SQUARE FUNCTIONS FOR DIRECTIONAL OPERATORS ON THE PLANE
FRANCESCO DI PLINIO, SHAOMING GUO, CHRISTOPH THIELE, AND PAVEL ZORIN-KRANICH
ABSTRACT. We show that the suitably truncated Hilbert transform along a variable monomial curve (x +
t, y + u(x, y)t α ) is almost diagonal with respect to the Littlewood–Paley decomposition in the second
variable provided that u is Lipschitz in the second variable. In particular, L p estimates for that operator
follow from L p estimates for a certain square function that have been recently obtained by Guo, Hickman,
Lie, and Roos in the curved case α 6= 1. In the flat case α = 1 we improve a conditional result due to Lacey
and Li by removing a C 1+ε regularity hypothesis on the function u. Still in the flat case, we estimate a
single scale version of our square function. The latter result almost (up to the endpoint and an ε loss in
the exponent) recovers the (log N )1/2 estimate for the single scale directional operator with N directions
due to Katz.
1. INTRODUCTION
Assume we are given some translation invariant operator T on functions on the real line, typical
examples are convolution by a bump function, the Hardy–Littlewood maximal operator, and the Hilbert
transform. Given in addition a field u of directions in the plane, one may form the corresponding
directional operator Tu on functions f on the plane as follows. At each point in the plane we apply the
operator T to the restriction of f to the line through the point in the direction dictated by the direction
field at the point, and then evaluate the operator at the given point.
Directional operators enjoy a rich body of research. Initial intuition on directional fields may suggest
that rotational symmetries play a role. However, a fruitful line of research has evolved rather along a biparameter approach. One restricts attention to a small sector of directions, fixes a direction transversal
to this sector, and considers the variety of directions as a result of shearing the plane while preserving
the fixed direction. Such point of view allows for a perturbed bi-parameter theory, putting particular
emphasis on the fixed direction.
Restriction to lines is tantamount to tensoring the operator in question with a Dirac delta in the fixed
direction. In this article we replace this Dirac delta by a natural square function construction. Such
construction appears in the literature, if implicitly at places. In the present article we observe two
remarkable properties of this construction.
To fix notation, assume the special direction is the vertical direction and assume the direction field
is given by the directions (1, u) for a measurable scalar function u in the plane, bounded in absolute
value by 1. Let Pt denote suitable Littlewood–Paley convolution in the vertical variable, that is
Z
Pt f (x, y) = f ∗2 ψ t (x, y) :=
f (x, y − z)ψ t (z)dz
R
where for simplicity we assume ψ is a Schwartz function with ψ(z) dz = 0 and ψ t (z) = t −1 ψ(t −1 z).
Our first main result concerns the directional averaging operator
Z +∞
(1.1)
Tu,φ f (x) :=
φ(r) f (x 1 + r, x 2 + u(x)r)dr.
−∞
associated to a Schwartz function φ. This operator is not bounded on L p unless p = ∞. Nevertheless,
we can estimate a corresponding square function.
Theorem 1.2. Let u : R2 → [−1, 1] be a measurable function. Then
Š1/2 X
® k f k , 2 < p < ∞.
(1.3)
|Tu,φ Pt f |2
p,φ
p
p
t∈2Z
2010 Mathematics Subject Classification. 42B25.
1
2
FRANCESCO DI PLINIO, SHAOMING GUO, CHRISTOPH THIELE, AND PAVEL ZORIN-KRANICH
As an application of this result, we elaborate on a remark made by Demeter in [Dem10]. There he
is concerned with direction fields u which take only finitely many values. Under this assumption it is
natural to ask for bounds on directional operators with optimal growth in the cardinality N of the range
of u. A theorem of Katz [Kat99] concerns the operator T given by convolution against the characteristic
function of the interval [−1, 1], and consequently Tu as above with φ replaced by that characteristic
function.
Theorem 1.4 ([Kat99]). Assume the measurable function u : R2 → [−1, 1] takes at most N different
values. Then Tu is bounded in L 2 (R2 ) with norm bounded by C log(N + 2)1/2 .
Demeter presents an alternative proof to the proof of Katz and suggests a further proof using an
inequality by Chang, Wilson, and Wolff [CWW85]. The latter approach naturally leads to the square
function above. We could not entirely reproduce Katz’ theorem in this fashion, but we produce an estimate with a loss of power ε in the logarithm, see Corollary 3.16.
