Orthogonal polynomials on the circle for the weight w satisfying conditions: w,1/w in BMO(T)

0.2 MB·

by  S. Denisov

ORTHOGONAL POLYNOMIALS ON THE CIRCLE FOR THE WEIGHT w
SATISFYING CONDITIONS w,w−1 BMO
∈
S.DENISOV,K.RUSH
6
1
0
2 Abstract. For the weight w satisfying w,w−1 ∈ BMO(T), we prove the asymptotics of {Φn(eiθ,w)} in
Lp[−π,π],26p<p0where{Φn(z,w)}aremonicpolynomialsorthogonalwithrespecttowontheunitcircle
v T. Immediate applications include the estimates on the uniform norm and asymptotics of the polynomial
o entropies. TheestimatesonhigherordercommutatorsbetweentheCalderon-ZygmundoperatorsandBMO
N functionsplaythekeyroleintheproofsofmainresults.
MSC AMS classification: 42C05, 33D45; key words: orthogonal polynomials, weight, bounded mean

2 oscillation.

]
A
C
. 1. Introduction
h
at Let σ be a probability measure on the unit circle. Define the monic orthogonal polynomials Φn(z,σ)
{ }
m by requiring
π
[ degΦ =n, coeff(Φ ,n)=1, Φ (eiθ,σ)Φ (eiθ,σ)dσ =0, m<n,
n n n m
2 Z−π
v wherecoeff(Q,j)denotesthecoefficientinfrontofzj inthe polynomialQ. We canalsodefine theorthonor-
7 mal polynomials by the formula
6 Φ (z,σ)
n
3 φn(z,σ)= Φ (eiθ,σ)
3 k n kL2σ
0 Later, we will need to use the following notation: for every polynomial Q (z)=q zn+...+q of degree at
n n 0
1. most n, we introduce the ( )–operation:
∗
0
6 Q (z) (∗) Q∗(z)=q¯ zn+...+q¯
n −→ n 0 n
1
v: TLphissta(∗n)dsdefporenLdps(Ton) onr. LInp[thπe,πp]a.peTrh,ewseyumsbeotlhseCs,hCortahraendreskefrkvped=fokrfkaLbps(oTl)u,tkefckoLnpwst=antsTw|fh(icθh)|pvwalduθe 1c/apn.
Xi change from one formula to−another. 1 (cid:0)R (cid:1)
r Thecurrentpaperismainlymotivatedbytwoproblems: theproblemofSteklovinthetheoryoforthogonal
a
polynomials [2] and the problem of the asymptotical behavior of the polynomial entropy [9].
The problem of Steklov [22] consists in obtaining the sharp estimates for φn(eiθ,σ) L∞[−π,π] assuming
k k
that the probability measure σ satisfies σ′ > δ/(2π) a.e. on [ π,π] and δ (0,1). This question attracted
− ∈
a lot of attention [1, 11, 12, 13, 14, 18, 19, 23] and was recently resolved in [2]. In particular, the following
stronger result was proved
Theorem 1.1 ([2]). Assume that the measure is given by the weight w: dσ = wdθ. Let p [1, ) and
∈ ∞
C >C (p,δ), then
0
C (p,δ)√n6 sup φ (eiθ,w) 6C (p,δ)√n
1 n ∞ 2
k k
w>δ/(2π),kwk1=1,kwkp6C
Remark. If the measure σ satisfies the Szego˝ condition [24]
π
logσ′(θ)dθ > (1)
−∞
Z−π
1
then Φ 1 and the polynomials φ and Φ are of the same size. In particular, φ can be replacedby
n L2 n n n
k k σ ∼
Φ in the previous Theorem.
n
Remark. In the formulation of the Steklov problem, the normalization that σ is a probability measure,
i.e.,
π
dσ =1
Z−π
isnotrestrictivebecauseofthefollowingscalings: φ (z,σ)=α1/2φ (z,ασ)andΦ (z,ασ)=Φ (z,σ), α>0.
n n n n
The previous Theorem handles all p< but not the case p= . That turns out to be essential: if the
∞ ∞
weight w is bounded, we get an improvement in the exponent.
