

w⁻

1 p

k t^λk

!_p¹0

.

We conclude Z

[0,1)

Xbnt^λn

p

dµ6 Z

[0,1)

X|bn|^pwn.w⁻

1 p

n t^λn X

k

w⁻

1 p

k t^λk

!p−1

dµ

=X

n

|b_n|^pw_nD^w,p_Λ,µ(n)^p.

Assume that D_Λ,µ^w,p(n)

nis a bounded sequence of real numbers. We may define the bounded diagonal operator

D^w,p_Λ,µ:`<sup>p</sup>(w)→`^p(w)

acting on the canonical basis of`^p(w) whose diagonal entries are the num- bersD^w,p_Λ,µ(n). In other wordsT_Λ,µ^w andD_Λ,µ^w,pare bounded, and we have

∀b∈`<sup>p</sup>(w), kT<sub>Λ,µ</sub><sup>w</sup> (b)k<sub>L</sub>p(µ)6kD<sub>Λ,µ</sub><sup>w,p</sup>(b)k<sub>`</sub>p(w).

This gives informations about the approximation numbers ofT_Λ,µ^w . Let us recall some definitions.

Definition 2.7. — For a bounded operator S : X → Y between two separable Banach spacesX, Y, the approximation numbers (an(S))n of S are defined forn>1by

a_n(S) = inf{kS−Rk,rank(R)< n}.

The essential normofS is defined by

kSke= inf{kS−Kk, K compact}.

It is the distance fromS to the compact operators.

We shall use in the sequel the following notions of operator ideals.

Definition 2.8.

• An operatorS:X →Y isnuclearif there is a sequence of rank-one operators(Rn)satisfyingS(x) =P

nRn(x)for every x∈ X with P

nkRnk<+∞.Thenuclear normofS is defined as kSk_N = inf

( X

n

kRnk,rank(Rn) = 1,X

n

Rn=S )

.

• An operator S : X → L^p(µ) is order bounded if there exists a positive function h ∈ L^p(µ) such that for every x ∈ B_X and for µ−almost everyt∈Ωwe have |S(x)(t)|6h(t).

• Forr >0 and whenX, Y are Hilbert spaces, we say that a (com- pact) operatorS:X →Y belongs to theSchatten classS^r if

X

n

a_n(S)^r

<+∞.

In this case, we define itsSchatten normbykSk_S^r= P

n an(S)r_r¹ . Recall that nuclear and Schatten class operators are always compact.

Of course, the Schatten norm is really a norm when r>1. TheS² class is also called the class ofHilbert–Schmidtoperators.

We shall be interested in how far from compact (the essential norm) or, on the contrary, how strongly compact (possibly Schatten in the Hilbert framework) are the Carleson embeddings.

For technical reasons, we introduce the following notation: for a bounded sequence (u_n)_n in R+, we define (u^∗_N)_N thedecreasing rearrangement of (un)n by

u^∗_N = inf

A⊂N

|A|=N

sup{un, n6∈A}.

We have lim_N→+∞u^∗_N = lim sup_n→+∞u_n. Now, we can state,

Proposition 2.9. — If D^w,p_Λ,µ(n)

nis a bounded sequence of real num- bers, then we have

(1) aN+1(T_Λ,µ^w )6 D_Λ,µ^w,p∗

(N).

(2) kT_Λ,µ^w kp6sup_n∈ND^w,p_Λ,µ(n).

(3) kT_Λ,µ^w ke6lim sup_n→+∞D_Λ,µ^w,p(n).

(4) ∀p>1,kT_Λ,µ^w kN 6P

n>0w⁻

1

np t^λn

_Lp(µ). (5) Ifp= 2, for any r >0,kT_Λ,µ^w kS^r 6 P

n>0 D^w,2_Λ,µ(n)^r1_r .

Proof. — We first prove (1). Forn∈N, we denoteϕ^∗_n :`<sup>p</sup>(w)→Cthe functional on`^p(w) defined byϕ^∗_n(b) =b_nfor a sequenceb= (b_n)_n ∈`^p(w).

We define alsogn ∈L^p(µ) bygn(t) =t^λn. For any integer N and A⊂N with|A|=N, we have

a_N+1(T_Λ,µ^w )6

T_Λ,µ^w −X

n∈A

ϕ^∗_n⊗g_n .

