Lacunary M\"untz spaces: isomorphisms and Carleson embeddings

0.3 MB·

by  Loic Gaillard

LACUNARY MU¨NTZ SPACES:
ISOMORPHISMS AND CARLESON EMBEDDINGS
LO¨ICGAILLARDANDPASCALLEFE`VRE
17 Abstract. In this paper we prove that MΛp is almost isometric to ℓp in the canonical
waywhenΛislacunarywithalargeratio.Ontheotherhand,ourapproachcanbeused
20 tostudyalsotheCarlesonmeasures forMu¨ntzspaces MΛp when Λislacunary. Wegive
some necessary and some sufficient conditions to ensure that a Carleson embedding is

n bounded or compact. In the hilbertiancase, the membershiptoSchatten classes is also
a studied. When Λ behaves like a geometric sequence the results are sharp, and we get
J somecharacterizations.

0
2

] 1. Introduction
A
F Let m be the Lebesgue measure on[0,1]. For p [1,+ ), Lp(m)=Lp([0,1],m)(some-
∈ ∞
. times denoted simply Lp when there is no ambiguity) denotes the space of complex-valued
h
at measurable functions on [0,1], equipped with the norm kfkp =( 01|f(t)|pdt)p1. In the same
way, =C([0,1])isthe spaceofcontinuousfunctions on[0,1]equippedwiththe usualsup-
m C R
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 associated Lp(µ) space. For a sequence w = (w )
n n
1 ofpositiveweights,we denote ℓp(w) the Banachspaceofcomplexsequences (bn)n equipped
7v with the norm kbkℓp(w) = ( n|bn|pwn)p1 and the vector space c00 consisting on complex
sequences with a finite number of non-zero terms. All along the paper, when p (1,+ ),
0 we denote as usual p = p Pits conjugate exponent. ∈ ∞
8 ′ p 1
5 The famous Mu¨ntz th−eorem ([BE, p.172],[GL, p.77]) states that if Λ = (λn)n N is an
0 increasingsequenceofnon-negativerealnumbers,thenthelinearspanofthemonom∈ialstλn
01. itshdatentsheeinMLu¨pnt(zrecsopn.dinitiCo)nifandon1ly<if +n≥1isλ1nfu=lfil+le∞d a(nrdespw.eanddefiλn0e=th0e).MWu¨entszhaslplaacsesuMmpe
17 as the closed linear space sPpann≥n1edλnby tPhe∞monomials tλn, where n ∈ N. We shall moreoveΛr
: assume that Λ satisfies the gapcondition: inf λn+1 λn >0. Under this later assumption
v n −
Xi the Clarkson-Erdo¨s theorem holds [GL, Th.6.(cid:0)2.3]: the fu(cid:1)nctions in MΛp are the functions f
inLp suchthatf(x)= anxλn (pointwiseon[0,1)).ThisgivesusaclassofBanachspaces
r Mp (Lp of analytic functions on (0,1).
a Λ P
In full generality, the Mu¨ntz spaces are difficult to study, but for some particular se-
quences Λ, we can find some interesting properties of the spaces Mp. Let us mention that
Λ
latelythese spacesreceivedanincreasingattentionfromthe pointofviewoftheir geometry
andoperators:themonographofGurariy-Lusky[GL],andvariousmoreorlessrecentpapers
(see for instance [AHLM],[AL],[CFT],[LL],[NT]).
We shall focus on two different questions on the Mu¨ntz spaces. The first one is linked
to an old result: Gurariy and Macaev proved in [GM] that, in Lp, the normalized sequence
((pλn+1)p1tλn)n is equivalent to the canonical basis of ℓp if and only if Λ is lacunary (see
Th.2.3 below). More recently, the monograph [GL] introduces the notion of quasi-lacunary
sequence (see definition 2.1 below), and states that Mp is still isomorphic to ℓp when Λ is
Λ
quasi-lacunary.Onthe otherhand,somerecentpapersdiscussaboutthe Carlesonmeasures
for the Mu¨ntz spaces. In [CFT], the authors introduced the class of sublinear measures
on [0,1), and proved that when Λ is quasi-lacunary, the sublinear measures are Carleson
2010 Mathematics Subject Classification. 30B10,47B10,47B38.
Key words and phrases. Mu¨ntzspaces,Carlesonembeddings,lacunarysequences, Schatten classes.