Theorem 1.2. ([8], Denisov-Nazarov) If T 1, we have
≫
C
sup kΦn(eiθ,w)kp0 6C(T), p0(T)=2+ T1, sup kΦn(eiθ,w)k∞ 6C(T)n12−CT
16w6T 16w6T
and, if 0<ǫ 1,
≪
C
sup Φ (eiθ,w) 6C(ǫ), p (ǫ)= 2, sup Φ (eiθ,w) 6C(ǫ)nCǫ
k n kp0 0 ǫ k n k∞
16w61+ǫ 16w61+ǫ
The uniform bound on the Lp norm suggests that maybe a stronger result on the asymptotical behavior
is true. It is well-known that for σ in the Szego˝ class (i.e., (1) holds), the following asymptotics is valid [10]
φ∗(eiθ,σ) (∗) S(eiθ,σ), π φ∗n(eiθ,σ) 1 2dθ 0, n (2)
n −→ S(eiθ,σ) − → →∞
Z−π(cid:12) (cid:12)
(cid:12) (cid:12)
where (∗) refers to weak-star convergence and S(z(cid:12)(cid:12),σ) is the Szeg(cid:12)(cid:12)o˝ function, i.e., the outer function in D
−→
which satisfies S(eiθ,σ)−2 = 2πσ′(θ),S(0,σ) > 0. In particular, if σ′ > (2π)−1δ, then φ∗ S 0.
| | k n − k2 →
Recall that φ (z,σ)=znφ∗(z,σ),z T.
n n ∈
The results stated above give rise to three questions: (a) What upper estimate can we get assuming
w BMO(T) [21] instead of w L∞(T)? Recall that L∞(T) BMO(T) Lp(T). (b) Is it possible
p<∞
∈ ∈ ⊂ ⊂ ∩
to relax the Steklov condition w >1? (c) Can one obtain an asymptotics of φ∗ in Lp classes with p>2?
{ n}
ThepartialanswerstothesequestionsarecontainedinthefollowingTheoremswhicharethemainresults
of the paper. We start with a comment about some notation. If α is a positive parameter, we write α 1
≪
to indicate the following: there is an absolute constant α (sufficiently small) such that α < α . Similarly,
0 0
we write α 1 as a substitute for: there is a constant α (sufficiently large) so that α > α . The symbol
0 0
≫
α α (α α ) will mean α /α 1 (α /α 1).
1 2 1 2 1 2 1 2
≪ ≫ ≪ ≫
Theorem 1.3. If w: w−1 BMO 6s, w BMO 6t, then there is Π Lp0[ π,π],p0 >2 such that
k k k k ∈ −
lim Φ∗ Π =0
n→∞k n− kp0
and we have for p :
0
C
1
2+ , ifst 1
(st)log2(st) ≫
p = (3)
0 C
 (st)21/4, if0<st≪1
We also have the bound for the uniformnorm
Φ∗ 6C n1/p0 (4)
k nk∞ (st)
where C denotes a function of u.
(u)
In the case when an additional information is given, e.g., w L∞ or w−1 L∞, this result can be
∈ ∈
improved.
Theorem 1.4. Under the conditions of the previous Theorem, we have
2
If w >1, then p can be taken as
0
•
C
1
2+ , ift 1
tlogt ≫
p =
0  C
2
 √t, if0<t≪1
If w 61, then we have 
•
C
1
2+ , ifs 1
slogs ≫
p =
0  C
2
 √s, if0<s≪1
We also have the bound for the uniform norm
Φ∗ 6C n1/p0 (5)
k nk∞ (t,s)
where C depends on t or s.
(t,s)
Remark. It is clear that the allowed exponent p is decaying in s and t so it can be chosen larger than
0
2 for all values of s and t.
