On sharp aperture-weighted estimates for square functions

0.2 MB·English

by  Andrei K. Lerner

ON SHARP APERTURE-WEIGHTED ESTIMATES FOR
SQUARE FUNCTIONS
3
1 ANDREI K. LERNER
0
2
Abstract. Let Sα,ψ(f) be the square function defined by means
n of the cone in Rn+1 of aperture α, and a standard kernel ψ. Let
a +
J [w]Ap denote the Ap characteristic of the weight w. We show that
8 for any 1<p< and α 1,
∞ ≥
1 kSα,ψkLp(w) .αn[w]mApax(12,p−11).
]
A Foreachfixedαthedependenceon[w]Ap issharp. Also,onallclass
A the result is sharp in α. Previously this estimate was proved
C p
in the case α = 1 using the intrinsic square function. However,
.
h thatapproachdoesnotallowtogetthe aboveestimate withsharp
at dependence on α. Hence we give a different proof suitable for all
m α 1 and avoiding the notion of the intrinsic square function.
≥
[

2
v
1 1. Introduction
5
0 Let ψ be an integrable function, ψ = 0, and, for some ε > 0,
1 Rn
.
1 c R
(1.1) ψ(x) and ψ(x+h) ψ(x) dx c h ε.
0
| | ≤ (1+ x )n+ε | − | ≤ | |
3 | | ZRn
1 Let Rn+1 = Rn R and Γ (x) = (y,t) Rn+1 : y x < αt . Set
v: + × + α { ∈ + | − | }
ψ (x) = t−nψ(x/t). Define the square function S (f) by
i t α,ψ
X
1/2
r dydt
a S (f)(x) = f ψ (y) 2 (α > 0).
α,ψ | ∗ t | tn+1
(cid:18)ZΓα(x) (cid:19)
We drop the subscript α if α = 1.
Given a weight w, define its A characteristic by
p
p−1
1 1
[w]Ap = sup Q wdx Q w−p−11 dx ,
Q (cid:18)| | ZQ (cid:19)(cid:18)| | ZQ (cid:19)
where the supremum is taken over all cubes Q Rn.
⊂
2010 Mathematics Subject Classification. 42B20,42B25.
Keywordsandphrases. Littlewood-Paleyoperators,sharpweightedinequalities,
sharp aperture dependence.
1
2 ANDREIK. LERNER
It was proved in [13] that for any 1 < p < ,
∞
max(1, 1 )
(1.2) S c [w] 2 p−1 ,
k ψkLp(w) ≤ p,n,ψ Ap
and this estimate is sharp in terms of [w] (we also refer to [13] for a
Ap
detailed history of closely related results).
Similarly one can show that
max(1, 1 )
(1.3) S c γ(α)[w] 2 p−1 (α 1,1 < p < );
k α,ψkLp(w) ≤ p,n,ψ Ap ≥ ∞
however, the sharp dependence on α in this estimate cannot be deter-
mined by means of the approach from [13]. The aim of this paper is to
find the sharp γ(α) in (1.3).
Let us explain first why the method from [13] gives a rough estimate
for γ(α). The proof in [13] was based on the intrinsic square function
G (f) by M. Wilson [19] defined as follows. For 0 < β 1, let be
α,β β
≤ C
the family of functions supported in the unit ball with mean zero and
such that for all x and x′, ϕ(x) ϕ(x′) x x′ β. If f L1 (Rn)
and (y,t) Rn+1, we define| A (f−)(y,t) =| ≤sup| − f| ϕ (y)∈anldoc
∈ + β ϕ∈Cβ | ∗ t |
1/2
dydt
2
G (f)(x) = A (f)(y,t) .
α,β β tn+1
(cid:18)ZΓα(x) (cid:19)
(cid:0) (cid:1)
Set G (f) = G (f).
1,β β
The intrinsic square function has several interesting features (estab-
lished in [19]). First, though G (f) is defined by means of kernels with
β
uniform compact support, it pointwise dominates S (f). Also there is
ψ
a pointwise relation between G (f) with different apertures:
α,β
(1.4) G (f)(x) α(3/2)n+βG (f)(x) (α 1).
α,β β
≤ ≥
Notice that for the usual square functions S (f) such a pointwise
α,ψ
relation is not available.