By Proposition 2.6, forb∈`^p(w),

T_Λ,µ^w (b)−X

n∈A

ϕ^∗_n(b)gn

=

X

n6∈A

bnt^λn _Lp(µ)

6sup

n6∈A

D_Λ,µ^w,p(n)

kbk_`p(w)

and so (1) holds.

Assertions (2) and (3) are direct consequences of (1).

Assertion (4) follows easily from the natural decomposition T_Λ,µ^w (b) = P

nϕ^∗_n(b)gn and the fact thatkϕ^∗_nk=w_n^−1p.

For (5): if (D^w,2_Λ,µ(n))n6∈`<sup>r</sup>then the result is obvious. If (D<sub>Λ,µ</sub><sup>w,2</sup>(n))n ∈`^r, we have in particularD_Λ,µ^w,2(n)→0 whenn→+∞. Since for allε >0, the set{n, D_Λ,µ^w,2(n)>ε} is finite, there exists a bijectionβ :N→Nsuch that for anyn∈N,D_Λ,µ^w,2(n)^∗=D^w,2_Λ,µ(β(n)). We have

X

N

aN+1(T_Λ,µ^w )^r6X

N

D_Λ,µ^w,2(N)^∗r

=X

n

D^w,2_Λ,µ(β(n))r

=X

n

D^w,2_Λ,µ(n)^r

.

Lemma 2.10. — Letα ∈R^∗+. Assume that Λ is a quasi-geometric se- quence. Then there are two constants C1, C2 ∈ R^∗+ such that for any t∈[0,1)we have

C1

1 1−t

α

6X

n

λ^α_nt^λn6C2

1 1−t

α

·

Proof. — Since Λ is quasi-geometric, it is r-lacunary for some r > 1, so there exists a constant C = (r−1)⁻¹ such that for any n ∈ N, λn 6 C(λn+1−λn). Moreover, there is a constantR >1 such that λn+16Rλn

and hence we have

λ^α_n ≈(λn+1−λn)^α≈λ^α_n+1

where the underlying constants do not depend onn. We obtain X

n

λ^α_nt^λn≈X

n

(λ_n+1−λ_n)^αt^λn≈X

n

X

λ_n6m<λ_n+1

(λ_n+1−λ_n)^α−1t^λn

≈X

n

X

λn6m<λn+1

m^α−1t^λn

Formsuch thatλn6m < λn+1, we havet^m6t^λn6t^mR and so we obtain X

n

λ^α_nt^λn. X

m>0

m^α−1t^mR . 1

1−t^R1 α

. 1

1−t α

·

On the other hand we have X

n

λ^α_nt^λn& X

m>1

m^α−1t^m&

1 1−t

^α

·

Remark 2.11. — If Λ is only lacunary, the right inequality in Lemma 2.10 still holds. Indeed, the above proof can be easily adapted. We can also notice that there exists a quasi-geometric sequence Λ⁰ = (λ⁰_n)_n which contains Λ, and we have

X

n∈N

λ^α_nt^λn6X

n∈N

λ^0α_nt^λ0n 6C2

1 (1−t)^α·

A new proof of the upper bound part in Gurariy–Macaev theorem (The- orem 2.3) follows from the next proposition.

Proposition 2.12. — Let p ∈ [1,+∞). Assume that the weights are given bywn=λ⁻¹_n or(pλn+ 1)⁻¹. IfΛ is lacunary andµis the Lebesgue measure, then D_Λ,µ^w,p(n)

n is a bounded sequence.

Proof. — From Lemma 2.10 and Remark 2.11 we get D^w,p_Λ,µ(n)^p

=λ

1

np

Z

t^λn X

k∈N

λ

1 p

kt^λk

!p−1

dt

.λ

1

np

Z 1 0

t^λn 1

1−t _p¹0

dt

=λ

1 p

n

Z 1−_λn¹ 0

t^λn 1

1−t _p¹0

dt+λ

1 p

n

Z 1 1−_λn¹

t^λn 1

1−t _p¹0

dt

6λ

1

npλ

1 p0

n

Z 1 0

t^λndt+λ

1

np

Z 1 1− ¹

λn

(1−t)^−p10dt

6 λn

λn+ 1+λ

1 p

n

p λ

1

np

6p+ 1.