Remark. As we have already mentioned, the following scaling invariance holds: Φ (z,σ) = Φ (z,ασ),
n n
α>0. TheBMOnormis1-homogeneous,e.g., αw =α w ,sotheestimatesintheTheorem1.3
BMO BMO
k k k k
are invariant under scaling w αw.
→
In the case when w =C, we get w = w−1 =0 and, although Φ∗(z,w)=1, we can not say
k kBMO k kBMO n
anything about the size of φ∗(z,w). The next Lemma explains how our results can be generalized to φ∗ .
n { n}
Lemma1.1. IntheTheorem1.3,ifonemakesanadditionalassumptionthat w =1,then φ∗ S 0
k k1 k n− kp0 →
with p given by (3).
0
Proof. Indeed, Lemma 3.3 from Appendix shows that
π
logwdθ >
−∞
Z−π
and thus the sequence Φ has a finite positive limit [10, 20]. Therefore, φ∗ = Φ∗n has an
{k nk2,w} { n} ||Φn||2,w
Lp0 limit by Theorem 1.3. By (2), φ∗ converges weakly to S and therefore we have thenstatemeont of the
{ n}
Lemma with Π being a multiple of S. (cid:3)
The polynomial entropy is defined as
E(n,σ)= φ 2log φ dσ
n n
T| | | |
Z
where φ are orthonormalwith respectto σ. In recentyears,a lotof efforts were made to understand the
n
{ }
asymptotics of E(n,σ) [3, 4, 5] as n . In [9], the sharp lower and upper bounds were obtained for σ in
→∞
the Szego˝class. In[2], itwasshownthatE(n,w) cannotexceedClogn ifw >1andw Lp[ π,π],p< ,
∈ − ∞
and that this bound saturates. This leaves us with very natural question: what are regularity assumptions
onw thatguaranteeboundednessofE(n,w)? ThefollowingcorollaryofLemma1.1givesthepartialanswer.
Corollary 1.1. If w :w,w−1 BMO(T) and w =1, then
1
∈ k k
1 π
lim E(n,w)= log(2πw)dθ
n→∞ −4π
Z−π
So far, the only classes in which the E(n,w) was known to be bounded were the Baxter’s class [20]:
dσ =wdθ, w W(T),w >0 (W(T) denotes the Wiener algebra)or the class givenby positive weights with
∈
a certain modulus of continuity [24]. Our conditions are obviously much weaker and, in a sense, sharp.
The structure of the paper is as follows: the main results are proved in the next section, the Appendix
contains auxiliary results from harmonic analysis.
3
We use the following notation: H refers to the Hilbert transform, P denotes the L2(dθ) projection to
[i,j]
the (i,...,j) Fourier modes. Given two non-negative functions f we write f .f is there is an absolute
1(2) 1 2
constant C such that
f 6Cf
1 2
for all values of the arguments of f . We define & similarly and say that f f if f . f and f . f
1(2) 1 2 1 2 2 1
∼
simultaneously. Given two operators A and B, we write [A,B]=AB BA for their commutator. If w is a
−
function, then in the expressionlike [w,A], the symbol w is identified with the operatorof multiplicationby
w. The Hunt-Muckenhoupt-Wheeden characteristic of the weight w A will be denoted by [w] . For the
∈ p Ap
basic facts about the BMO class, A and their relationship, we refer the reader to, e.g., the classical text
p
[21]. If A is a linear operator from Lp(T) space to Lp(T), then A denotes its operator norm.
p,p
k k
2. Proofs of main results
Beforeprovingthemainresult,Theorem1.3,weneedsomeauxiliaryLemmas. Westartwiththefollowing
observation which goes back to S. Bernstein [6, 24].