1
2 LO¨ICGAILLARDANDPASCALLEFE`VRE
embeddings for M1. In [NT], the authors extended this result to the case p = 2 but only
Λ
when the sequence Λ is lacunary.
In this paper, we introduce another method to study the lacunary Mu¨ntz spaces: for a
weight w and a measure µ on [0,1), we define Tµ : ℓp(w) → Lp(µ) by Tµ(b) = nbntλn
for b = (b ) ℓp(w). The operator T depends on w,µ,p and Λ, and when it is bounded
n µ
∈ P
we shall denote by T its norm. We shall see that an estimation of T can be used
µ p µ p
k k k k
to improve the theorem of Gurariy-Macaev, and to generalize former Carleson embedding
results to lacunary Mu¨ntz spaces Mp for any p 1.
Λ ≥
The paper is organized as follows: in part 2, we specify the missing notations and some
usefull lemmas. The main result gives an upper bound for the approximation numbers of
T (see Prop.2.9). In section 3, we focus on the classical case: we fix w = (pλ +1) 1
µ n n −
and we define JΛ : ℓp(w) → MΛp by JΛ(b) = nbntλn. It is the isomorphism underlying in
the theorem of Gurariy-Macaev.For p>1, we prove that J is bounded exactly when Λ is
Λ
P
quasi-lacunary.Ontheotherhand,whenΛislacunarywithalargeratio,wealsogetasharp
bound for kJΛ−1kp (see Th.3.6 below). Our approach leads to an asymptotically orthogonal
version of Gurariy-Macaev theorem exactly for the super-lacunary sequences. In section 4,
weapplytheresultsofsection2forapositiveandfinitemeasureµon[0,1)withtheweights
w = λ 1. To treat the Carleson embedding problem, we shall give an estimation of the
n −n
approximationnumbers of the embedding operator ip :Mp Lp(µ). In section5, we focus
µ Λ →
onthecompactnessofip usingthesametoolsasinsection4.Inthecasep=2,thisleadsto
µ
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 that
A cB. This constant c may depend along the paper on Λ (or sometimes only on its ratio
≤
oflacunarity),onp....Weshallspecify thisdependence toavoidanyambiguousstatement.
In the same way, we shall use the notations A B or A&B.
≈
2. Preliminary results
Beforegivingpreliminaryresults,letusgiveafewwordsofexplanationaboutourchoice
ofmeasures on[0,1).This comes fromthe fact that the measuresinvolved(if consideredon
[0,1])mustsatisfyµ( 1 )=0.Indeed,wefocuseitherontheLebesguemeasurem(satisfying
of course m( 1 ) = 0{)}or on measures such that the Carleson embedding f Mp f
{ } ∈ Λ 7→ ∈
Lp(µ) is (defined and) bounded, so that testing a sequence of monomials gn(t) = tλn we
must have
µ( 1 )=lim g p .lim g p =0.
{ } k nkLp(µ) k nkLp(m)
Therefore practically, we shall consider in the whole paper measures on [0,1). Moreover,
thanks to the result of Clarkson-Erdo¨s,the value at any point of [0,1) of any function of a
Mu¨ntz space can be defined without ambiguity.
We shall need several notions of growth for increasing sequences.
Definition 2.1. A sequence u = (u ) of positive numbers is said to be lacunary
n n
•
if there exists r >1 such that u ru , for every n N. We shall say that such
n+1 n
≥ ∈
a sequence is r-lacunary and that r is a ratio of lacunarity of this sequence.
The sequence u is called quasi-lacunary if there is an extraction (n ) such that
k k
•
sup(n n )<+ , and (u ) is lacunary.
k N k+1− k ∞ nk k
T∈hesequenceuiscalledquasi-geometriciftherearetwoconstantsrandRsuchthat
• u
we have 1 < r n+1 R < + , for every n N. In particular, these sequences
≤ u ≤ ∞ ∈
n
are lacunary.
u
The sequence u is called super-lacunary if n+1 + .
• u −→ ∞
n
Remark 2.2. It is provedin [GL, Prop.7.1.3p.94] that a sequence is quasi-lacunaryif and
only if it is a finite union of lacunary sequences.
3
The following result is due to Gurariy and Macaev.