Our second main result concerns situations where one seeks to reduce bounds for Tu to bounds for
the square function. Note that for suitable generating function ψ for Pt we have
X
X
X
|Tu f | = |
Tu Pt f | ≤
|(1 − Pt )Tu Pt f | + |
Pt Tu Pt f |
t∈2Z
t∈2Z
t∈2Z
The latter term can be estimated for 1 < p < ∞ via
X
X
X
X
2 1/2
2 1/2
k
Pt Tu Pt f k p ® k(
|Pt Tu Pt f | ) k p ® k(
|MV Tu Pt f | ) k p ® k(
|Tu Pt f |2 )1/2 k p
t∈2Z
t∈2Z
t∈2Z
t∈2Z
where MV denotes the maximal operator in the second variable and we have applied Littlewood–Paley
theory and the Fefferman–Stein maximal estimate.
The situation we are concerned with here is where the one-dimensional operator is the truncated
Hilbert transform and it is applied along a monomial curve, so that
Z1
dr
(α)
(1.5)
Tu f (x) :=
f (x 1 + r, x 2 + u(x)r α ) .
r
−1
Here α > 0 and r α can be interpreted as either |r|α or sgn(r)|r|α .
Due to the prensence of curvature, the behaviour of the operators Tu(α) with α 6= 1 is very different
from the case α = 1. In [LL10] the question as to whether Tu(1) has weak type (2, 2) provided that u is
Lipschitz (still unresolved) is attributed to Stein in [Ste87]. We prove the following theorem:
Theorem 1.6. Let 0 < α < ∞. Assume that u(x, ·) has Lipschitz constant ≤ 1/100 for each x ∈ R.
Assume that ψ̂ identically equals 1 on ±[99/100, 103/100] and vanishes outside of ±[98/100, 104/100].
Let Ψ be another Schwartz function on R such that Ψ̂ is supported on ±[1, 101/100]. Then
X
(α)
|(1 − Pt )Tu (Ψ t ∗2 f )| ® k f k p , 1 < p < ∞.
t∈2Z
p
It it important that ψ̂ identically equals 1 on a neighborhood of the support of Ψ̂ whose size is related
to the Lipschitz constant of the direction field. Theorem 1.6 turns out to be essentially a one dimensional
result (Theorem 2.24) applied in the vertical direction.
The L 2 bound for the square function is reduced to a single scale operator and one may use Theorem 1.6 to obtain the following strengthening of a theorem of Lacey and Li [LL10, Theorem 1.18].
Theorem 1.7. Suppose that kukLip is sufficiently small and [LL10, Conjecture 1.14] holds for the vector
field v = (1, u). Then Tu(1) is bounded on L 2 (R2 ).
In [LL10, Theorem 1.18] the L 2 estimate is proved (conditionally on [LL10, Conjecture 1.14]) for
the single band operators Tu(1) ◦ Pt , and it is observed that the single band estimates can be assembled
into an estimate for the full operator Tu(1) provided that the function u is C 1+η -regular for any η > 0.
Theorem 1.7 removes this additional regularity hypothesis.
SQUARE FUNCTIONS FOR DIRECTIONAL OPERATORS ON THE PLANE
3
We recall that the Lipschitz regularity hypothesis in Theorem 1.7 cannot be substantially relaxed (e.g.
to Hölder continuity of lower degree or even to Lipschitz continuity with too large Lipschitz constant).
This can be seen by testing on characteristic functions of Perron trees (the trees used to construct Kakeya
sets, see e.g. [Ste93, Section X.1]).
A maximal variant of Tu(α) is given by
Zε
1
(α)
(1.8)
Mu f (x) := sup
| f (x 1 + r, x 2 + u(x)r α )|dr.
0<ε≤1 2ε −ε
It is a long-standing conjecture attributed to Zygmund that the operator Mu(1) should be bounded for
some p 6= ∞.
In the curved case α 6= 1 the operator Mu(α) has been proved to be bounded on L p (R2 ) for every
p > 1, provided that kukLip is small enough, in [Guo+16]. The corresponding problem about Tu(α) has
been left open, and we resolve it here.
Theorem 1.9. For every 0 < α < ∞, α 6= 1, and every 1 < p < ∞, there exits ε0 > 0 such that for every
Lipschitz function u with kukLip ≤ ε0 , we have
kTu(α) f k p ® k f k p .
