On maximal functions for Mikhlin-Hoermander multipliers

0.21 MB·

by  Loukas Grafakos

ON MAXIMAL FUNCTIONS FOR
MIKHLIN-HO¨RMANDER MULTIPLIERS
5
0
0 LOUKASGRAFAKOS,PETR HONZ´IK,ANDREASSEEGER
2

n Abstract. Given Mikhlin-H¨ormander multipliers m , i = 1,...,N,
i
a
with uniform estimates we prove an optimal log(N+1) bound in Lp
J
for themaximal function sup |F−1[m f]| and related boundsfor maxi-
8 i i p
mal functions generated by dilations. These improveresults in [7].

] b
A
C
. 1. Introduction
h
t
a Given a symbol m satisfying
m
(1.1) ∂αm(ξ) C ξ −α
α
[ | | ≤ | |
for all multiindices α, then by classical Calder´on-Zygmund theory the oper-
1
ator f −1[mf] defines an Lp bounded operator. We study two types of
v
7→ F
4 maximal operators associated to such symbols.
1 First we considber N multipliers m ,...,m satisfying uniformly the con-
1 N
1
ditions (1.1) and ask for bounds
1
0 (1.2) sup −1[m f] A(N) f ,
5 |F i | p ≤ k kp
1≤i≤N
0
(cid:13) (cid:13)
/ for all f . (cid:13) b (cid:13)
h ∈S
Secondlyweformtwomaximalfunctionsgeneratedbydilationsofasingle
t
a
multiplier,
m
: (1.3) dyadf(x)= sup −1[m(2k )f]
v Mm k∈Z|F · |
i
X (1.4) f(x)= sup −1[m(t )f]b
m
r M t>0 |F · |
a
and ask under what additional conditions on m thbese define bounded ope-
rators on Lp.
Concerning (1.3), (1.4) a counterexample in [7] shows that in general
additional conditions on m are needed for the maximal inequality to hold;
moreover positive results were shown using rather weak decay assumptions
on m. The counterexample also shows that the optimal uniform bound in
(1.2) satisfies
(1.5) A(N) c log(N +1).
≥
Date: October26, 2004. p
Grafakos and Seegerweresupportedin partbyNSFgrants. Honz´ık wassupportedby
201/03/0931 Grant Agency of theCzech Republic.
1
2 LOUKASGRAFAKOS,PETRHONZ´IK,ANDREASSEEGER
The extrapolation argument in [7] only gives the upper bound A(N) =
O(log(N +1)) and the main purpose of this paper is to close this gap and
to show that the upper bound is indeed O( log(N +1)).
We will formulate our theorems with minimal smoothness assumptions
p
that will be described now.
Let φ C∞(Rd) be supported in ξ :1/2 < ξ < 2 so that
∈ 0 { | | }
φ(2−kξ)= 1
k∈Z
X
for all ξ Rd 0 . Let η C∞(Rd) so that η is even, η (x) = 1
∈ \ { } 0 ∈ c 0 0
for x 1/2 and η is supported where x 1. For ℓ > 0 let η (x) =
0 ℓ
| | ≤ | | ≤
η (2−ℓ(x)) η (2−ℓ+1x) and define
0 0
−
H [m](x) = η (x) −1[φm(2k )](x).
k,ℓ ℓ
F ·
In what follows we set
m := sup 2ℓα H [m] .
Y(q,α) k,ℓ Lq
k k k∈Z k k
ℓ≥0
X
Using the Hausdorff-Young inequality one gets
(1.6) kmkY(r′,α) . sku∈pZkφm(2k·)kBαr,1, if 1 ≤r ≤ 2
where Br is the usual Besov space (cf. Lemma 3.3 below). Thus if m
α,1
belongsto Y(2,d/2) thenm is aFourier multiplier of Lp(Rd), for 1 < p <
∞
(this follows from a slight modification of Stein’s approach in [16], ch. IV.3,
see also [15] for a related endpoint bound).
Theorem 1.1. Suppose that 1 r < 2 and suppose that the multipliers m ,
i
≤
i= 1,...,N satisfy the condition
(1.7) sup mi Y(r′,d/r) B < .
k k ≤ ∞
i
Then for r < p <
∞
sup −1[m f] C B log(N +1) f .