Theorem 2.3. [GL, Corollary 9.3.4, p.132]
For p [1,+ ), the following are equivalent:
∈ ∞
(i) The sequence Λ is lacunary.
tλn
(ii) The sequence in Lp is equivalent to the canonical basis of ℓp.
tλn p
(cid:16)k k (cid:17)
In particular, since tλn p =(pλn+1)−p1, we have for any b c00
k k ∈
b p 1
bntλn | n| p
p ≈ pλn+1
(cid:13)X (cid:13) (cid:16)X (cid:17)
when Λ is lacunary, and wher(cid:13)e the unde(cid:13)rlying constants depend on p and Λ only.
(cid:13) (cid:13)
We shall recover and generalize partially this result: for a given sequence of weights
(w ) and a positive finite measure µ on [0,1), we study the boundedness of the operator
n n
ℓp(w) Lp(µ)
Tµ : b −→ bntλn .
(cid:26) 7−→
Example 2.4. In the case of the Lebesgue meaPsure µ = m and when the weights are
w =(pλ +1) 1 or in a simpler way (when we do not care on the value of the constants)
n n −
w =λ 1, Th.2.3 states in particular that T is bounded when Λ is lacunary.
n −n m
Remark 2.5. In the case p > 1, a (rough) sufficient condition to ensure the boundedness
of T is
wn−pp′tp′λn pp′dµ< .
∞
Z[0,1)(cid:16)Xn (cid:17)
Indeed, this is just the consequence of the majorization
1 1
sup sup bnwn−ptλng(t)dµ sup g(t) sup bnwn−ptλn dµ.
bb∈∈Bc0ℓ0p g∈BLp′(µ)(cid:12)(cid:12)Z[0,1)Xn (cid:12)(cid:12)≤g∈BLp′(µ)Z[0,1)| | bb∈∈Bc0ℓ0p (cid:12)(cid:12)Xn (cid:12)(cid:12)
Point out tha(cid:12)t in the case of standard(cid:12)weights w λ 1 and for a(cid:12)quasi-geometr(cid:12)ic
n ≈ −n
sequence Λ, this condition can be reformulated with the help of Lemma 2.10 below as
1 1
dµ dµ<
1 t ≈ 1 tp′ ∞
Z[0,1) − Z[0,1) −
but we shall come back to that kind of condition later (see Prop.5.5 below for instance).
To get a sharper estimation, we introduce the sequence (D (p)) defined for n N and
n n
∈
p 1, with a priori value in R + by
+
≥ ∪{ ∞}
p 1 1
1 1 − p
Dn(p)= wn−ptλn wk−ptλk dµ .
Z[0,1) k 0 ! !
X≥
Proposition 2.6. Let p [1,+ ). Assume that (D (p)) is a bounded sequence of real
n n
∈ ∞
numbers. Then we have for every b ℓp(w),
∈
1
bntλn bn pwnDn(p)p p .
Lp(µ) ≤ | |
(cid:13)(cid:13)nX≥0 (cid:13)(cid:13) (cid:16)nX≥0 (cid:17)
Proof. If p=1 the resu(cid:13)lt is obviou(cid:13)s. Assume now that p>1. For any t [0,1) and n N,
∈ ∈
we have:
bntλn =bnwnp1p′tλpn wn−p1p′tλpn′ ,
×
we apply Ho¨lder’s inequality and get:
bntλn ≤ |bn|pwnp1′tλn p1 wk−p1tλk p1′ .
(cid:12)X (cid:12) (cid:16)Xn (cid:17) (cid:16)Xk (cid:17)
(cid:12) (cid:12)
(cid:12) (cid:12)
4 LO¨ICGAILLARDANDPASCALLEFE`VRE
We obtain:
p 1 1 p 1
bntλn dµ≤ |bn|pwn.wn−ptλn wk−ptλk − dµ
Z[0,1)(cid:12)X (cid:12) Z[0,1)X (cid:16)Xk (cid:17)
(cid:12) (cid:12)
(cid:12) (cid:12) = b pw D (p)p .
n n n
| |
n
X
(cid:3)
If (D (p)) is a bounded sequence of real numbers, we define the bounded diagonal
n n
operator
:ℓp(w) ℓp(w)
D →
acting on the canonical basis of ℓp(w) whose diagonal entries are the numbers D (p). In
n
other words, in that case, T and are bounded, and we have
µ
D
b ℓp(w), T (b) (b) .
µ Lp(µ) ℓp(w)
∀ ∈ k k ≤kD k
This givesinformations about the approximationnumbers of T . Let us specify this notion.
