Lipschitz linearization of the maximal hyperbolic cross multiplier

0.38 MB·

by  Olli Saari

LIPSCHITZ LINEARIZATION OF THE MAXIMAL
HYPERBOLIC CROSS MULTIPLIER
OLLI SAARI AND CHRISTOPH THIELE
Abstract. We study the linearized maximal operator associated
with dilates of the hyperbolic cross multiplier in dimension two.
Assuming a Lipschitz condition and a lower bound on the lin-
7
1 earizing function, we obtain Lp(R2) → Lp(R2) bounds for all
0 1<p<∞. We discuss various related results.
2

n
a
J 1. Introduction

8
1 Given a bounded measurable function m on Rd, we define the mul-

] tiplier operator T by setting T(cid:100)f = mf(cid:98)for every Schwartz function
m m
A
f. We also define for t > 0 the dilated operators
C
(cid:91)
. T f(η) = m(tη)f(cid:98)(η),
h m,t
t
a the corresponding maximal operator
m
T f(x) = sup|T f(x)|,
[ m,∗ m,t
t>0

1 and the corresponding linearized maximal operator
v
3 T f(x) = T f(x),
9 m,V m,V(x)
0 for a measurable function V : Rd → (0,∞). For example, in the case of
5
Bochner-Riesz multipliers m, such operators have been studied in [1].
0
. In this paper we are concerned with the family of hyperbolic cross
1
0 multipliers, see [2, 5, 6]. A hyperbolic cross multiplier on R2 takes
7
the form m(|ηξ|) for some compactly supported smooth function m.
1
: By the Marcinkiewicz multiplier theorem, this multiplier is bounded
v
on Lp(R2) for all 1 < p < ∞. Whether the corresponding maximal
i
X operator satisfies any Lp bounds remains open, this question has been
r
posed in [4, 14]. In general, a supremum over an infinite family of
a
Marcinkiewicz or H¨ormander-Mikhlin multipliers does not admit such
bounds [3, 9, 11].
We propose a Lipschitz assumption on the linearizing function V, in
analogytoconjecturesfordirectionaloperators, see[13]. Tobeefficient
in the otherwise dilation invariant setting, this assumption has to come
withatruncationofthemaximaloperator,whichinthetheoremsbelow
2010 Mathematics Subject Classification. Primary: 42B15, 42B25.
Keywordsandphrases. Hyperboliccross,multipliers,maximalfunctions,square
functions.
1
2 OLLI SAARI AND CHRISTOPH THIELE
takes either the form of a truncation of the multiplier on the Fourier
side or a cutoff for the linearization function.
The hyperbolic cross has spikes in vertical and horizontal directions.
Splitting the multiplier into separate pieces, each spike needs a Lips-
chitz assumption only in one variable corresponding to the direction of
the spike.
Theorem 1.1. Let m ∈ C3(R) be a compactly supported function of
one variable and β ∈ R. Let V : R2 → [0,∞) be one-Lipschitz in the
first variable, that is
(1.1) |V(x,y)−V(x(cid:48),y)| ≤ |x−x(cid:48)|
for all x,x(cid:48),y ∈ R. Then
(cid:90)(cid:90)
T Π f(x,y) := m(V(x,y)|ξ||η|β)f(cid:98)(ξ,η)e2πi(xξ+yη)dηdξ
m,V β
{|η|β≤1}
satisfies
(cid:107)T Π f(cid:107) ≤ C(p,m,β)(cid:107)f(cid:107) .
m,V β Lp(R2) Lp(R2)
for all 1 < p < ∞ and all Schwartz functions f. Here we denoted
Π = T .
β 1
{|η|β≤1}
The case β = 0 is fairly easy, the maximal operator can then be dom-
inated by the Hardy–Littlewood maximal operator in the first variable.
The Lipschitz assumption is not needed in this case. For β (cid:54)= 0, the
truncation to |η| ≤ 1 or |η| ≥ 1 depending on the sign of β is the one
alluded to above that is necessary to make the Lipschitz assumption
effective. Without truncation one could change the Lipschitz constant
by scaling the otherwise scaling invariant problem, and by a limiting
process, letting the constant tend to infinity, get rid of the Lipschitz
condition.