F i p ≤ p,r k kp
i=1,...,N
(cid:13) (cid:12) (cid:12)(cid:13) p
By (1.6) w(cid:13)e immed(cid:12)iately getb(cid:12)(cid:13)
Corollary 1.2. Suppose that 1 < r < 2, and
(1.8) sup sup φmi(t ) Br A.
1≤i≤N t>0 k · k d/r,1 ≤
Then for r < p <
∞
sup −1[m f] C A log(N +1) f .
F i p ≤ p,r k kp
i=1,...,N
(cid:13) (cid:12) (cid:12)(cid:13) p
(cid:13) (cid:12) b(cid:12)(cid:13)
ON MAXIMAL FUNCTIONS FOR MIKHLIN-HO¨RMANDER MULTIPLIERS 3
Remark. If one uses Y( ,d + ε) in (1.7) or B1 in (1.8) one can use
∞ d+ε,1
Calder´on-Zygmund theory (see [8], [7]) to prove the H1 L1 boundedness
−
and the weak type (1,1) inequality, both with constant O( log(N +1)).
dyadp
Our second result is concerned with the operators , generated
m m
M M
by dilations.
Theorem 1.3. Suppose 1 < p< , q = min p,2 .
∞ { }
(i) Suppose that
(1.9) kφm(2k·)kLqα ≤ ω(k), k ∈ Z,
holds for α > d/q and suppose that the nonincreasing rearrangement ω∗
satisfies
∞ ω∗(l)
(1.10) ω∗(0)+ < .
l√logl ∞
l=2
X
Then dyad is bounded on Lp(Rd).
m
M
(ii) Suppose that (1.10) holds and (1.9) holds for α > d/p+1/p′ if 1 <
p 2 or for α > d/2+1/p if p > 2. Then is bounded on Lp(Rd).
m
≤ M
If (1.9), (1.10) are satisfied with q = 1, α > d then is of weak type
m
M
(1,1), and maps H1 to L1.
m
M
Thisimproves theearlier resultin[7]wheretheconclusion isobtainedun-
der the assumption ∞ ω∗(l)/l < , however somewhat weaker smooth-
l=2 ∞
ness assumptions were made in [7].
P
In 2 we shall discuss model cases for Rademacher expansions. In 3 we
§ §
shall give the outline of the proof of Theorem 1.1 which is based on the
exp(L2) estimate by Chang-Wilson-Wolff [5], for functions with bounded
Littlewood-Paley square-function. The proof of a critical pointwise inequal-
ity is given in 4. The proof of Theorem 1.3 is sketched in 5. Some open
§ §
problems are mentioned in 6.
§
Acknowledgement: The second named author would like to thank Luboˇs
Pick for a helpful conversation concerning convolution inequalities in re-
arrangement invariant function spaces.
2. Dyadic model cases for Rademacher expansions
Before we discuss the proof of Theorem 1.1 we give a simple result on
expansions for Rademacher functions r on [0,1] which motivated the proof.
j
Proposition 2.1. Let ai ℓ2. and let
∈
F (s) = air (s), s [0,1].
i j j ∈
j
X
Then
sup F . sup ai log(N +1).
| i| L2[0,1] k kℓ2
i<N
(cid:13) (cid:13) p
(cid:13) (cid:13)
4 LOUKASGRAFAKOS,PETRHONZ´IK,ANDREASSEEGER
Proof. We use the well known estimate for the distribution function of the
Rademacher expansions ([16], p. 277),
(2.1) meas s [0,1] : F (s) >λ 2exp λ2
{ ∈ | i | } ≤ − 4kaik2ℓ2
Set u = (4log(N(cid:0)+1))1/2sup ai (cid:1). Then (cid:0) (cid:1)
N 1≤i≤N ℓ2
k k
N ∞
sup F 2 u2 +2 λmeas s : F (s) > λ dλ
| i| 2 ≤ N { | i | }
i=1,...,N i=1ZuN
(cid:13) (cid:13) X (cid:0) (cid:1)
(cid:13) (cid:13)N ∞
≤ u2N +4 λe−λ2/(4kaik2ℓ2)dα ≤ u2N +4 sup kaik2ℓ2Ne−u2N/4
i=1ZuN i=1,...,N
X
which is bounded by (1+4log(N +1))sup ai 2 . The claim follows. (cid:3)
ik kℓ2
There is a multiplier interpretation to this inequality. One can work
with a single function f = a r and a family of bounded sequences (or
j j
multipliers) bi and one forms F (s) = bia r (s). The norm then grows
{ } P i j j j j
as a square root of the logarithm of the number of multipliers; i.e. we have
P
Corollary 2.2.