µ
Weshallbeinterestedinhowfarfromcompact(theessentialnorm)or,onthecontrary,how
stronglycompact(possiblySchattenintheHilbertframework)aretheCarlesonembeddings.
A way to measure this is to estimate the approximation numbers:
Definition 2.7. For a bounded operator S :X Y between two separable Banach spaces
→
X,Y, the approximation numbers (a (S)) of S are defined for n 1 by
n n
≥
a (S)=inf S R ,rank(R)<n .
n
{k − k }
The essential norm of S is defined by
S =inf S K ,K compact .
e
k k {k − k }
It is the distance from S to the compact operators.
We shall use in the sequel the following notions of operator ideal.
Definition 2.8.
An operator S :X Y is nuclear if there is a sequence of rank-oneoperators (R )
n
• →
satisfying S(x)= R (x) for everyx X with R <+ .The nuclear norm
n n
∈ k k ∞
n n
of S is defined asP P
S =inf R ,rank(R )=1, R =S .
n n n
k kN k k
nXn Xn o
An operator S : X Lp(µ) is order bounded if there exists a positive function
• →
h Lp(µ) such that for every x B and for µ almost every t Ω we have
X
∈ ∈ − ∈
S(x)(t) h(t).
| |≤
For r > 0 and when X,Y are Hilbert spaces, we say that a (compact) operator
•
S :X Y belongs to the Schatten class r if
→ S
a (T)r <+ .
n
∞
n
X
1
In this case, we define its Schatten norm by S = a (S)r r.
r n
k kS n
(cid:16) (cid:17)
P
Recall that nuclear and Schatten class operators are always compact.
Of course, the Schatten norm is really a norm when r 1. The 2 class is also called
≥ S
the class of Hilbert-Schmidt operators.
Fortechnicalreasons,weintroducethefollowingnotation:foraboundedsequence(u )
n n
in R , we define (u ) the decreasing rearrangement of (u ) by
+ ∗N N n n
u = inf sup u ,n A .
∗N A N { n 6∈ }
A⊂=N
| |
5
We have lim u =limsupu .
∗N n
N + n +
Now, w→e ca∞n state, → ∞
Proposition 2.9. If (D (p)) is a bounded sequence of real numbers, then we have
n n
(i) a (T ) D (p) .
N+1 µ N ∗
≤
(ii) T supD (p).
µ p n
k k ≤n N
(iii) T li∈msupD (p).
µ e n
k k ≤ n +
→ ∞ 1
(iv) ∀p≥1, kTµkN ≤ wn−p tλn Lp(µ).
n 0
X≥ (cid:13) (cid:13)
(v) If p=2, then for any r >(cid:13)0, (cid:13)T D (2)r r1 .
µ r n
k kS ≤ n 0
(cid:16) ≥ (cid:17)
P
Proof. We first prove (i). For n N, we denote ϕ : ℓp(w) C the functional on ℓp(w)
∈ ∗n →
defined by ϕ (u) = u for a sequence u = (u ) ℓp(w). We define also g Lp(µ) by
∗n n n n ∈ n ∈
gn(t)=tλn. For any integer N and A N with A =N, we have:
⊂ | |
a (T ) T ϕ g .
N+1 µ ≤ µ− ∗n⊗ n
We fix b ℓp(w) and apply Prop.2.6: (cid:13)(cid:13)(cid:13) nX∈A (cid:13)(cid:13)(cid:13)
∈
Tµ(b)− ϕ∗n(b)gn = bntλn Lp(µ) ≤nsupADn(p)kbkℓp(w)
(cid:13)(cid:13) nX∈A (cid:13)(cid:13) (cid:13)(cid:13)nX6∈A (cid:13)(cid:13) 6∈
and so (i) hol(cid:13)ds. (cid:13) (cid:13) (cid:13)
The points (ii) and (iii) are direct consequences of (i).
The assertion (iv) follows easily from the natural decomposition Tµ(b) = ϕ∗n(b)tλn
n
and the fact that kϕ∗nk=wn−p1. P
For(v):if(D (2)) ℓr thentheresultisobvious.Else,wehaveinparticularD (2) 0
n n n
6∈ →
when n + . Since for all ε > 0, the set n,D (2) ε is finite, there exists a bijection
n
→ ∞ { ≥ }
ϕ:N N such that for any n N, D (2) =D (2). We have:
n ∗ ϕ(n)
→ ∈
a (T )r (D (2) )r = D (2)r = D (2)r.