The next theorem has a truncation in the linearizing function V, and
oneneedstheLipschitzconditiononlyfordistantpoints. Forsimplicity
we formulate it only in the case β = 1 of the hyperbolic cross.
Theorem 1.2. Let m ∈ C3(R) be a compactly supported function of
one variable. Let L > 0 and let V : R2 → [L2,∞) satisfy
(1.2) |V(z)−V(z(cid:48))| ≤ max(L2,L|z −z(cid:48)|)
for all z,z(cid:48) ∈ R2. Then
(cid:90)(cid:90)
T f(x,y) = m(V(x,y)|ξ||η|)f(cid:98)(ξ,η)e2πi(xξ+yη)dξdη
m,V
R2
satisfies
(cid:107)T f(cid:107) ≤ C(p,m)(cid:107)f(cid:107) .
m,V Lp(R2) Lp(R2)
for all 1 < p < ∞ and all Schwartz functions f.
LIPSCHITZ HYPERBOLIC CROSS 3
It actually suffices to demand (1.1) just for L = |z−z(cid:48)|, because for
other pairs of points one can reduce to this case by a suitable chain of
points connecting z and z(cid:48). Questions of sharp modulus of continuity
of V therefore do not arise in this Theorem.
In the case of directional operators, the multiplier m may be chosen
to be a characteristic function of a half line, essentially yielding what is
called the directional Hilbert transform. In the case of the hyperbolic
cross, a characteristic function m is problematic since the boundary of
the hyperbolic cross is curved, and with the methods of the celebrated
disc multiplier theorem [7] one sees that the operator T cannot be
m
bounded in Lp(R2) if p (cid:54)= 2. When p = 2, this obstruction disappears,
and we have the following variant of our result:
Theorem 1.3. Let V : R2 → {2k}k∈Z+ be such that V = 2(cid:98)log2v(cid:99) for a
function v that is one-Lipschitz. Then
(cid:90)(cid:90)
T f(x,y) = 1 (|η|)(v(x,y)|ξη|)f(cid:98)(ξ,η)e2πi(xξ+yη)dηdξ
V [0,1)
R2
is bounded in L2(R2).
Thistheoremshouldbecomparedwiththemainresultin[10]byGuo
and the second author. It is an analogous estimate for the directional
HilberttransformandshowsthatiftheLipschitzcontinuityisenhanced
by a lacunarity-type condition on V, the Lp bounds do follow for all
1 < p < ∞. It is a well known open problem whether or not V being
Lipschitz alone is enough to imply Lp boundedness of the truncated
directional Hilbert transform. On the other hand, a counterexample
by Karagulyan [12] shows that without the Lipschitz assumption, the
mere assumption of V taking values in a lacunary set such as {2k}
k∈Z
is not sufficient.
It is tempting to investigate whether Karagulyan’s counterexample
can be modified to have some implications in the case of the hyper-
bolic cross multiplier, such as proving unboundedness for the maxi-
mal linearized hyperbolic cross multiplier in the absence of a Lipschitz
condition. However, we have not been able to implement this idea.
Karagulyan’s example uses the fact that every sector between two lines
emanating from the origin contains discs of arbitrarily large diame-
ter. The set between two dilates of hyperbolas however do not contain
arbitrarily large discs.
The rest of the present paper is structured as follows. After in-
troducing some notation, we prove a dyadic model of our theorems.
This part is parallel to [10] and further highlights the connection to
the directional Hilbert transform. Then we prove technical versions,
Lemma 4.4 and Lemma 4.1, of the main theorems, and the last section
is devoted to deriving the theorems from the technical Lemmas.