sup bia r . sup bi log(N +1) a r .
i=1,...,N j j j L2([0,1]) i k k∞ j j L2([0,1])
(cid:13) (cid:12)Xj (cid:12)(cid:13) p (cid:13)Xj (cid:13)
(cid:13) (cid:13) (cid:13) (cid:13)
(cid:12) (cid:12)
(cid:13)We shall now consi(cid:13)der a dyadic model case for the(cid:13)maximal(cid:13)operators
generated by dilations.
Proposition 2.3. Consider a sequence b = bi i∈Z which satisfies
{ }
A
b∗(l) .
≤ (log(l+2))1/2
Then for any sequence a = a ∞ we have
{ n}n=1
∞
sup b a r CA a .
j−k j j 2
k∈Z 2 ≤ k k
(cid:13) (cid:12)Xj=0 (cid:12)(cid:13)
(cid:13) (cid:13)
(cid:12) (cid:12)
Proof. We may assum(cid:13)e that both a and(cid:13)b are real valued sequences. Let
∞
H (s)= b a r (s).
k j−k j j
j=1
X
Then by orthogonality of the Rademacher functions
∞
H 2 = [b a ]2.
k kk2 j−k j
j=1
X
WeshallusearesultofCalder´on[4]whichstatesthatifsomelinearoperator
is boundedon L1(µ) and on L∞(µ) on a space with σ-finite measure µ, then
it is bounded on all rearrangement invariant function spaces on that space.
ON MAXIMAL FUNCTIONS FOR MIKHLIN-HO¨RMANDER MULTIPLIERS 5
In our case the intermediate space is the Orlicz space expℓ, which coincides
with the space of all sequences γ = γj j∈Z that satisfy the condition
{ }
C
(2.2) γ∗(l) , l 0,
≤ log(l+2) ≥
and the best constant in 2.2 is equivalent to the norm in exp(ℓ). We apply
Calder´on’s result to the operator T defined by
∞
[Tγ] = γ a2
k j−k j
j=1
X
and get
suplog(l+2)(Tγ)∗(l) C a2 suplog(l+2)γ∗(l).
≤ { n} ℓ1
l≥0 l≥0
(cid:13) (cid:13)
Let c = H ([T(b2)] )1/2 where(cid:13)b2 st(cid:13)ands for the sequence b2 ; then
k k kk2 ≡ k { j}
by our bound for Tγ and the assumption on b it follows that
(2.3) c∗(l) C A a log(2+l) −1/2.
1 ℓ2
≤ k k
We can proceed with the proof as (cid:0)in Proposit(cid:1)ion 2.1, using again (2.1),
i.e.
meas( s [0,1] : Hk(s) > α ) 2e−α2/4c2k.
{ ∈ | | } ≤
Then we obtain for u> 0
∞
sup|Hk| 2 ≤ u2+4 αe−α2/4c2k
k k Zu
(cid:13) (cid:13) X
(cid:13) (cid:13) ≤ u2+8 c2ke−u2/(4c2k)
k
X
= u2+8 (c∗(l))2e−u2/4(c∗(l))2.
l≥0
X
We set the cutoff level to be u= 10C A a and obtain
1 2
k k
sup H 2 u2+C2A2 (2+l)−5/2 . A2 a 2
k | k|k2 ≤ 1 k k2
k
l≥0
X
which is what we wanted to prove. (cid:3)
Remark: Since the Lp norm of a r is equivalent to the ℓ2 norm of a
j j j
{ }
one can also prove Lp analogues of the two propositions, for 0 < p < .