Lemma 2.1. For a monic polynomial Q of degree n, we have:
Q(z)=Φ (z,w) if and only if P (wQ)=0. (6)
n [0,n−1]
Proof. It is sufficient to notice that (6) is equivalent to
π
Q(eiθ)e−ijθw(θ)dθ =0, j =0,...,n 1
−
Z−π
which is the orthogonality condition. (cid:3)
Lemma 2.2. If f L2(T) is real-valued function, Q L∞(T), then
∈ ∈
znP (fznQ)=P (fQ)
[0,n−1] [1,n]
In particular, for a polynomial P of degree at most n with P(0)=1, we have:
P(z)=Φ∗(z,w) if and only if P (wP)=0.
n [1,n]
Proof. The first statement is immediate. The second one follows from the Lemma above and the formula
Φ =znΦ∗, z T. (cid:3)
n n ∈
We have the following three identities for Φ∗(z,w); the first one was used in [8] recently. They are
n
immediately implied by the Lemma above.
Φ∗ =1+P (1 αw)Φ∗ , α R (7)
n [1,n] − n ∈
Φ∗ =1+(cid:16)w−1[w,P (cid:17)]Φ∗ (8)
n [1,n] n
Φ∗ =1 [w−1,P ](wΦ∗) (9)
n − [1,n] n
Denote the higher order commutators recursively:
C =P , C =[w,P ], C =[w,C ], l =2,3,...
0 [1,n] 1 [1,n] l l−1
Define the multiple commutators of w−1 and P (in that order!) by C .
[1,n] j
Lemma 2.3. The following representations hold
e
j
j 1
wjP Φ∗ = − C wj−lΦ∗ (10)
[1,n] n l 1 l n
l=1(cid:18) − (cid:19)
X
and
j
j
w−jP Φ∗ = C w−(j−l)(wΦ∗) (11)
[1,n] n − l l+1 n
l=0(cid:18) (cid:19)
X
where j =1,2,.... e
4
Proof. We will prove (10), the other formula can be obtained in the similar way. The case j = 1 of this
expressionis our familiar formula wP Φ∗ =[w,P ]Φ∗. Now the proof proceeds by induction. Suppose
[1,n] n [1,n] n
we have
k−1
k 2
wk−1P Φ∗ = − C wk−1−lΦ∗
[1,n] n l 1 l n
l=1(cid:18) − (cid:19)
X
Multiply both sides by w and write
k−1 k−1
k 2 k 2
wkP Φ∗ = − wC wk−1−lΦ∗ = − C wk−1−lΦ∗ +C wk−lΦ∗ =
[1,n] n l 1 l n l 1 l+1 n l n
l=1(cid:18) − (cid:19) l=1(cid:18) − (cid:19)
X X (cid:0) (cid:1)
k−1 k k
k 2 k 2 k 1
− C wk−lΦ∗ + − C wk−lΦ∗ = − C wk−lΦ∗
l 1 l n l 2 l n l 1 l n
l=1(cid:18) − (cid:19) l=2(cid:18) − (cid:19) l=1(cid:18) − (cid:19)
X X X
because
k 1 k 2 k 2
− = − + −
l 1 l 2 l 1
(cid:18) − (cid:19) (cid:18) − (cid:19) (cid:18) − (cid:19)
. (cid:3)
Motivatedby the previous Lemma, we introduce certainoperators. Givenf Lp, define y recursively
j
∈ { }
by
j−1
j 1
y =f, y =wj + − C y
0 j l+1 j−1−l
l
l=0(cid:18) (cid:19)
X
Then, we let
j
j
z =w−j C z
j l+1 j−l−1
− l
l=0(cid:18) (cid:19)
X
where z−1 = y1,z0 = y0 = f. Notice that for fixed j botheyj and zj are affine linear transformations in f.