In [13], (1.2) with G (f) instead of S (f) was obtained. Combining
β ψ
this with (1.4), we would obtain that one can take γ(α) = α(3/2)n+β
in (1.3) assuming that ψ is compactly supported. For non-compactly
supported ψ some additional ideas from [19] can be used that lead to
even worst estimate on γ(α). Observe also that it is not clear to us
whether (1.4) can be improved.
It is easy to see that the dependence γ(α) = α(3/2)n+β in (1.3) is far
from the sharp one. For instance, it is obvious that the information on
β should not appear in (1.3). All this indicates that the intrinsic square
function approach is not suitable for our purposes in determining the
sharp γ(α).
SHARP APERTURE-WEIGHTED ESTIMATES 3
Suppose we seek for γ(α) in the form γ(α) = αr. Then a simple
observation shows that r n for any 1 < p < . Indeed, consider the
≥ ∞
Littlewood-Paley function g∗ (f) defined by
µ,ψ
µn 1/2
t dydt
g∗ (f)(x) = f ψ (y) 2 .
µ,ψ t+ x y | ∗ t | tn+1
Rn+1
(cid:18)ZZ + (cid:18) | − |(cid:19) (cid:19)
Using the standard estimate
∞
g∗ (f)(x) S (f)(x)+ 2−kµn/2S (f)(x),
µ,ψ ≤ ψ 2k+1,ψ
k=0
X
we obtain that (1.3) for some p = p and γ(α) = αr0 implies
0
∞
max(1, 1 )
(1.5) kgµ∗,ψkLp0(w) . 2−kµn/22kr0 [w]Ap0 2 p0−1 .
(cid:16)Xk=0 (cid:17)
Thismeansthatifµ > 2r /n,theng∗ isboundedonLp0(w),w A .
0 µ,ψ ∈ p0
From this, by the Rubio de Francia extrapolation theorem, g∗ is
µ,ψ
bounded on the unweighted Lp for any p > 1, whenever µ > 2r /n.
0
But it is well known [8] that g∗ is not bounded on Lp if 1 < µ < 2
µ,ψ
and 1 < p 2/µ. Hence, if r < n, we would obtain a contradiction to
0
≤
the latter fact for p sufficiently close to 1.
Our main result shows that for any 1 < p < one can take the
∞
optimal power growth γ(α) = αn.
Theorem 1.1. For any 1 < p < and for all 1 α < ,
∞ ≤ ∞
max(1, 1 )
S c αn[w] 2 p−1 .
k α,ψkLp(w) ≤ p,n,ψ Ap
By (1.5), we immediately obtain the following.
Corollary 1.2. Let µ > 2. Then for any 1 < p < ,
∞
max(1, 1 )
g∗ (f) c [w] 2 p−1 .
k µ,ψ kLp(w) ≤ p,n,µ,ψ Ap
Observe that if µ = 2, then g∗ is also bounded on Lp(w) for w A
2,ψ ∈ p
(see [17]). However, the sharp dependence on [w] in the correspond-
Ap
ing Lp(w) inequality is unknown to us.
We emphasize that the growth γ(α) = αn is best possible in the
weighted Lp(w) estimate for w A . In the unweighted case a better
p
∈ n
dependenceonαisknown, namely, Sα,ψ Lp cp,n,ψαmin(p,2),see[1,18].
k k ≤
Some words about the proof of Theorem 1.1. As in [13], we use here
the local mean oscillation decomposition. But in [13] we worked with
the intrinsic square function, and due to the fact that this operator
4 ANDREIK. LERNER
is defined by uniform compactly supported kernels, we arrived to the
operator
1/2
(f)(x) = (f )2χ (x) ,
A γQkj Qkj
(cid:16)Xj,k (cid:17)
where Qk is a sparse family and γ > 1. This operator can be handled
j
sufficiently easy.
Here we work with the square function S (f) directly, more pre-
α,ψ
cisely we consider its smooth variant S (f). Applying the local mean
α,ψ
oscillation decomposition to S (f), we obtain that S (f) is essen-
α,ψ α,ψ
tially pointwise bounded by αn (f), ewhere
B
e
∞
1 1/2
(f)(x) = (f )2χ (x) (δ > 0).