N+1 µ N ∗ ϕ(n) n
≤
N N n n
X X X X
(cid:3)
Lemma 2.10. Let α R . Assume that Λ is a quasi-geometric sequence. Then there are
∈ ∗+
two constants C ,C R such that for any t [0,1) we have:
1 2 ∈ ∗+ ∈
1 α 1 α
C λαtλn C
1 1 t ≤ n ≤ 2 1 t ·
(cid:16) − (cid:17) Xn (cid:16) − (cid:17)
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,λ C(λ λ ). Moreover, there is a constant
− n n+1 n
− ∈ ≤ −
R>1 such that λ Rλ and hence we have:
n+1 n
≤
λα (λ λ )α λα
n ≈ n+1− n ≈ n+1
where the underlying constants do not depend on n. We obtain:
λαntλn ≈ (λn+1−λn)αtλn ≈ (λn+1−λn)α−1tλn
Xn Xn Xn λn≤mX<λn+1
mα−1tλn
≈
Xn λn≤mX<λn+1
For m such that λn m<λn+1, we have tm .tλn .tmR and so we obtain:
≤
1 α 1 α
Xn λαntλn .mX≥0mα−1tmR .(cid:16)1−tR1 (cid:17) .(cid:16)1−t(cid:17) ·
6 LO¨ICGAILLARDANDPASCALLEFE`VRE
On the other hand we have
1 α
λαntλn & mα−1tm & 1 t ·
Xn mX∈N (cid:16) − (cid:17)
(cid:3)
Remark 2.11. If Λ is only lacunary, the majorization part of the result above still holds.
Indeed, the proof above can be easily adapted, but anyway, we can also notice that there
exists a quasi-geometric sequence Λ =(λ ) which contains Λ, and we have
′ ′n n
λαntλn ≤ λ′nαtλ′n ≤C2(1 1t)α·
n N n N −
X∈ X∈
We can give a new proof of the majorization part of the theorem of Gurariy-Macaev
(Th.2.3). It follows from the next proposition:
Proposition 2.12. Let p [1,+ ). Assume that the weights are given by w = λ 1 or
∈ ∞ n −n
(pλ +1) 1. If Λ is lacunary and µ is the Lebesgue measure, then (D (p)) is a bounded
n − n n
sequence.
Proof. From Lemma 2.10 and Remark 2.11 we get:
1 1 p 1
Dn(p)p =λnp tλn λkptλk − dt
Z (cid:16)kX∈N (cid:17)
.λn1p 1tλn 1 p1′dt
1 t
Z0 (cid:16) − (cid:17)
=λn1p 1−λ1n tλn 1 p1′dt+λnp1 1 tλn 1 p1′dt
1 t 1 t
Z0 (cid:16) − (cid:17) Z1−λ1n (cid:16) − (cid:17)
λnp1λnp1′ 1tλndt+λnp1 1 (1 t)−p1′dt
≤ −
Z0 Z1−λ1n
λn +λp1 p .
≤ λ +1 n 1
n λp
n
We obtain that D (p) is a bounded sequence of real numbers. (cid:3)
n
From Prop.2.6, we obtain as claimed:
b p 1
b tλn . | n| p ,
n
(cid:13)(cid:13)nX∈N (cid:13)(cid:13)p (cid:16)nX∈N λn (cid:17)
for any b c00, when Λ is lacu(cid:13)nary. (cid:13)
∈
Let us mention that from Lemma 2.10 and the Gurariy-Macaev’s Theorem, one can
easily get an estimation of the point evaluation on Mp:
Λ
Proposition 2.13. Let Λ be a quasi-geometric sequence and p 1. For any t [0,1), the
point evaluation f Mp δ (f)=f(t) satisfies ≥ ∈
∈ Λ 7−→ t
1
δ
t (MΛp)∗ ≈ (1 t)p1 ·
(cid:13) (cid:13) −
(cid:13) (cid:13) 1
A fortiori, when Λ is lacunary, we have δ .
t (MΛp)∗ (1 t)p1 ·
(cid:13) (cid:13) −
Proof. We fix p>1. Since Λ is in particular(cid:13)lac(cid:13)unary, the Gurariy-Macaev theorem gives:
δt (Mp)∗ = sup |f(t)|≈ sup λnp1antλn = λnpp′tp′λn p1′
where the u(cid:13)(cid:13)nde(cid:13)(cid:13)rlyiΛng conf∈stBaMnΛtps dependao∈nBℓppa(cid:12)(cid:12)(cid:12)nnX≥d0Λ. We co(cid:12)(cid:12)(cid:12)nclu(cid:16)denX≥w0ith Lemm(cid:17)a 2.10.