We can write
y =y′ +y′′
j j j
where
y′ =f, y′′ =0
0 0
and, recursively,
j−1 j−1
j 1 j 1
y′ = − C y′ , y′′ =wj + − C y′′
j l l+1 j−1−l j l l+1 j−1−l
l=0(cid:18) (cid:19) l=0(cid:18) (cid:19)
X X
Similarly, we write z =z′ +z′′ where
j j j
z′ =y′, z′′ =y′′, z′ =f, z′′ =0
−1 1 −1 1 0 0
and
j j
j j
z′ = C z′ , z′′ =w−j C z′′ ,
j − l l+1 j−l−1 j − l l+1 j−l−1
l=0(cid:18) (cid:19) l=0(cid:18) (cid:19)
X X
Let us introduce linear operators: B ef =y′, D f =z′. We need an impeortant Lemma.
j j j j
Lemma 2.4.
wjΦ∗ =y′′+B Φ∗, w−jΦ∗ =z′′+D Φ∗
n j j n n j j n
Proof. This follows from
wjΦ∗ =wj +wjP Φ∗, w−jΦ∗ =w−j +w−jP Φ∗
n [1,n] n n [1,n] n
and the previous Lemma. (cid:3)
The next Lemma, in particular, provides the bounds for B and D .
j j
5
Lemma 2.5. Assume w>0, w =t, w−1 =s, w =1, and p [2,3]. Then,
BMO BMO 1
k k k k k k ∈
B 6(Ctj)j, D 6(1+st)(Csj)j
j p,p j p,p
k k k k
Moreover,
y′′ 6(Ctj)j, z′′ 6st(Csj)j
k jkp k jkp
with
e ee e
t=max t,1 , s=max s,1
{ } { }
Proof. We will prove the estimates for B and y′′ only, the bounds for D , z′′ are shown
e k jkp,p kejkp k jkp,p k jkp
similarly. By John-Nirenberg inequality ([21], p.144), we get
π ∞
w (2π)−1 jpdθ .j xjp−1exp( Cx/t)dx=j(Ct)jpΓ(jp)6(C tj)pj
1
| − | −
Z−π Z0
where Stirling’s formula was used for the gamma function Γ.
Since
wjp 6(w (2π)−1 +(2π)−1)jp 6Cjp(w (2π)−1 jp+1)
| | | − | | − |
we have
π
wjpdθ 6Cjp(1+(tj)jp)6(C tj)jp
1
| |
Z−π
Lemma 3.2 yields e
j−1 (j 1)! j−1 (Ct)−k
y′ 6 − (C(l+1)t)l+1 y′ 6(Ct)jj! y′
k jkp l!(j 1 l)! k j−1−lkp k! k kkp
l=0 − − k=0
X X
e y′
Divide both sides by (Ct)jj! and denote β = k jkp . Then,
j (Ct)jj!
j−1
β 6 β
j l
l=0
X
Since β = f , we have β 6 3j f by induction and thus y′ 6 (Ctj)j f . The estimates for
0 k kp j k kp k jkp k kp
y′′ , z′ , z′′ can be obtained similarly. (cid:3)
k jkp k jkp k jkp
Lemma 2.6. If w =1, w =t, w−1 =s, and p [2,3], then
1 BMO BMO
k k k k k k ∈
CΛ
min Λ−l B 6exp
l p,p
l∈N k k − t
(cid:16) (cid:17) (cid:18) (cid:19)
and
C
min ǫj D (1+st)exp
j p,p
j∈N k k ≤ −ǫs
(cid:16) (cid:17) (cid:18) (cid:19)
provided that Λ t and ǫ s−1.
≫ ≪
Proof. By the previous Lemma, we have
l
Ctl
Λ−l B 6
l p,p
k k Λ
(cid:16) (cid:17) (cid:18) (cid:19)
Optimizing in l we get l∗ CΛ/(te) and it gives the first estimate. The proof for the second one is
∼
identical. (cid:3)
Now we are ready to prove the main results of the paper.
6
Proof. (Theorem 1.3). Notice first that (4) follows from the Nikolskii inequality
Q <Cn1/p0 Q , degQ=n, p >2
k k∞ k kp0 0
as long as the Lp0 norms are estimated.
By scaling invariance, we can assume that w = 1. We consider two cases separately: st 1 and
1
k k ≫
st 1. The proofs will be different.
≪
1. The case st 1.
≫
Let p =2+δ with δ <1. Take two n-independent parameters ǫ and Λ such that ǫs 1 and Λt−1 1.