4 OLLI SAARI AND CHRISTOPH THIELE
Acknowledgement. The authors would like to thank Andreas Seeger
for bringing this problem to their attention, and Joris Roos for suggest-
ing to look at general β. Part of the research was done during the first
author’s stay at the Mathematical Institute of the University of Bonn,
which he wishes to thank for its hospitality. The first author acknowl-
edges support from the Va¨is¨ala¨ Foundation, the Academy of Finland,
and the NSF Grant no. DMS-1440140 (the paper was finished while
in residence at MSRI, Berkeley, California, during the Spring 2017 se-
mester). The second author acknowledges support from the Hausdorff
center of Mathematics and DFG grant CRC 1060.
2. Notation
Most of the notation we use is standard. For a Schwartz function f,
we define the Fourier transform as
(cid:90)(cid:90)
f(cid:98)(ξ,η) = f(x,y)e−2πi(xξ+yη)dxdy.
R2
We denote by C the positive constants that only depend on parameters
we do not keep track of, p and β. Dependency on m is also sometimes
hidden. If A ≤ CB for positive numbers A and B, we write A (cid:46) B.
For a set E, the characteristic function 1 takes the value 1 in E and
E
equals zero elsewhere. For a measurable set E, we denote its Lebesgue
measure, whose dimension is always clear from the context, by |E|.
3. The dyadic models
In this section, I and J will denote dyadic intervals, that is, intervals
of the form 2k((0,1]+n) with n,k ∈ Z and n ≥ 0. With each dyadic
interval I we associate the L2 normalized Haar function
h = |I|−1/2(1 −1 )
I I− 1+
where I± denote the left and right halves of I. We consider the basis
{h ⊗ h } in the plane, and (cid:104)·,·(cid:105) means the L2(R2) inner product.
I J I,J
Occasional use of the inner product in one coordinate is denoted by a
subscript.
The dyadic metric is defined by d(x,x(cid:48)) = inf{|I| : x,y ∈ I} where
the infimum is over all dyadic intervals. The two dimensional dyadic
metric is defined analogously with dyadic squares. The following the-
orem is the dyadic model for Theorem 1.2.
Theorem 3.1. Let L > 0, and let
V : (0,∞)2 → {2k} , V1/2 > L
k∈Z
be L-Lipschitz with respect to the dyadic metric. Then
(cid:88)
f (cid:55)→ (cid:104)h ⊗h ,f(cid:105)h (x)h (y)
I J I J
|I||J|≤V(x,y)
LIPSCHITZ HYPERBOLIC CROSS 5
with the sum over dyadic intervals I and J is a bounded operator from
Lp((0,∞)2) into itself for all 1 < p < ∞.
Proof. Inanalogyto[10], theheartofthematteristhefollowingpropo-
sition.
Proposition 3.2. Take (x,y) ∈ (0,∞)2. Let I (cid:51) x and J (cid:51) y with
|I| ≤ |J|. If |I||J| ≤ V(x,y), then |I||J| ≤ V(x(cid:48),y) for all x(cid:48) ∈ I.
Proof. If there existed x(cid:48) ∈ I with |I||J| > V(x(cid:48),y), then V(x(cid:48),y) <
|I||J| ≤ V(x,y) would hold. The assumptions together with this ob-
servation immediately imply
V(x,y) = d(V(x(cid:48),y),V(x,y)) ≤ Ld(x,x(cid:48)) ≤ L|I|
(cid:112) (cid:112)
≤ L |I||J| ≤ L V(x,y) < V(x,y),
which is a contradiction. (cid:3)
We can divide the sum in the definition of the dyadic operator of
Theorem 3.1 into two parts and without loss of generality concentrate
on
(cid:88) (cid:88)
(cid:104)h ⊗h ,f(cid:105)h (x)h (y).
I J I J
I J:|I||J|≤V(x,y)
|I|≤|J|
Let J(I,y) be the collection of dyadic intervals J with y ∈ J and
|I| ≤ |J| for which there exists x so that |I||J| ≤ V(x,y). Suppose
then that J ∈ J(I,y). If there exists x ∈ I such that |I||J| ≤ V(x,y),
the above proposition implies that this inequality is true for all x ∈ I.