In the case p=1, we can easily adapt the argument, without using Lemma 2.10. (cid:3)
7
3. Revisiting the classical case
In this section, we focus mainly on the case p > 1 and we shall consider the Lebesgue
measure µ=m on [0,1]. We define the operator
ℓp(ω) Mp
JΛ : b −→ bnΛtλn
( 7−→
n
P
where the weights ω = (ωn) are given by ωn = (pλn+1)−1 = ktλnkpp. In particular, if we
denote by (e ) the canonical basis of ℓp(ω), we have
k k
k N, J (e ) = e .
Λ k p k ℓp(ω)
∀ ∈ k k k k
The theorem of Gurariy-Macaev says that J is an isomorphism if and only if Λ is
Λ
lacunary. Our Proposition 2.12 proves as well that J is bounded when Λ is lacunary.
Λ
WearegoingtorecovertheboundednessofJ refiningthemethodusedforProp.2.12,in
Λ
ordertogetasharperestimateofthenorm.Actually,weprovethatJ isboundedifandonly
Λ
if Λ is quasi-lacunary or p = 1. Our approach is different from the one of Gurariy-Macaev
(which was based on some slicing of the interval (0,1)), that is why we are able to control
theconstantsofthenormswithexplicitquantitiesdependingontheratiooflacunarity(and
p)only.As aconsequence,weshallgetthatforp (1,+ ),J isanasymptoticalisometry
Λ
∈ ∞
if and only if Λ is super-lacunary.
Lemma 3.1. Let α (0,+ ), p (1,+ ) and (q ) be an r-lacunary sequence. We have
n n
∈ ∞ ∈ ∞
qp1qp1′ α pα pα
nsu∈pNkkX=∈Nn qpnn+kqpk′ ! ≤ rαp ′−1 + rpα′ −1·
6
Proof. Let n N. For k<n, we have qqnp1qkp1q′ p qk p1′ pr−np−′k We obtain:
∈ n + k ≤ qn ≤ ·
p p (cid:16) (cid:17)
′
n−1 qnp1qkp1′ α pαn−1 1 pα
kX=0(cid:16)qpn + qpk(cid:17) ≤ Xk=0 r(n−p′k)α ≤ rpα′ −1·
′
Whenk>n,wehave qqnp1qkp1q′ p′ qn p1 p′r−k−pn and,summingoverthek’s,weobtain
n + k ≤ qk ≤
p p (cid:16) (cid:17)
′
the majorization. (cid:3)
For p [1,+ ) we consider the sequence D (p) defined in section 2:
n
∈ ∞
1
1 p 1 p
Dn(p)= (pλn+1)p1tλn (pλk+1)p1tλk − dt .
Z0 (cid:16)Xk (cid:17) !
Proposition 3.2. Let p 2 and Λ be a (lacunary) sequence such that (pλ +1) is r-
n n
≥
lacunary. Then we have:
2pp−11 p1′
J 1+ .
k Λkp ≤ rp(p1−1) 1!
−
1 tλj
Proof. For j ∈N, we denote qj =(pλj +1) and fj(t)=qjptλj = tλj p· We have:
k k
1 p 1 p 1
D (p)p = f f − dt= f − .
n n k k
Z0 (cid:16)Xk (cid:17) (cid:13)Xk (cid:13)Lp−1(fndt)
(cid:13) (cid:13)
(cid:13) (cid:13)
8 LO¨ICGAILLARDANDPASCALLEFE`VRE
Since p 1 1, the triangle inequality gives:
− ≥
Dn(p)p′ ≤ kfkkLp−1(fndt) = qnp1qkp1′ 1tλn+(p−1)λkdt p−11.
Xk Xk (cid:16) Z0 (cid:17)
For n,k N, we have :
∈
qnp1qkp1′ Z01tλn+(p−1)λkdt= λn+(pqnp1−qk1p1′)λk+1 = qpqnnp1+qkp1qp′k ·
′
We apply Lemma 3.1 and we obtain for any n N:
∈
Dn(p)p′ ≤kX∈N(cid:16)qpqnnp1+qkp1qp′k′ (cid:17)p−11 ≤1+ rp(2p1p−1p)−11−1
since p p and using that the term for n=k is 1. Thanks to Prop.2.6, we have
′
≥
J = T supD (p).