≪ ≫
Consider the following sets Ω = x:w6ǫ , Ω = x:ǫ<w <Λ ,Ω = x:w>Λ . Notice that
1 2 3
{ } { } { }
ǫs 1,tΛ−1 1 = (ǫs)(tΛ−1) 1 = ǫΛ−1 (st)−1 1 = ǫ Λ
≪ ≪ ⇒ ≪ ⇒ ≪ ≪ ⇒ ≪
From (7), we have
Φ∗ =1+P (1 w/Λ)Φ∗
n [1,n] − n
The idea of our proof is to rewrite this identity in the form
Φ∗ =f +O(n)Φ∗
n n n
where f <C(s,t) and O(n) is a contraction in Lp for the suitable choice of p. To this end, we consider
n p
k k
operators
w j
O (n)f =ǫjP (1 w/Λ)χ D f
1 [1,n] − Ω1 ǫ j
O (n)f =P (1 (cid:16)w/Λ(cid:17))χ f
2 [1,n] − Ω2
O (n)f =Λ−lP (1 w/Λ)(Λ/w)lχ B f
3 [1,n] − Ω3 l
where j and l will be fixed later, they will be n-independent. Let us estimate the (Lp,Lp) norms of these
operators. Since P 61+Cδ (see Lemma 3.1), we choose j and l as in Lemma 2.6 to ensure
[1,n] p,p
k k
C
O (n) 6stexp
1 p,p
k k −ǫs!
b
O (n) 6(1+Cδ)(1 ǫΛ−1)
2 p,p
k k −
CΛ
O (n) 6exp
3 p,p
k k − t !
b
Lemma 2.4 now yields
Φ∗ =1+f (n)+f (n)+(O (n)+O (n)+O (n))Φ∗
n 1 3 1 2 3 n
where
w j
f (n)=ǫjP (1 w/Λ)χ z′′, f (n)=Λ−lP (1 w/Λ)(Λ/w)lχ y′′
1 [1,n] − Ω1 ǫ j 3 [1,n] − Ω3 l
Let (cid:16) f((cid:17)n)=1+f (n)+f (n)
1 3
Then Lemma 2.5 provides the bound
f(n) 6C(s,t) (12)
p
k k
uniforminn. DenoteO(n)=O (n)+O (n)+O (n)andselectparametersǫ,Λ,δsuchthat O(n) <1 Cδ.
1 2 3 p,p
k k −
To do so, we first let δ =cǫΛ−1 with small positive absolute constant c. Then, we consider
C CΛ c ǫ
1
stexp +exp =
−ǫs! − t ! Λ
b b
with c again being a small constant. Now, solving equations
1
stexp( C/(ǫs))=exp( CΛ/t), c ǫ/Λ=2exp( CΛ/t)
1
− − −
we get the statement of the Theorem. Indeed, we have two equations:
b b b
Ct
ǫ=
s(CΛ+tlog(st))
b
7
b
and
Λ 1 Λ log(st)
= C+log(sΛ)+log +
t C t C
(cid:18) (cid:16) (cid:17)(cid:19)
Denote
b u=CΛ/t b
and then
b log(st)
u=C+log(st)+2logu+log 1+
u
Tofindtherequiredroot,werestricttherangeofutoc log(st)(cid:16)<u<c lo(cid:17)g(st)forc 1,c 1. Rewrite
1 2 1 2
≪ ≫
the equation above as
log(st)
u 2logu log 1+ =log(st)+C
− − u
(cid:18) (cid:19)
Differentiating the left hand side in u, we see that l.h.s.′ 1 within the given range. Therefore, there is
∼
exactly one solution u and u log(st). Then, since
∼
log(st)
log 1+
u
(cid:18) (cid:19)
is O(1), we get
u=log(st)+2logu+O(1)=log(st)+2loglog(st)+O(1)
by iteration. Thus,
ǫ 1
=Ce−u
Λ ∼ stlog2(st)
and δ 1 . Now that we proved that O(n) 61 Cδ <1, we can rewrite
∼ stlog2(st) k kp,p −
∞
Φ∗ =f(n)+ Oj(n)f(n)
n
j=1
X
and the series converges geometrically in Lp with tail being uniformly small in n due to (12).