Hence the inner sum can be viewed as sum over J ∈ J(I,y), in effect
removing the x dependency from the inner sum. We then estimate the
Lp norm of the last display by means of the dyadic Littlewood–Paley
inequality:
(cid:12) (cid:12)p
(cid:90)(cid:90) (cid:12) (cid:12)
(cid:12)(cid:88) (cid:88) (cid:12)
(cid:12) (cid:104)h ⊗h ,f(cid:105)h (x)h (y)(cid:12) dxdy
I J I J
(cid:12) (cid:12)
(cid:12) I J∈J(I,y) (cid:12)
 (cid:12) (cid:12)2p/2
(cid:90)(cid:90) (cid:12) (cid:12)
(cid:46) (cid:88)(cid:12)(cid:12) (cid:88) (cid:104)hI ⊗hJ,f(cid:105)hI(x)hJ(y)(cid:12)(cid:12)  dxdy.
(cid:12) (cid:12)
I (cid:12)J∈J(I,y) (cid:12)
For a fixed I, if J,J(cid:48) ∈ J(I,y) are nested, then also J(cid:48)(cid:48) with J(cid:48) ⊂ J(cid:48)(cid:48) ⊂
J is in J(I,y). Hence the inner sum telescopes into the difference of
two dyadic martingale averages in the second variable. We estimate
the above display by
(cid:32) (cid:33)p/2
(cid:90)(cid:90)
(cid:88)
|M ((cid:104)h ,f(cid:105) h (x))(y)|2 dxdy
2 I 1 I
I
6 OLLI SAARI AND CHRISTOPH THIELE
(cid:32) (cid:33)p/2
(cid:90)(cid:90)
(cid:88)
(cid:46) |(cid:104)h ,f(·,y)(cid:105) h (x)|2 dydx (cid:46) (cid:107)f(cid:107)p
I 1 I Lp
I
where M is the dyadic maximal function with respect to the second
2
variable; it is controlled by the vector valued Fefferman–Stein maximal
function theorem. The last step in the above estimation relies again
on the dyadic Littlewood–Paley inequality. (cid:3)
ToobtainadyadicmodelofTheorem1.1, wereplacethelowerbound
of V by a truncation of the admissible scales. Note that the choice
β = −1 in the resulting theorem corresponds to the directional Hilbert
transform (or its dyadic model to be precise).
Theorem 3.3. Let L > 0 and β ∈ R, and let
V : (0,∞)2 → {2k}
k∈Z
be L-Lipschitz in the first variable with respect to the dyadic metric.
Then
(cid:88)
f (cid:55)→ (cid:104)h ⊗h ,f(cid:105)h (x)h (y)
I J I J
|I||J|β≤V(x,y)
|J|β≥L
is a bounded operator from Lp((0,∞)2) to itself for all 1 < p < ∞.
The proof of the theorem is almost identical to that of Theorem 3.1.
The only difference is that where we previously used Proposition 3.2,
we will now use the following fact.
Proposition 3.4. Take (x,y) ∈ (0,∞)2. Let I (cid:51) x and J (cid:51) y. If
|I||J|β ≤ V(x,y), then |I||J|β ≤ V(x(cid:48),y) for all x(cid:48) ∈ I.
Proof. If there were x(cid:48) ∈ I with |I||J|β > V(x(cid:48),y), then V(x(cid:48),y) <
|I||J|β ≤ V(x,y) would hold. Since |J|β ≥ L, we see that
V(x,y) = d(V(x(cid:48),y),V(x,y)) ≤ Ld(x,x(cid:48)) ≤ L|I| < |J|β|I| ≤ V(x,y),
which is a contradiction. (cid:3)
Note that in this case the two variables behave differently. We did
not assume any ordering of |I| and |J| so we need not decompose the
operator to two sums with |I| ≤ |J| and |J| < |I| in the proof of the
theorem.
4. The technical lemmas
The following lemma captures the essence our main results.