Λ p m p n
k k k k ≤ n
(cid:3)
Remark 3.3. For p (1,2), we can apply the same method and it would lead to:
∈
1
2p p
J 1+ ′ .
k Λkp ≤ rp1′ 1!
−
But this bound is not sharp when p is close to 1. For instance, it tends to + when
∞
p 1 and r is fixed. But J is always 1, without any assumption on Λ.
Λ 1
→ k k
Point out that the operators J : ℓp(ω) Mp Lp(m) are not defined on the same
Λ → Λ ⊂
scale of Lp-spaces, since the weight ω actually depends on p. We cannot apply directly
Riesz-Thorin theorem for this problem, even not the weighted versions of the literature.
Nevertheless, we shall adapt the proof in the next result and it gives the expected bound.
Proposition 3.4. Let p [1,2] and let Λ be a (lacunary) sequence such that (pλ +1) is
n n
∈
r-lacunary. Then we have:
4 p1′
J 1+
k Λkp ≤ r12 1! ·
−
Proof. Our proof is adapted from the classical proof of Riesz-Thorin theorem, with an ad-
ditional trick.
2 1 θ
Let θ = (0,1). We have = 1 As usual, for z C such that 0 Re(z) 1,
p ∈ p − 2· ∈ ≤ ≤
′
1 z 1 z
we define =1 and = We have p(θ)=p and p(θ)=p. We fix a=(a )
p(z) − 2 p(z) 2· ′ ′ n n
′
a sequence in R with a finite number of non-zero terms and g Lp′ positive, such that
+
∈
a ℓp(ω) = g p′ =1. Finally we define
k k k k
F(z)= anp(pz) 1tp(pz)λng(t)p′p(′z)dt.
n N Z0
X∈
Pointoutthatwe actuallyhaveafinite sum,andF is anholomorphicfunction onthe band
z C Re(z) (0,1) . For x R, we have
{ ∈ | ∈ } ∈
1 ap
F(ix) ap tpλndt= n =1.
| |≤ n pλ +1
n N Z0 n N n
X∈ X∈
9
On the other hand, for every real number x:
F(1+ix) apn(1−12) 1tp(1−12)λng(t)p2′dt
| |≤
n N Z0
X∈
1 ′
= g(t)p2 bntψndt
Z0 n N
X∈
where bn =anp2 and Ψ=(ψn)n = pλn . Since (2ψn+1)n is also r-lacunarywe canapply
2 n
Prop.3.2. in the hilbertian case: (cid:16) (cid:17)
1 1
4 2 b 2 2
b tψn = J (b) 1+ | n|
(cid:13)(cid:13)nX∈N n (cid:13)(cid:13)2 k Ψ k2 ≤ r21 −1! Xn 2ψn+1! ·
1 (cid:13) 1 (cid:13)
Since = and b 2 = a p, we have
n n
2ψ +1 pλ +1 | | | |
n n
b 2 a p
n n
| | = | | =1
2ψ +1 pλ +1 ·
n n
n n
X X
We apply the Cauchy-Schwarzinequality and get:
|F(1+ix)|≤kgp2′k2×(cid:13)Xn bntψn(cid:13)2 ≤(cid:16)1+ r124−1(cid:17)21 .
(cid:13) (cid:13)
Now, the proof finishes in a standard(cid:13)way and th(cid:13)e three lines theorem gives
θ
4 2
F(θ) 1+ .
| |≤ r12 1!
−
From this, we conclude easily that for arbitrary a ℓp(ω), we have
∈
J (a) 1+ 4 p1′ a .
k Λ kp ≤ r12 1 k kℓp(ω)
(cid:16) − (cid:17)
(cid:3)
Now we can give a characterizationof the boundedness of J .
Λ
Theorem 3.5. Let p (1,+ ). The following are equivalent:
∈ ∞
(i) The sequence Λ is quasi-lacunary ;
(ii) The operator J is bounded on ℓp(ω).