From Proposition 2.6, we obtain as claimed

X

n∈N

bnt^λn p

. X

n∈N

|bn|^p λn

!_p¹ ,

for anyb∈c00, when Λ is lacunary.

From Lemma 2.10 and Gurariy–Macaev’s Theorem, one can easily get an estimate of point evaluations onM_Λ^p.

Proposition 2.13. — LetΛbe a quasi-geometric sequence andp>1.

For anyt∈[0,1), the point evaluationf ∈M_Λ^p7−→δt(f) =f(t)satisfies δ_t

_(Mp

Λ)^∗≈ 1 (1−t)^p1

· WhenΛis only lacunary, we only have

δt

_(Mp

Λ)^∗. ¹

(1−t)^1p ·

Proof. — Since Λ is in particular lacunary, Gurariy–Macaev theorem gives

δt

_(Mp

Λ)^∗= sup

f∈B_Mp Λ

|f(t)| ≈ sup

a∈B_`p

X

n>0

λ

1

npant^λn

where the underlying constants depend onpand Λ. Ifp >1, the last term is

P

n>0λ

p0 p

n t^p0λn_p¹0

and we conclude with Lemma 2.10.

Ifp= 1, we have fortclose to 1 (sayt>exp(−1/λ₁)) δt

_(M1

Λ)^∗≈ sup

a∈B_`1

X

n>0

λnant^λn

= sup

n>0

λnt^λn

6 sup

s>λ₀

st^s= 1

e|ln(t)| ≈ 1 (1−t)· Moreover, if Λ is quasi-geometric then there exists someδ∈(0,1) (depend- ing on Λ only) and some integern_tsuch thatλ_nt ∈ δ/|ln(t)|,1/|ln(t)|

so that

sup

n>0

λ_nt^λn >λ_ntt^λnt > δ

e|ln(t)| ≈ 1

(1−t)·

3. Revisiting the classical case

We consider the Lebesgue measure µ=mon [0,1]. We define the oper- ator

T_Λ^w(p):

`^p(w(p)) −→ M_Λ^p

b 7−→ P

nbnt^λn

where the weightsw(p) = (w_n(p))_n∈Nare given byw_n(p) = (pλ_n+ 1)⁻¹= kt^λnk^p_p. In particular, if we denote by (ek)k the canonical basis of`^p(w(p)), we have

∀k∈N, kT_Λ^w(p)(e_k)k_p=ke_kk_`p(w(p)).

Gurariy–Macaev theorem says that T_Λ^w(p) is an isomorphism if and only if Λ is lacunary. By our Propositions 2.6 and 2.12, we recover thatT_Λ^w(p)is bounded when Λ is lacunary.

Since T_Λ^w(1)

₁ = 1, we focus mainly on the case p > 1. We shall also prove thatT_Λ^w(p)is bounded if and only if Λ is quasi-lacunary (for p >1).

We shall refine the method used in Proposition 2.12 and get a sharper estimate of the norm. Our approach is different from the original one (which was based on some slicing of the interval (0,1)). We control the norm with explicit quantities depending only on the ratio of lacunarity (andp). As a

consequence, we shall get that for p∈ (1,+∞), the operator T_Λ^w(p) is an asymptotical isometry if and only if Λ is super-lacunary.

Lemma 3.1. — Let α ∈ (0,+∞), p ∈ (1,+∞) and (q_n)_n be an r- lacunary sequence. We have

sup

n∈N

X

k∈N k6=n



 q

1

npq

1 p0

k q_n

p +^q_p^k0





α

6 p^0α

r^αp −1+ p^α r^pα0 −1

·

Proof. — Letn∈N. For k < n, we have ^q

1 p nq

1 p0 qn k

p+^qk

p0 6p

qk

q_n

_p¹0

6pr^−n−kp0 · We obtain

n−1

X

k=0



 q

1 p

nq

1 p0

k q_n

p +^q_p^k0





α

6p^α

n−1

X

k=0

1 r

(n−k)α p0

6 p^α r^pα0 −1

·

Similarly, whenk > n, we use ^q

1 p nq

1 p0 qn k

p+^qk

p0 6p⁰

qn

qk

¹_p

6p⁰r^−k−np . For p ∈ [1,+∞) we consider the sequence D^w,p_Λ,µ(n)

n defined in Sec- tion 2, but since we focus on the caseµ=mandw=w(p), we lighten the notation and write

D^(p)_Λ (n) = Z 1

0

(pλ_n+ 1)^1pt^λn X

k

(pλ_k+ 1)^1pt^λk

!p−1

dt

!¹_p .