B 2mδ 2mQkj Qkj
mX=0 (cid:16)Xj,k (cid:17)
Observe that this pointwise aperture estimate is interesting in its own
right. In order to handle , we use a mixture of ideas from recent
B
papers on a simple proof of the A conjecture [14] and sharp weighted
2
estimates for multilinear Caldero´n-Zygmund operators [5]. In particu-
lar, similarly to [14], we obtain the X(2)-norm boundedness of by
B A
on an arbitrary Banach function space X.
Thepaperisorganizedasfollows. Nextsectioncontainssomeprelim-
inary information. In Section 3, we obtain the main estimate, namely,
the local mean oscillation estimate of S (f). The proof of Theo-
α,ψ
rem 1.1 is contained in Section 4. Section 5 contains some concluding
remarks concerning the sharp aperture-weeighted weak type estimates
for S (f).
α,ψ
2. Preliminaries
2.1. A weak type (1,1) estimate for square functions. It is well
known that the operator S is of weak type (1,1). However, we could
α,ψ
not find in the literature the sharp dependence on α in the correspond-
ing inequality. Hence we give below an argument based on general
square functions.
For a measurable function F on Rn+1 define
+
1/2
dydt
S (F)(x) = F(y,t) 2 .
α | | tn+1
(cid:18)ZΓα(x) (cid:19)
Lemma 2.1. For any α 1,
≥
(2.1) Sα(F) L1,∞ cnαn S1(F) L1,∞.
k k ≤ k k
SHARP APERTURE-WEIGHTED ESTIMATES 5
Proof. We will use the following estimate which can be found in [18,
p.315]: ifΩ Rn isanopensetandU = x Rn : Mχ (x) > 1/2αn ,
Ω
⊂ { ∈ }
where M is the Hardy-Littlewood maximal operator, then
S (F)(x)2dx 2αn S (F)(x)2dx
α 1
≤
ZRn\U ZRn\Ω
(observe that the definitions of S (F) here and in [18] are differ by the
α
factor αn/2.)
Let Ω = x : S (F)(x) > ξ . Using the weak type (1,1) of M,
ξ 1
{ }
Chebyshev’s inequality and the above estimate, we obtain
x Rn : S (F)(x) > ξ
α
|{ ∈ }|
U + x Rn U : S (F)(x) > ξ
ξ ξ α
≤ | | |{ ∈ \ }|
1
c αn x : S (F)(x) > ξ + S (F)(x)2dx
≤ n |{ 1 }| ξ2 α
ZRn\Uξ
2αn
c αn x : S (F)(x) > ξ + S (F)(x)2dx.
≤ n |{ 1 }| ξ2 1
ZRn\Ωξ
Further,
ξ
S1(F)(x)2dx 2 λ x : S1(F)(x) > λ dλ 2ξ S1(F) L1,∞.
≤ |{ }| ≤ k k
ZRn\Ωξ Z0
Combining this with the previous estimate gives
4αn
x : Sα(F)(x) > ξ cnαn x : S1(F)(x) > ξ + S1(F) L1,∞,
|{ }| ≤ |{ }| ξ k k
which proves (2.1). (cid:3)
Note that the sharp strong Lp estimates related square functions of
different apertures were obtained recently in [1].
By Lemma 2.1 and by the weak type (1,1) of S (f) [9],
ψ
(2.2) Sα,ψ(f) L1,∞ cn,ψαn f L1.
k k ≤ k k
2.2. Dyadic grids and sparse families. Recall that the standard
dyadic grid in Rn consists of the cubes
2−k([0,1)n +j), k Z,j Zn.
∈ ∈
Denote the standard grid by .
By a general dyadic grid DDwe mean a collection of cubes with the
following properties: (i) for any Q D its sidelength ℓ is of the form
Q
2k,k Z; (ii) Q R Q,R, fo∈r any Q,R D; (iii) the cubes of a
fixed∈sidelength 2∩k fo∈rm{a part∅i}tion of Rn. ∈
Given a cube Q , denote by (Q ) the set of all dyadic cubes with
0 0
D
respect to Q , that is, the cubes from (Q ) are formed by repeated
0 0
D
6 ANDREIK. LERNER
subdivision of Q and each of its descendants into 2n congruent sub-
0
cubes. Observe that if Q D, then each cube from (Q ) will also
0 0
belong to D. ∈ D
We will use the following proposition from [10].
Proposition 2.2. There are 2n dyadic grids D such that for any cube
i
Q Rn there exists a cube Q D such that Q Q and ℓ 6ℓ .