To show that Φ∗ convergesin Lp as n , it is sufficient to prove that Oj(n)f(n) converges for each j.
n →∞
This, however,is immediate. Indeed,
P f P f, as n
[1,n] [1,∞]
→ →∞
in Lq for all f Lq,1 < q < . Since w,w−1 BMO Lp ([21], this again follows from John-
p>1
∈ ∞ ∈ ⊂ ∩
Nirenbergestimate),wesee thatmultiplicationby w±j mapsLp1 to Lp2 continuouslyby Ho¨lder’sinequality
provided that p <p and j Z. Therefore, if µ L∞,j =1,...,k, then
2 1 j
∈ ∈
µ w±j1P µ w±j2...µ w±jk−1P µ w±jk (13)
1 [1,n] 2 k−1 [1,n] k
has the limit in each Lp,p < when n . Notice that each f(n) and Oj(n)f(n) can be written as a
∞ → ∞
linear combination of expressions of type (13) ( µ taken as the characteristic functions). Now that δ is
j
{ }
chosen, we define p in the statement of the Theorem as p =2+δ.
0 0
2. The case st 1.
≪
The proof in this case is much easier. Let us start with two identities
Φ∗ =1+w−1[w,P ]Φ∗, Φ∗ =1+[P ,w−1]wΦ∗
n [1,n] n n [1,n] n
which can be recast as
wΦ∗ =w+[w,P ]Φ∗, Φ∗ =1+[P ,w−1]wΦ∗
n [1,n] n n [1,n] n
Substitution of the first formula into the second one gives
Φ∗ =1+[P ,w−1]w+G Φ∗
n [1,n] n n
where
G =[P ,w−1][w,P ]
n [1,n] [1,n]
We have
1+[P ,w−1]w 6C(s,t,p)
[1,n] p
k k
8
and
G .stp4
n p,p
k k
by Lemma 3.4. Taking p < p (st)−1/4 we have that G is a contraction. Now, the convergence of all
0 n
∼
terms in the geometric series can be proved as before. (cid:3)
Let us give a sketch of how the arguments can be modified to prove Theorem 1.4.
Proof. (Theorem 1.4). Consider the case w >1 first.
1. The case t 1.
≫
The proof is identical except that we can chose ǫ=1/2 so that Ω = . We get an equation for Λ
1
∅
C CΛ
=exp , Λ=C−1t(logΛ logC)
Λ − t ! −
b
b
Denote CΛ/t=u, then
u=logt+logu+O(1), u=logt+loglogt+O(1)
and δ b(tlogt)−1.
∼
2. The case t 1.
≪
We have
Φ∗ =1+L Φ∗, L f =w−1[w,P ]f
n n n n [1,n]
and Lemma 3.4 yields
L .p2t<0.5
n p,p
k k
for p<p =O(t−1/2).
0
The case w 6 1 can be handled similarly. When s is large, we take Λ = 1 in the proof of the previous
Theorem and get an equation for ǫ:
C
Cǫ=sexp
−ǫs!
b
Its solution for large s gives the required asymptotics for ǫ and, correspondingly,for δ and p . If s is small,
0
it is enough to consider the equation
Φ∗ =1 [w−1,P ]wΦ∗
n − [1,n] n
where the operator [w−1,P[1,n]]w is contraction in Lp0 for the specified p0. (cid:3)
Now we are ready to prove Corollary 1.1.