Lemma 4.1. Let m ∈ C3(R) satisfy 1 ≤ m ≤ 1 for dyadic
(−(cid:15),(cid:15)) (−2(cid:15),2(cid:15))
(cid:15) ∈ (0,1]. Let L > 0, β ∈ R and let V : R2 → [0,∞) satisfy
|V(x,y)−V(x,y(cid:48))| ≤ L|y −y(cid:48)|
LIPSCHITZ HYPERBOLIC CROSS 7
for all x,y,y(cid:48) ∈ R. We define
(cid:90) (cid:90)
T Π f(x,y) := m(V(x,y)|ξ|β|η|)e2πi(ξx+ηy)dξdη.
m,V β
R |ξ|β≤L−1
Then
(cid:18) (cid:19)
1
(cid:107)T Π f(cid:107) ≤ C(p,β)A log +1 (cid:107)f(cid:107)
m,V β Lp(R2) (cid:15) Lp(R2)
for all 1 < p < ∞ and all Schwartz functions f. Here
3
(cid:88)
A = sup|tim(i)(t)|.
t∈R
i=0
Proof. Let f be a Schwartz function. The proof consists of representing
f with the aid of the Calder´on reproducing formulas; reducing the
question to a cut-off of scales at the price of a Marcinkiewicz multiplier,
and finally proving the theorem for the scale cut-off. For simplicity, we
write T for T throughout the proof. We start the proof by first
m,V
assuming that (cid:15) = 1.
4.1. Caldero´n formulae. Let φ ,φ ∈ S(R) be even and real. Let
1 2
φ(cid:98) be supported in the annulus [−21/|β|,21/|β|] \ (−2−1/|β|,2−1/|β|) and
1
φ(cid:98) in [−2,2]\(−1,1). Let ψ with space support in (−2−5,2−5) be one
2 2
more Schwartz function. We require the normalization
(cid:90) ∞ dt (cid:90) ∞ dt
1 = φ(cid:98)(t) = φ(cid:98)(t)ψ(cid:99)(t)
1 2 2
t t
0 0
and the mean zero condition φ(cid:98)(0) = ψ(cid:99)(0) = 0.
2 2
A subscript as in φ (x) = tφ (tx) for t > 0 denotes a dilation, and
1t 1
the same notation is also used for φ and ψ . We define the Littlewood-
2 2
Paley operators P , P and P for s,t > 0 by
1s 2t 3t
(cid:90)
P f(x,y) = φ (x−z)f(z,y)dz,
1s 1s
R
(cid:90)
P f(x,y) = φ (y −z)f(x,z)dz,
2t 2t
R
(cid:90)
P f(x,y) = ψ (y −z)f(x,z)dz.
3t 2t
R
Under the conditions stated above, the Caldero´n reproducing formulae
corresponding to each variable hold true:
(cid:90) ∞ ds (cid:90) ∞ dt
f(x,y) = P f(x,y) = P P f(x,y) .
1s 2t 3t
s t
0 0
In particular, we can write
(cid:90) ∞(cid:90) ∞ dtds
f(x,y) = P P P f(x,y) .
1s 2t 3t
t s
0 0
8 OLLI SAARI AND CHRISTOPH THIELE
4.2. Decomposition of the operator. Consider λ > 0. We let
T(cid:99)(ξ,η) = m(λ|ξη|)f(cid:98)(ξ,η). Taking the Fourier transform and using
λ
the Calder´on formula, we note that
T(cid:100)f(ξ,η) = m(λ|ξη|)f(cid:98)(ξ,η)
λ
(cid:90) ∞(cid:90) ∞ dtds
= m(λ|ξη|)φ(cid:98)(ξ/s)φ(cid:98)(η/t)ψ(cid:99)(η/t)f(cid:98)(ξ,η)
1 2 2
t s
0 0
(cid:90) ∞(cid:90) 4/(λs) dtds
= m(λ|ξη|)φ(cid:98)(ξ/s)φ(cid:98)(η/t)ψ(cid:99)(η/t)f(cid:98)(ξ,η) .