Proposition 3.2. — Letp >2 and Λ be a (lacunary) sequence such that(pλ_n+ 1)_n isr-lacunary. Then we have

kT_Λ^w(p)k_p6 1 + 2p^p−11 r^p(p−1)1 −1

!_p¹0

.

Proof. — Forj ∈N, we denoteqj =pλj+ 1 andfj(t) =q

1 p

j t^λj = ^tλj

kt^λjkp

· We have

D_Λ^(p)(n)^p

= Z 1

0

f_n X

k

f_k

!p−1

dt=

X

k

f_k

p−1

L^p−1(fndt)

.

Sincep−1>1, the triangle inequality gives D^(p)_Λ (n)^p0

6X

k

kfkk_Lp−1(f_ndt)=X

k

q

1

npq

1 p0

k

Z 1 0

t^{λn+(p−1)λk}dt p−1¹

.

Forn, k∈N, we have q

1

npq

1 p0

k

Z 1 0

t^{λn+(p−1)λk}dt= q

1

npq

1 p0

k

λn+ (p−1)λk+ 1 = q

1

npq

1 p0

k q_n

p +^q_p^k0

·

By Lemma 3.1, we obtain for anyn∈N D^(p)_Λ (n)p⁰

6X

k∈N

q

1

npq

1 p0

k q_n

p +^q_p^k0

!p−1¹

61 + 2p^p−11 r^p(p−1)1 −1

sincep>p⁰and the term indexed byn=kis 1. Thanks to Proposition 2.6, we have

kT_Λ^w(p)kp6sup

n

D_Λ^(p)(n).

Let us point out that the operatorsT_Λ^w(p):`^p(w(p))→M_Λ^p ⊂L^p(m) are not defined on the same scale ofL^p-spaces, since the weightw(p) depends on p. We cannot apply directly the Riesz–Thorin theorem withT_Λ^w(1)andT_Λ^w(2) to estimate the norm ofT_Λ^w(p)whenp∈(1,2), even not its weighted version.

The next result gives a bound different from the one in Proposition 3.2;

they coincide whenp= 2.

Proposition 3.3. — Letp∈[1,2]andΛbe a (lacunary) sequence such that(pλn+ 1)n isr-lacunary. Then we have

kT_Λ^w(p)kp6 1 + 4 r¹² −1

!_p¹0

·

Proof. — Our proof is adapted from the classical proof of Riesz–Thorin theorem, with an additional trick.

Let θ = _p²0 ∈ (0,1). We have ¹_p = 1− ^θ₂· As usual, for z ∈ C such that 0 6 Re(z) 6 1, we define _p(z)¹ = 1− ^z₂ and _p0¹(z) = ^z₂· We have p(θ) = p and p⁰(θ) = p⁰. We fix a = (an)n a sequence in R+ with a finite number of non-zero terms and a positive functiong∈L^p0, such that kak`^p(w(p))=kgkp⁰ = 1.Finally we define

F(z) =X

n∈N

a

p p(z)

n

Z 1 0

t^{p(z)p λn}g(t)

p0 p0(z)dt .

This is actually a finite sum, andF is a holomorphic function on the band {z∈C|Re(z)∈(0,1)}. Forx∈R, we have

|F(ix)|6X

n∈N

a^p_n Z 1

0

t^pλndt=X

n∈N

a^p_n

pλn+ 1 = 1.