⊂ i ∈ i ⊂ i Qi ≤ Q
We say that Qk is a sparse family of cubes if: (i) the cubes Qk are
{ j} j
disjoint in j, with k fixed; (ii) if Ω = Qk, then Ω Ω ; (iii)
k ∪j j k+1 ⊂ k
Ω Qk 1 Qk .
| k+1 ∩ j| ≤ 2| j|
2.3. A “localmean oscillation decomposition”. Thenon-increasing
rearrangement of a measurable function f on Rn is defined by
f∗(t) = inf α > 0 : x Rn : f(x) < α < t (0 < t < ).
{ |{ ∈ | | }| } ∞
Given a measurable function f on Rn and a cube Q, the local mean
oscillation of f on Q is defined by
∗
ω (f;Q) = inf (f c)χ λ Q (0 < λ < 1).
λ Q
c∈R − | |
By a median value of f(cid:0)over Q we(cid:1)m(cid:0)ean a(cid:1) possibly nonunique, real
number m (Q) such that
f
max x Q : f(x) > m (Q) , x Q : f(x) < m (Q) Q /2.
f f
|{ ∈ }| |{ ∈ }| ≤ | |
It i(cid:0)s easy to see that the set of all median values of f is(cid:1)either one
point or the closed interval. In the latter case we will assume for the
definiteness that m (Q) is the maximal median value. Observe that it
f
follows from the definitions that
(2.3) m (Q) (fχ )∗( Q /2).
f Q
| | ≤ | |
Given a cube Q , the dyadic local sharp maximal function m#,d f is
0 λ;Q0
defined by
m#,d f(x) = sup ω (f;Q′).
λ;Q0 x∈Q′∈D(Q0) λ
The following theorem was proved in [15] (its very similar version
can be found in [12]).
Theorem 2.3. Let f be a measurable function on Rn and let Q be a
0
fixed cube. Then there exists a (possibly empty) sparse family of cubes
Qk (Q ) such that for a.e. x Q ,
j ∈ D 0 ∈ 0
f(x) m (Q ) 4m#,d f(x)+2 ω (f;Qk)χ (x).
| − f 0 | ≤ 2n1+2;Q0 2n1+2 j Qkj
k,j
X
SHARP APERTURE-WEIGHTED ESTIMATES 7
3. A key estimate
Inthissectionwewillobtainthemainlocalmeanoscillationestimate
of S . We consider a smooth version of S defined as follows. Let
α,ψ α,ψ
Φ be a Schwartz function such that
χ (x) Φ(x) χ (x).
B(0,1) B(0,2)
≤ ≤
Define
1/2
x y dydt
S (f)(x) = Φ − f ψ (y) 2 (α > 0).
α,ψ t
Rn+1 tα | ∗ | tn+1!
ZZ + (cid:16) (cid:17)
It iseeasy to see that
S (f)(x) S (f)(x) S (f)(x).
α,ψ α,ψ 2α,ψ
≤ ≤
Hence, by (2.2),
e
(3.1) Sα,ψ(f) L1,∞ cn,ψαn f L1.
k k ≤ k k
Lemma 3.1. For any cube Q Rn,
e ⊂
∞ 2
1 1
(3.2) ω (S (f)2;Q) c α2n f ,
λ α,ψ ≤ n,λ,ψ 2kδ 2kQ | |
k=0 (cid:18)| | Z2kQ (cid:19)
X
where δ = ε froem condition (1.1) if ε < 1, and δ < 1 if ε = 1.
Proof. Given a cube Q, let T(Q) = (y,t) : y Q,0 < t < ℓ , where
Q
{ ∈ }
ℓ denotes the side length of Q. For x Q we decompose S (f)(x)2
Q α,ψ
∈
into the sum of
x y dydt e
I (f)(x) = Φ − f ψ (y) 2
1 t
tα | ∗ | tn+1
ZZT(2Q) (cid:16) (cid:17)
and
x y dydt
I (f)(x) = Φ − f ψ (y) 2 .
2 t
tα | ∗ | tn+1
ZZRn++1\T(2Q) (cid:16) (cid:17)
Let us show first that
∞ 2
1 1
(3.3) (I (f)χ )∗(λ Q ) c α2n f .