1 2 2
t s
0 0
The discarded part of the t-integral gives zero contribution because the
following conditions
• φ(cid:98)(η/t) (cid:54)= 0 only if 1|η| ≤ t ≤ |η|,
2 2
• φ(cid:98)(ξ/s) (cid:54)= 0 only if 2−1/|β||ξ| ≤ s ≤ 21/|β||ξ|,
1
• m(λ|ξη|) (cid:54)= 0 only if |ξ|β|η| < 2λ−1,
imply that whenever the integrand of the last display does not vanish
we have
2 4
t ≤ |η| ≤ ≤ .
λ|ξ|β λsβ
Using the fact that m(λ|ξ|β|η|) = 1 when |ξ|β|η| < λ−1 (in particular
when 4sβt ≤ λ−1), we can write
T(cid:100)f(ξ,η) = S(cid:100)f(ξ,η)+E(cid:100)f(ξ,η).
λ λ λ
These operators are defined as follows. We let λ˜ = min{2j+2 : 2j >
λ,j ∈ Z}. Then
(cid:90) ∞(cid:90) (λ˜sβ)−1 dtds
S f(x,y) = P P P f(x,y) ,
λ 1s 2t 3t
t s
0 0
is the principal term and
(cid:90) ∞(cid:90) 4(λsβ)−1 dtds
E(cid:100)f = m(λ|ξη|)φ(cid:98)(ξ/s)φ(cid:98)(η/t)ψ(cid:99)(η/t)f(cid:98)(ξ,η)
λ 1 2 2
t s
0 (λ˜sβ)−1
is an error term.
A similar decomposition is valid with λ replaced by V(x,y). For
every integer j, we denote
Ω = {(x,y) ∈ R2 : 2j ≤ V(x,y) < 2j+1}.
j
We have that
Tf = Sf +Ef
where
(cid:90) ∞(cid:90) (2j+3sβ)−1 dtds
(4.1) 1 Sf(x,y) = 1 (x,y)· P P P f(x,y)
Ωj Ωj 1s 2t 3t t s
0 0
LIPSCHITZ HYPERBOLIC CROSS 9
is the principal part of the operator, and the error part Ef is defined
as
(cid:90) ∞(cid:90) 4(V(x,y)sβ)−1 dtds
(4.2) 1 Ef(x,y) = TP P P f(x,y)
Ωj 1s 2t 3t t s
0 (2j+3sβ)−1
for all (x,y) ∈ Ω . Further, it is possible to write
j
(cid:90) V(x,y)
Ef(x,y) = E f(x,y)+ ∂ E f(x,y)dτ
2j τ τ
2j
(cid:90) 2j+1
≤ |E f(x,y)|+ |∂ E f(x,y)|dτ.
2j τ τ
2j
We call these two components large variation error and small variation
error and estimate them separately.
4.3. The large variation error. The multiplier E has symbol
2j
(cid:90) ∞(cid:90) (2j−2sβ)−1 dtds
m(2j|ξη|)φ(cid:98)(ξ/s)φ(cid:98)(η/t)ψ(cid:99)(η/t)
1 2 2
t s
0 (2j+3sβ)−1
that is supported in
{(ξ,η) ∈ R2 : 2−j−5 ≤ |ξ|β|η| ≤ 2−j+4}.
Since Ω are pairwise disjoint, we see that
j
(cid:88) (cid:88)
(cid:107) 1 E f(cid:107) ≤ (cid:107)( |E f|2)1/2(cid:107) .
Ωj 2j Lp(R2) 2j Lp(R2)
j∈Z j∈Z
As j runs over all integer values, the number of E having non-zero
2j
symbol at any point in the frequency plane is uniformly bounded.
(cid:80)
Hence one can verify that α E is a Marcinkiewicz multiplier with
j j 2j
uniform symbol estimates for any choice of |α | ≤ 1.
j
By the Marcinkiewicz multiplier theorem and Khinchin’s inequality
(cid:88) (cid:88)
(cid:107)( |E f|2)1/2(cid:107) (cid:46) sup (cid:107) α E f(cid:107) ≤ A(cid:107)f(cid:107)
2j Lp(R2) j 2j Lp(R2) Lp(R2)
j∈Z (αj)j∈Z j∈Z
which proves the claimed bound for the large variation error.