On the other hand, for every real number x,

|F(1 +ix)|6X

n∈N

a^p(1−n ¹²⁾

Z 1 0

t^p(1−12)λng(t)^p

0 2dt

= Z 1

0

g(t)^p

0

2 X

n∈N

b_nt^ψndt

wherebn=a

p

n2 and Ψ = (ψn)n= ^pλ₂ⁿ

n. Since (2ψn+1)nis alsor-lacunary we can apply Proposition 3.2. in the hilbertian case

X

n∈N

bnt^ψn

₂6 1 + 4 r¹² −1

!¹₂ X

n

|bn|² 2ψ_n+ 1

!¹₂

·

By Cauchy–Schwarz inequality, we get

|F(1 +ix)|6kg^p

0 2k2×

X

n

b_nt^ψn ₂6

1 + 4 r¹² −1

¹₂ .

Now, the proof ends in a standard way and the three lines theorem gives

|F(θ)|6 1 + 4 r¹² −1

!^θ₂ .

From this, we conclude easily that for arbitrarya∈`^p(w(p)), we have kT_Λ^w(p)(a)kp6

1 + 4 r¹² −1

_p¹0

kak`^p(w(p)).

Now we can give a characterization of the boundedness of T_Λ^w(p). Theorem 3.4. — Letp∈(1,+∞). The following are equivalent

(1) The sequenceΛ is quasi-lacunary ;

(2) The operatorT_Λ^w(p):`^p(w(p))→M_Λ^p is bounded.

Proof. — Assume that Λ is a quasi-lacunary sequence. Using Remark 2.2, there existK >1 and lacunary sets Λj ⊂Λ (with j ∈ {1, . . . , K}) such that Λ = Λ1∪ · · · ∪ΛK. We define the operators

T^(j):

`^p(w(p)) −→ M_Λ^p

b 7−→ P

nbnt^λn1Λj(λn) where1Λj is the indicator function of the set Λ_j.

We haveT_Λ^w(p)=PK

j=1T^(j). Since each Λj is lacunary, Proposition 3.2 and Proposition 3.3 (or Gurariy–Macaev theorem) imply that T_Λ^w(p) is bounded.

For the converse, we assume that Λ is not quasi-lacunary. We denoteq_k= pλk+ 1. For an arbitrarily largeN ∈N, the sequenceqN k is not lacunary since (N k)_k∈Nhas bounded gaps. This implies lim inf_k→+∞^q(k+1)N_q

kN = 1, so there existsk0 such that it is less than 2. Forn0=k0N we have

q_n0+N 62q_n0.

Let A={n0, . . . , n0+N −1}. Thanks to the inequality between arith- metic and geometric means, we have

kT_Λ^w(p)(1^A)k^p_p= Z 1

0

X

j∈A

t^λj

p

dt>

Z 1 0

N^p Y

j∈A

t^pλjN dt.

We obtain

kT_Λ^w(p)(1^A)k^p_p> N^p P

j∈A q_j N

> N^p

q_n0+N > N^p 2q_n0· On the other hand,k1Ak^p_`p(w(p)) =P

j∈A 1

q_j 6 q^N_n0· SinceN is arbitrarily

large andp >1,T_Λ^w(p)is not bounded.

The following is a refinement of Gurariy–Macaev theorem for lacunary sequences with a large ratio.

Theorem 3.5. — Letp >1. For anyε∈(0,1), there existsrε>1with the following property:

For any Λsuch that(pλ_n+ 1)n isr_ε-lacunary, we have

∀a∈`<sup>p</sup>(w(p)), (1−ε)kak`^p(w(p))6

T_Λ^w(p)(a)

_p6(1 +ε)kak`^p(w(p)). Remark 3.6. — If we denote q = max{p, p⁰}, the parameter r_ε =

1 +^4q

1 q−1

ε

q(q−1)

is suitable for Theorem 3.5.

Proof. — Letq= max{p, p⁰}>2 andrε=

1 +^4q

1 q−1

ε

q(q−1)

. In order to lighten the computation below, we shall writeω instead ofw(p) so that ω_n=w_n(p) = _pλ¹

n+1·

Let a be a sequence with kak_`p(ω) = 1. Thanks to the above choice of rε, we have that

T_Λ^ω

_p 6 1 + ^ε_2p¹0

61 +₂^ε either by Proposition 3.2 if p>2, or by Proposition 3.3 ifp62.