1 Q | | ≤ n,λ,ψ 2kε 2kQ | |
k=0 (cid:18)| | Z2kQ (cid:19)
X
Using that (a+b)2 2(a2 +b2), we get
≤
I1(f)(x) 2 I1(fχ4Q)(x)+I1(fχRn\4Q)(x) .
≤
Hence,
(cid:0) (cid:1)
(3.4) (I (f)χ )∗(λ Q ) 2 (I (fχ ))∗(λ Q /2)
1 Q 1 4Q
| | ≤ | |
+ (I(cid:0)1(fχRn\4Q)χQ)∗(λ Q /2) .
| |
(cid:1)
8 ANDREIK. LERNER
By (3.1),
(3.5) (I (fχ ))∗(λ Q /2) (S (fχ ))∗(λ Q /2)2
1 4Q α,ψ 4Q
| | ≤ | |
2
1
cne,λ,ψα2n f .
≤ 4Q | |
(cid:18)| | Z4Q (cid:19)
Further, by (1.1), for (y,t) T(2Q),
∈
1
|(fχRn\4Q)∗ψt(y)| ≤ cψtε |f(ξ)|(t+ y ξ )n+εdξ
ZRn\4Q | − |
∞
1 1
c (t/ℓ )ε f .
≤ n,ψ Q 2kε 2kQ | |
k=0 | | Z2kQ
X
Hence, usingChebyshev’s inequalityandthat Φ x−y dx c (tα)n,
Rn tα ≤ n
we have (cid:16) (cid:17)
R
(I1(fχRn\4Q)χQ)∗(λ Q /2)
| |
2 x y dydt
≤ λ Q Φ t−α dx |(fχRn\4Q)∗ψt(y)|2tn+1
| | ZZT(2Q)(cid:16)ZRn (cid:16) (cid:17) (cid:17)
∞ 1 1 2 2ℓQ
c αn(1/ℓ )2ε f t2ε−1dt
≤ n,λ,ψ Q 2kε 2kQ | |
k=0 | | Z2kQ ! Z0
X
∞ 2
1 1
c αn f .
≤ n,λ,ψ 2kε 2kQ | |
k=0 | | Z2kQ !
X
By H¨older’s inequality,
∞ 2 ∞ ∞ 2
1 1 1 1 1
f f .
2kε 2kQ | | ≤ 2kε 2kε 2kQ | |
k=0 | | Z2kQ ! k=0 !k=0 (cid:18)| | Z2kQ (cid:19)
X X X
Combining this with the previous estimate and with (3.5) and (3.4)
proves (3.3).
Let x,x Q, and let us estimate now I (f)(x) I (f)(x ) . We
0 2 2 0
∈ | − |
have
I (f)(x) I (f)(x )
2 2 0
| − |
∞
x y x y dydt
Φ − Φ 0 − f ψ (y) 2 .
t
≤ tα − tα | ∗ | tn+1
Xk=1ZZT(2k+1Q)\T(2kQ)(cid:12) (cid:16) (cid:17) (cid:16) (cid:17)(cid:12)
(cid:12) (cid:12)
Suppose (y,t) T(2k+1Q(cid:12)) T(2kQ). If y 2kQ, t(cid:12)hen t 2kℓ . On
Q
∈ \ ∈ ≥
the other hand, if y 2k+1Q 2kQ, then for any x Q, y x
∈ \ ∈ | − | ≥
SHARP APERTURE-WEIGHTED ESTIMATES 9
(2k 1/2)ℓ . Hence, if t < 1 (2k 1/2)ℓ , then y x /αt > 2 and
− Q 2α − Q | − |
y x /αt > 2, and therefore,
0
| − |
x y x y
0
Φ − Φ − = 0.
tα − tα
(cid:16) (cid:17) (cid:16) (cid:17)
Using also that
x y x y √nℓ
0 Q
Φ − Φ − Φ L∞,
tα − tα ≤ αt k∇ k
(cid:12) (cid:16) (cid:17) (cid:16) (cid:17)(cid:12)
we get
(cid:12) (cid:12)
(cid:12) (cid:12)
x y x y
0
Φ − Φ − χ (y,t)
tα − tα {T(2k+1Q)\T(2kQ)}
(cid:12) (cid:16) ℓ (cid:17) (cid:16) (cid:17)(cid:12)
(cid:12) Q (cid:12)
c χ (y,t).