Λ
Proof. Assume that Λ is a quasi-lacunary sequence. Using Remark 2.2, there exist K 1
≥
andlacunarysets Λ Λ (with j 1, ,K )suchthatΛ=Λ Λ . We define the
j 1 K
⊂ ∈{ ··· } ∪···∪
operators
ℓp(ω) Mp
J(j) :( b −7−→→ bntλn1ΛIΛj(λn)
n
where 1I is the indicator function of the set ΛP.
Λj j
K
We have J = J(j). Moreover, for any j, the norm J(j) = J < + thanks
Λ k kp k Λjkp ∞
j=1
to Prop.3.4 and ProPp.3.2. Therefore, J is bounded.
Λ
Forthe converse,weassumethatΛisnotquasi-lacunary.We denoteq =(pλ +1).For
k k
anarbitrarilylargeN N we considerthe extraction(Nk)k N. It hasbounded gaps,sothe
∈ q ∈
sequence q is not lacunary. This implies liminf (k+1)N =1, so there exists k such that
Nk k + qkN 0
it is less than 2. For n =k N we have → ∞
0 0
q 2q .
n0+N ≤ n0
10 LO¨ICGAILLARDANDPASCALLEFE`VRE
Let A = n ,...,n +N 1 . Thanks to the inequality of arithmetic and geometric
0 0
{ − }
means, we have:
kJΛ(1IA)kpp = 1 tλj pdt≥ 1Np tpNλjdt.
Z0 (cid:12)(cid:12)jX∈A (cid:12)(cid:12) Z0 jY∈A
(cid:12) (cid:12)
We obtain
Np Np Np
J (1I ) p
k Λ A kp ≥ qj ≥ q ≥ 2q ·
N n0+N n0
j A
∈
P
1 N
On the other hand, 1I p = Since N is arbitrarily large and p>1, J is
k Akℓp(ω) q ≤ q · Λ
j A j n0
not bounded. X∈ (cid:3)
Thefollowingisarefinementofthe Gurariy-Macaevtheoremforthelacunarysequences
with a large ratio.
Theorem 3.6. Let p>1.For any ε (0,1), thereexists r >1 with thefollowing property:
ε
∈
For any Λ such that (pλ +1) is r -lacunary, we have:
n n ε
a ℓp(ω), (1 ε) a J (a) (1+ε) a .
ℓp(ω) Λ p ℓp(ω)
∀ ∈ − k k ≤k k ≤ k k
1
4qq−1 q(q 1)
Remark3.7. Ifwedenoteq =max p,p ,theparameterr = 1+ − issuitable
{ ′} ε ε
for Th.3.6. (cid:16) (cid:17)
1
4qq−1 q(q 1)
Proof. Let q =max p,p 2 and r = 1+ − .
{ ′}≥ ε ε
We fix a sequence a ℓp(ω) with a(cid:16)ℓp(ω) = 1. T(cid:17)hanks to the choice of rε, when p 2
∈ k k ≥
weapply Prop.3.2andwegetthat J 1+ε p1′ 1+ε. Whenp 2,Prop.3.4gives
Λ p
k k ≤ 2 ≤ 2 ≤
ε ε
also JΛ p 1+ )p1′ 1+ In the two(cid:16)cases,(cid:17)the majorization part holds.
k k ≤ 2 ≤ 2·
For the m(cid:16)inoration part, we consider a sequence b ∈ ℓp′(ω) such that kbkℓp′(ω) = 1. We
pλ
define Ψ=(ψ ) by ψ = n =(p 1)λ . We have:
n n n n
p −
′
1
JΛ(a).JΨ(b) 1 = anbktλn+(p−1)λk dt
k k
Z0 (cid:12)Xn,k (cid:12)
(cid:12) (cid:12)
+∞(cid:12) anbn (cid:12) an .bk
| || |
≥ pλ +1 − λ +(p 1)λ +1·
(cid:12)(cid:12)nX=0 n (cid:12)(cid:12) nkX,k=∈nN n − k
(cid:12) (cid:12) 6
We introduce the sequence (q ) =(pλ +1) =(ω 1) . Since a =1 and by duality
n n n n n− n k kℓp(ω)
we have
a b
sup pλn+n1,kbkℓp′(ω) =1 =1.
n
nXn o
We now majorize the second term. For any n,k, Young’s inequality gives:
a b = a ωp1b ωp1′ qp1qp1′
| n k| | n n k k |× n k
1 a pω + 1 b p′ω qp1qp1′ .
≤ p| n| n p | k| k × n k
′
(cid:16) (cid:17)