P ∞
3. Proof of Theorem 1.1
To prove (1.2) we may assume that f is compactly supported in Rd 0
\{ }
and thus we may assume that the multipliers m are compactly supported
i
on a finite union of dyadic annuli. Inbview of the scale invariance of the
assumptions we may assume without loss of generality that
(3.1) m (ξ) = 0, ξ 2N,i = 1,...,N.
i
| | ≤
6 LOUKASGRAFAKOS,PETRHONZ´IK,ANDREASSEEGER
In the case of Fourier multipliers the inequality (2.1) will be replaced by
a “good-λ inequality” involving square-functions for martingales as proved
by Chang, Wilson and Wolff [5]. To fix notation let, for any k 0, Q
k
≥
denote the family of dyadic cubes of sidelength 2−k; each Q is of the form
d [n 2−k,(n +1)2−k). Denote by E the conditional expectation,
i=1 i i k
1
Q E f(x)= χ (x) f(y)dy
k Q
Q
QX∈Qk | | ZQ
and by D the martingale differences,
k
D f(x)= E f(x) E f(x).
k k+1 k
−
The square function for the dyadic martingale is defined by
1/2
S(f)= D f(x)2 ;
k
| |
(cid:16)Xk≥0 (cid:17)
one has the inequality S(f) C f for 1 < p < (see [3], [2] for the
p p p
k k ≤ k k ∞
general martingale case, and for our special case cf. also Lemma 3.1 below).
The result from [5] says that there is a constant c > 0 so that for all
d
λ> 0, 0 < ε< 1, one has
(3.2) meas x:sup E g(x) E g(x) > 2λ, S(g) < ǫλ )
k 0
| − |
k≥0
(cid:0)(cid:8) c (cid:9)(cid:1)
Cexp( d)meas x: sup E g(x) > ελ ;
≤ −ǫ2 | k |
k≥0
(cid:0)(cid:8) (cid:9)(cid:1)
see [5] (Corollary 3.1 and a remark on page 236). To use (3.2) we need a
pointwise inequality for square functions applied to convolution operators.
Choose a radial Schwartz function ψ which equals 1 on the support of φ
(defined in the introduction) and is compactly supported in Rd 0 , and
\{ }
define the Littlewood-Paley operator L by
k
(3.3) L f(ξ) = ψ(2−kξ)m(ξ)f(ξ)
k
Let M be the Hardy-Littlewood maximal operator and define the operator
d b
M by
r
M = (M(f r))1/r.
r
| |
Denote byM =M M M thethree-fold iteration of themaximal operator.
◦ ◦
Now define
1/2
(3.4) G (f) = M[L f r] 2/r .
r k
| |
(cid:16)Xk∈Z(cid:0) (cid:1) (cid:17)
From the Fefferman-Stein inequality for vector-valued maximal functions
[9],
(3.5) G (f) C f , 1 < r < 2, r < p < .
r p p,r p
k k ≤ k k ∞
ON MAXIMAL FUNCTIONS FOR MIKHLIN-HO¨RMANDER MULTIPLIERS 7
Lemma 3.1. Let Tf = −1[mf] and let 1 <r . Then for x Rd,
F ≤ ∞ ∈
(3.6) S(Tf)(x) Ar m Y(r′,d/r)Gr(f)(x).
≤b k k
The proof will be given in 4.
§
We shall also need
Lemma 3.2. Let Tf = −1[mf] and suppose that m(ξ) = 0 for ξ 2N.
F | | ≤
Then
(3.7) E0Tf(x) C2−Nb/rCr m Y(r′,d/r)(M(f r))1/r.
| |≤ k k | |
We now give the proof of Theorem 1.1. Let T f = −1[m f]. We need to
i i
F
estimate
∞ b 1/p
sup T f = p4p λp−1meas( x : sup T f(x) > 4λ )dλ .
| i | p { | i | }
(cid:13)1≤i≤N (cid:13) (cid:16) Z0 i (cid:17)
Now(cid:13) by Lemma(cid:13)3.1 one gets the pointwise bound
(3.8) S(T f) A BG (f).
i r r
≤
We note that
x : sup T f(x) > 4λ E E E
i λ,1 λ,2 λ,3
{ | | } ⊂ ∪ ∪
1≤i≤N
where with
c 1/2
d
(3.9) ε :=
N
10log(N +1)
(cid:16) (cid:17)
we have set
ε λ
E = x: sup T f(x) E T f(x) >2λ,G (f)(x) N ,
λ,1 i 0 i r
{ | − | ≤ A B}
1≤i≤N r
ε λ
N
E = x: G (f)(x) > ,
λ,2 r
{ A B}
r
E = x: sup E T f(x) > 2λ .