Proof. (of Corollary 1.1). The following inequality follows from the Mean Value Formula
x2logx y2logy 6C(1+x logx +y logy )x y , x,y >0
| − | | | | | | − |
Since w Lp, the Theorem 1.3 yields
p<∞
∈∩
π π
φ 2log φ S 2log S wdθ . (1+ φ logφ + SlogS ) φ∗ S wdθ 0, n
|| n| | n|−| | | || | n n| | | || n|−| || → →∞
Z−π Z−π
by applying the trivial bound: u logu 6 C(δ)(1+u1+δ),δ > 0 and the generalized Ho¨lder’s inequality to
| |
φ 1+δ(or S 1+δ), φ∗ S , and w. To conclude the proof, it is sufficient to notice that
| n| | | || n|−| ||
π 1 π
S 2log S wdθ = log(2πw)dθ
| | | | −4π
Z−π Z−π
because S −2 =2πw. (cid:3)
| |
9
3. Appendix
In this Appendix, we collect some auxiliary results used in the main text.
Lemma 3.1. For every p [2, ),
∈ ∞
P 61+C(p 2). (14)
[1,n] p,p
k k −
Proof. If P+ is the projection of L2(T) onto H2(T) (analytic Hardy space), then
P+ =0.5(1+iH)+P ,
0
where H is the Hilbert transform on the circle and P denotes the Fourier projection to the constants, i.e.,
0
P f =(2π)−1 f(x)dx (15)
0
T
Z
We therefore have a representation
P =zP+z−1 zn+1P+z−(n+1) =0.5i(zHz−1 zn+1Hz−(n+1))+zP z−1 zn+1P z−(n+1). (16)
[1,n] 0 0
− − −
Since H =cot(π/(2p)) [17], we have
p,p
k k
π
P 6cot +2 (17)
k [1,n]kp,p 2p
(cid:18) (cid:19)
by triangle inequality. On the other hand, P = 1 so by Riesz-Thorin theorem, we can interpolate
[1,n] 2,2
k k
between p=2 and, e.g., p=3 to get
P 61+C(p 2), p [2,3].
[1,n] p,p
k k − ∈
Noticing that cot(π/(2p)) p, p>3, we get the statement of the Lemma. (cid:3)
∼
Remark. In the proof above,we could have used the expressionfor the norm P+ obtained in [15].
p,p
k k
The proof of the following Lemma uses some standard results of Harmonic Analysis.
Lemma 3.2. If w =t and p [2,3], then we have
BMO
k k ∈
C 6(Cjt)j
j p,p
k k
Proof. Consider the following operator-valued function
F(z)=ezwP e−zw
[1,n]
If we can prove that F(z) is weakly analytic around the origin (i.e., analyticity of the scalar function
F(z)f ,f with fixed f C∞), then
1 2 1(2)
h i ∈
1 F(ξ)
F(z)= dξ, z B (0)
ǫ
2πi ξ z ∈
Z|ξ|=ǫ −
understood in a weak sense. By induction, one can then easily show the well-known formula
j! F(ξ)
C =∂jF(0)= dξ
j 2πi ξj+1
Z|ξ|=ǫ
which explains that we can control C by the size of F(ξ) on the circle of radius ǫ. Indeed,
j p,p p,p
k k k k
C = sup C f ,f
j p,p j 1 2
k k f1(2)∈C∞,kf1kp61,kf2kp′61|h i|≤
j! F(ξ)f ,f j!
sup h 1 2idξ 6 max F(ξ)
2π f1(2)∈C∞,kf1kp61,kf2kp′61(cid:12)(cid:12)Z|ξ|=ǫ ξj+1 (cid:12)(cid:12) ǫj |ξ|=ǫk kp,p
The weak analyticity of F(z) aroundthe origin(cid:12)follows immediately(cid:12)from, e.g., the John-Nirenbergestimate
(cid:12) (cid:12)
([21],p.144). Tobound F ,weusethefollow(cid:12) ingwell-knownresul(cid:12)t(whichisagainanimmediatecorollary
p,p
k k
from John-Nirenberg inequality, see, e.g., [21], p.218).
There is ǫ such that
0
w <ǫ = [ewe] 6[ewe] <C, p>2
k kBMO 0 ⇒ Ap A2
e 10