(cid:12)≤ nαt {(y,t):y∈2k+1Q,2k−2ℓQ(cid:12)/α≤t≤2k+1ℓQ}
Hence,
x y x y dydt
Φ − Φ 0 − f ψ (y) 2
tα − tα | ∗ t | tn+1
ZZT(2k+1Q)\T(2kQ)(cid:12) (cid:16) (cid:17) (cid:16) (cid:17)(cid:12)
ℓ 2k+1ℓQ (cid:12) dydt (cid:12)
c Q (cid:12) f ψ (y) 2 (cid:12)c (J +J ),
≤ n α | ∗ t | tn+2 ≤ n 1 2
Z2k−2ℓQ/αZ2k+1Q
where
ℓ 2k+1ℓQ dydt
J = Q (fχ ) ψ (y) 2
1 α | 2k+2Q ∗ t | tn+2
Z2k−2ℓQ/αZ2k+1Q
and
ℓ 2k+1ℓQ dydt
J = Q (fχ ) ψ (y) 2 .
2 α | Rn\2k+2Q ∗ t | tn+2
Z2k−2ℓQ/αZ2k+1Q
Let us first estimate J . Using Minkowski’s integral inequality, we
1
obtain
2
ℓ 2k+1ℓQ dydt 1/2
J Q f(ξ) ψ (y ξ)2 dξ .
1 ≤ α | | t − tn+2
Z2k+2Q (cid:16)Z2k−2ℓQ/αZ2k+1Q (cid:17) !
Since
ψ (y ξ)2dy kψkL∞ ψ = kψkL∞kψkL1,
t − ≤ tn k tkL1 tn
Z2k+1Q
we get
ℓ 2 ∞ dt
Q
J c f(ξ) dξ
1 ≤ ψ α | | t2n+2
(cid:16)Z2k+2Q (cid:17) Z2k−2ℓQ/α
1 2
c α2n2−k f(ξ) dξ .
≤ n,ψ 2k+2Q | |
(cid:16)| | Z2k+2Q (cid:17)
10 ANDREIK. LERNER
We turn to the estimate of J . By (1.1), for (y,t) T(2k+1Q),
2
∈
1
(fχ ) ψ (y) c tε f(ξ) dξ
| Rn\2k+2Q ∗ t | ≤ ψ | |(t+ y ξ )n+ε
ZRn\2k+2Q | − |
∞
1 1
c (t/ℓ )ε f .
≤ n,ψ Q 2iε 2iQ | |
i=k | | Z2iQ
X
Therefore,
ℓ ∞ 1 1 2 1 2k+1ℓQ dydt
Q
J c f
2 ≤ n,ψ α 2iε 2iQ | | ℓ2ε tn+2−2ε
(cid:16)Xi=k | | Z2iQ (cid:17) Q Z2k−2ℓQ/αZ2k+1Q
∞
1 1 2
c αn−2ε2(2ε−1)k f .
≤ n,ψ 2iε 2iQ | |
(cid:16)Xi=k | | Z2iQ (cid:17)
Combining the estimates for J and J , we obtain
1 2
∞
1 1 2
I (f)(x) I (f)(x ) c α2n f(ξ) dξ
2 2 0 n,ψ
| − | ≤ 2k 2kQ | |
Xk=1 (cid:16)| | Z2kQ (cid:17)
∞ 22εk ∞ 1 1 2
+ c αn−2ε f .
n,ψ 2k 2iε 2iQ | |
Xk=1 (cid:16)Xi=k | | Z2iQ (cid:17)
By H¨older’s inequality,
∞ 22εk ∞ 1 1 2
f
2k 2iε 2iQ | |
k=1 i=k | | Z2iQ !
X X
∞ 2εk ∞ 1 1 2
c f
≤ ε 2k 2iε 2iQ | |
k=1 i=k (cid:18)| | Z2iQ (cid:19)
X X
∞ 2
1
c γ(k,ε) f ,
ε
≤ 2kQ | |
k=1 (cid:18)| | Z2kQ (cid:19)
X
where
1 , ε < 1
γ(k,ε) = 2εk
k , ε = 1.
(2k
Therefore,
∞ 2
1
I (f)(x) I (f)(x ) c α2n γ(k,ε) f .
2 2 0 n,ψ
| − | ≤ 2kQ | |
k=1 (cid:18)| | Z2kQ (cid:19)
X