4.4. The small variation error. To estimate
(cid:13) (cid:13)
(cid:13)(cid:88) (cid:90) 2j+1 (cid:13)
(cid:13) 1 |∂ E f(x,y)|dτ(cid:13) ,
(cid:13) Ωj τ τ (cid:13)
(cid:13) 2j (cid:13)
j Lp(R2)
we write E(cid:48) for the operator ∂ E f(x,y). Changing variables in the τ-
τ τ τ
integral, using disjointness of Ω , and applying Minkowski’s inequality,
j
we obtain
(cid:13) (cid:13)
(cid:13)(cid:90) 2 (cid:88) (cid:13)
(cid:13) | 1 2jE(cid:48) f(x,y)|dτ(cid:13)
(cid:13) Ωj 2jτ (cid:13)
(cid:13) 1 (cid:13)
j Lp(R2)
10 OLLI SAARI AND CHRISTOPH THIELE
(cid:13) (cid:13)
(cid:90) 2(cid:13) (cid:88) (cid:13) dτ
≤ (cid:13)( |2jτE(cid:48) f(x,y)|2)1/2(cid:13) .
(cid:13) 2jτ (cid:13) τ
1 (cid:13) (cid:13)
j Lp(R2)
The symbol of the multiplier 2jτE(cid:48) is supported in
2jτ
{(ξ,η) ∈ R2 : 2−j−5τ ≤ |ξ|β|η| ≤ 2−j+4τ},
and it satisfies the estimates required for the Marcinkiewicz multiplier
theorem uniformly in 2jτ. Hence
(cid:13) (cid:13)
(cid:13)(cid:88) (cid:13)
(cid:13) α 2jτE(cid:48) f(x,y)(cid:13) ≤ A
(cid:13) j 2jτ (cid:13)
(cid:13) (cid:13)
j Lp(R2)
with A independent of |α | ≤ 1 so we can again use Khinchin’s inequal-
j
ity to conclude the desired bound. The use of Marcinkiewicz multiplier
theorem here and in the previous subsection yields the constant A ap-
pearing in the claim. See e.g. [8].
Remark 4.2. If we were only to prove L2 bounds and we assumed that
V took values in a lacunary set like {2k} , we could work with a non-
k∈Z
smooth m like 1 . To handle this case, we note that Ω = {V(x,y) =
[0,1) j
2j} and that, consequently, there is no small variation error term. The
operators E have bounded symbols and bounded overlap as j takes
2j
all integer values. Hence we can use orthogonality to conclude
(cid:88) (cid:88)
(cid:107)Ef(cid:107) = (cid:107) 1 E f(cid:107) ≤ (cid:107) E f(cid:107) (cid:46) (cid:107)f(cid:107) .
L2 Ωj 2j L2 2j L2 L2
j j
This observation is needed to prove Theorem 1.3. The rest of its proof
isidenticaltotheproofofLemma4.1(thecurrentLemmawithRemark
4.3).
4.5. The principal term. Let v(x,y) = min{2j+2 : 2j > V(x,y),j ∈
Z}. We will estimate
(cid:90) ∞(cid:90) (v(x,y)sβ)−1 dtds
Sf(x,y) = P P P f(x,y) .
1s 2t 3t
t s
0 0
Take a Schwartz function g. Aiming at a bound only depending on
(cid:107)g(cid:107) and (cid:107)f(cid:107) , we write the P convolution out and divide the
Lp(cid:48) Lp 3t
t-integral into two parts
(cid:90)(cid:90)
Sf(x,y)g(x,y)dxdy = I +II
R2
where
(cid:90) (cid:90) ∞(cid:90) (v(x,z)sβ)−1 dtds
I = P P f(x,z)ψ (y −z)g(x,y) dzdxdy,
1s 2t 2t
t s
R3 0 0
(cid:90) (cid:90) ∞(cid:90) (v(x,y)sβ)−1 dtds
II = P P f(x,z)ψ (y −z)g(x,y) dzdxdy.
1s 2t 2t
t s
R3 0 (v(x,z)sβ)−1