λ,3 0 i
{ | | }
1≤i≤N
By (3.8),
N
(3.10) E x : T f(x) > 2λ,S(T f) ε λ ,
λ,1 i i N
⊂ { | | ≤ }
i=1
[
and thus using the good-λ inequality (3.2) we obtain
N
meas(E ) meas x : T f(x) E T f(x) > 2λ,S(T f) ε λ
λ,1 i 0 i i N
≤ { | − | ≤ }
i=1
X (cid:0) (cid:1)
N
Cexp( cd )meas( x :sup E (T f) > λ ).
≤ −ε2 { | k i | }
N k
i=1
X
8 LOUKASGRAFAKOS,PETRHONZ´IK,ANDREASSEEGER
Hence
∞ 1/p
p λp−1meas(E )dλ
λ,1
(cid:16) Z0 (cid:17)
N
1/p
. exp( cd ) sup E (T f) p
−ε2 | k i | p
N k
(cid:16)Xi=1 (cid:13) (cid:13) (cid:17)
N (cid:13) (cid:13)
1/p
. exp( cd ) T f p
−ε2 i p
N
(cid:16)Xi=1 (cid:13) (cid:13) (cid:17)
(3.11) . B N exp( cd )(cid:13)1/p (cid:13)f . B f
−ε2 k kp k kp
N
(cid:0) (cid:1)
uniformly in N (by our choice of ε in (3.9)).
N
Next, by a change of variable,
∞ 1/p A B
p λp−1meas(E )dλ = r G (f)
λ,2 ε r p
(cid:16) Z0 (cid:17) N (cid:13) (cid:13)
(3.12) . B lo(cid:13)g(N +(cid:13)1) f p
k k
Finally, from Lemma 3.2 and the Feffermanp-Stein inequality
N
meas(E ) meas x: E T f(x) > 2λ
λ,3 0 i
≤ { | | }
i=1
X (cid:0) (cid:1)
and thus
∞ 1/p
p λp−1meas(E )dλ = 2 sup E (T f)
λ,3 | 0 i | p
(cid:16) Z0 (cid:17) (cid:13)i=1,...,N (cid:13)
N 1/p (cid:13) (cid:13)
(3.13) 2 E (T f) p . BN1/p2−N/r f . B f .
≤ 0 i p k kp k kp
(cid:16)Xi=1(cid:13) (cid:13) (cid:17)
(cid:13) (cid:13)
The asserted inequality follows from (3.11), (3.12), and (3.13). (cid:3)
For completeness we mention the well known relation of the Y(r′,α) con-
ditions with Besov and Sobolev norms.
Lemma 3.3. Let 1 r 2 and α > d/r. Then
≤ ≤
m Y(r′,d/r) . sup φm(2k ) Br
k k k · k d/r,1
k
. sup φm(2k ) . sup φm(2k )
k · kLrα k · kL2α
k k
Proof. By the Hausdorff-Young inequality and the definition of the Besov
space we have
∞ ∞
2ℓd/r Hk,ℓ r′ . 2ℓd/r [φm(2k )] ηℓ r . φm(2k ) Br .
k k k · ∗ k k · k d/r,1
ℓ=0 ℓ=0
X X
b
ON MAXIMAL FUNCTIONS FOR MIKHLIN-HO¨RMANDER MULTIPLIERS 9
By elementary imbedding properties g Br . g Lr if γ > d/r. Finally
k k d/r,1 k k γ
φm(2k ) . C′ φm(2k ) , if 1 < r 2. In this last inequality we
k · kLrγ rk · kL2γ ≤
used that for χ ∈ Cc∞ we have kχgkLrγ0 . kgkLrγ1 for r0 ≤ r1, γ ≥ 0; this is
trivial for integers γ from H¨older’s inequality and follows for all γ 0 by
≥
interpolation. (cid:3)
4. Proofs of Lemma 3.1 and Lemma 3.2
ChoosearadialSchwartzfunctionβ withthepropertythatβ issupported
in x : x 1/4 so that β(ξ) = 0 in ξ :1/4 ξ 4 and β(0) = 0. Now
{ | |≤ } 6 { ≤ | |≤ }
choose a function ψ C∞ so that ψ(ξ)(β(ξ))2 = 1 for all ξ bsupp φ, here
∈ c ∈
φ is as in the formulation of the theorem. Define operators T , B , L by
k k k
e e
T f(ξ) = φ(2−kξ)m(ξ)f(ξ)
k
e
B f(ξ) = β(2−kξ)f(ξ)
dk b
L f(ξ) = ψ(2−kξ)f(ξ).
dk b
Then T = T = B2L T L and we write
k k k edk k k ke b
(4.1) P DkTfP= e(DkBk+n)(Bk+nLk+n)Tk+nLk+nf.
n∈Z
X
Sublemma 4.1. e
(4.2) B L f(x) . Mf(x).
k k
| |
Proof. Immediate. (cid:3)
e
Sublemma 4.2. For s 0,
≥
(4.3) E B f(x) + E B f(x) . 2−s/q′M f(x)
k+1 k+s k k+s q
| | | |
and
(4.4) D B f(x) . 2−sMf(x).
k k−s
| |
Proof. We give the proof although the estimates are rather standard (for
similar calculations in other contexts see for example [6], [12], [10], [13]).
For (4.3) first note this inequality is trivial if s is small and assume, say,
s 10. For Q Q , s > 0 let b (Q) be the set of all x Q for which the
k s
≥ ∈ ∈
ℓ∞ distance to the boundary of Q is 2−k−s+1.
≤
Fix a cube Q Q . If Q′ is a dyadic subcube of sidelength 2−k−s+1
0 k+1
∈
subcube which is not contained in bs(Q) then Bk+s[fχQ′] is supported in
Q0 and usingthe cancellation of −1[β] we see that Ek+1Bk+s[χQ′g] = 0 for
F
all g. Let (Q ) be the union over all dyadic cubes of sidelength 2−k−s+1
s 0
V
whose closures intersect the boundary of Q . Then
0
E B [χ g] = E B [gχ ]
k+1 k+s Q0 k+1 k+s Vs(Q0)
forallg. Inviewofthesupportpropertiesofβ wenotethatB [gχ ]is
k+s Vs(Q0)
also supported in (Q ). Observe that this set has measure O(2−kd2−s).
s−1 0
V
b
10 LOUKASGRAFAKOS,PETRHONZ´IK,ANDREASSEEGER
It follows that for x Q
0
∈
E B f(x) 2d Q −1 B [χ f](y)dy
| k+1 k+s | ≤ | 0| | k+s Vs(Q0) |
ZVs−1(Q0)
. Q −1 f(y)qdy 1/q2−(kd+s)/q′
0
| | | |
ZQ0
. 2−s/q′ (cid:0)M(f q) 1/q (cid:1)
| |
By the same argument one obtai(cid:0)ns this bo(cid:1)und also for E B f and thus
k k+s
| |
(4.3) follows.
The inequality (4.4) D B f is a simple consequence of the smoothness
k k−s
of the convolution kernel of B and the cancellation properties of the
k−s
operator D = E E . (cid:3)
k k+1 k
−
Sublemma 4.3. Let 1< r < . We have
∞
(4.5) Tkf(x) C m Y(r′,d/r)Mrf(x).
| | ≤ k k
Proof. We may decompose T using the kernels H and obtain
k k,l
∞
T f(x) = 2kdH (2ky)f(x y)dy
k k,ℓ
| | −
(cid:12)Xℓ=0Z (cid:12)
(cid:12)(cid:12)∞ 2kd H (2ky)r′dy 1/r′(cid:12)(cid:12)2kd f(x y)rdy 1/r
k,ℓ
≤ Xℓ=0(cid:16) Z | | (cid:17) (cid:16) Z|y|≤2−k+ℓ| − | (cid:17)
∞
2ℓd/r Hk,ℓ r′ M(f r)(x) 1/r. (cid:3)
≤ k k | |
ℓ=0
X (cid:0) (cid:1)
Proof of Lemma 3.1. To estimate the terms in (4.1) we use Sublemma 4.1
to bound B L , Sublemma 4.2 to bound D B and Sublemma 4.3
k+n k+n k k+n
to bound T . This yields that
k+n
e
|DkBk2+nLk+nTk+nLk+nf(x)| . kmkY(r′,d/r)
2−n/q′M M M (L f)(x) if n 0
e q ◦ ◦ r k+n ≥
×(2nM M Mr(Lk+nf)(x) if n < 0,
◦ ◦
and straightforward estimates imply the asserted bound. (cid:3)
Proof of Lemma 3.2. We split E Tf = E B2L T , and by the
0 k≥N−2 0 k k k
sublemmas we get
P
e
|E0Bk2LkTkf(x)|. 2−k/rkmkY(r′,d/r)Mr ◦M ◦Mr(f)(x)
which implies the assertion. (cid:3)
e