(1.10)
2. LIPSCHITZ
VECTOR FIELDS
2.1. Carleson embeddings with compactly supported test functions. We refer to [DT15, Section 2
and 3] for the general theory of outer measure spaces. In this section we use the outer measure space
X = Rd × (0, ∞) with the collection of distinguished sets E consisting of the tents
T (x, s) = {( y, t) : kx − yk + t ≤ s}
and an outer measure µ generated by σ(T (x, s)) = s d .
Let ω be a Dini modulus of continuity, that is, ω : [0, ∞) → [0, ∞) is a function that is subadditive
in the sense
u ≤ s + t =⇒ ω(u) ≤ ω(s) + ω(t)
R1
and has finite Dini norm kωkDini = 0 ω(t) dtt . Let C be the class of testing functions φ : Rd → C that
satisfy
R
(2.1)
φ(z)dz = 0,
(2.2)
(2.3)
supp φ ⊂ B(0, 1)
|φ(z) − φ(z 0 )| ≤ ω(kz − z 0 k) for all z, z 0 ∈ Rd .
For locally integrable functions f we define the embeddings
Z
Ac f (x, t) := t −d
| f |,
B(x,t)
Dc f (x, t) := sup t −d
φ∈C
Z
f (z)φ(t −1 ( y − z))dz .
Theorem 2.4 (cf. [DT15, Theorem 4.1]). For every 1 < p ≤ ∞ we have
kAc f k L p (S ∞ ) ® k f k L p (Rd ) ,
kDc f k L p (S 2 ) ® k f k L p (Rd ) .
Moreover, we have the endpoint estimates
kAc f k L 1,∞ (S ∞ ) ® k f k L 1 (Rd ) ,
kDc f k L 1,∞ (S 2 ) ® k f k L 1 (Rd ) .
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FRANCESCO DI PLINIO, SHAOMING GUO, CHRISTOPH THIELE, AND PAVEL ZORIN-KRANICH
The main difference from [DT15, Theorem 4.1] is the supremum over φ ∈ C in the definition of Dc ,
whereas [DT15, Theorem 4.1] uses a fixed φ. This supremum does not affect the proof strongly, but is
important for our application. The Dini regulartiy condition plays the same role as in [Zor16].
We linearize the supremum in the definition of Dc f by choosing for each pair ( y, t) a function φ ∈ C
for which the supremum is almost attained. Denote then φ y,t (z) = t −d φ(t −1 ( y − z)). This is an L 1
normalized wave packet at scale t. The almost orthogonality of these wave packets is captured by the
following estimate.
Lemma 2.5. If t ≤ t 0 then
|〈φ y,t , φ y 0 ,t 0 〉| ® (t 0 )−d ω(t/t 0 )
Proof. Using the cancellation condition (2.1) and the support condition we write
Z
Z
φ y,t (z)(φ y 0 ,t 0 (z) − φ y 0 ,t 0 ( y))dz φ y,t (z)φ y 0 ,t 0 (z)dz = Rd
B( y,t)
Z
|φ y,t (z)|(t 0 )−d ω(t/t 0 )dz ® (t 0 )−d ω(t/t 0 ).
≤
B( y,t)
We use the almost orthogonality statement in Lemma 2.5 to deduce a square function estimate for
p = 2.
Lemma 2.6.
Z
|Dc f ( y, t)|2 d y
(2.7)
Rd ×R>0
dt
® k f k22 .
t
Proof. We begin with a measurable selection of functions φ y,t that almost extremize Dc f ( y, t). Expand
the square of the left hand side of (2.7)
Œ2
Œ
Z
2 ‚Z ‚Z
dt
dt
|〈 f , φ y,t 〉|2 d y
f (z)dz
=
〈 f , φ y,t 〉φ y,t (z)d y
t
t
Rd
Rd ×R>0
Z
dt 2
≤
〈 f , φ y,t 〉φ y,t (z)d y 2 k f k22
t
Rd ×R
>0
We further expand the square from the former term
ZZ
=
〈 f , φ y,t 〉〈φ y,t , φ y 0 ,t 0 〉〈φ y 0 ,t 0 , f 〉d y
Z
≤
|〈 f , φ y,t 〉|2
Z
|〈φ y,t , φ y 0 ,t 0 〉|d y 0
dt 0 dt 0
d y 0 k f k22
t
t
dt 0
dt
d y k f k22 ,
0
t
t
using the estimate
2|〈 f , φ y,t 〉〈φ y 0 ,t 0 , f 〉| ≤ |〈 f , φ y,t 〉|2 + |〈 f , φ y 0 ,t 0 〉|2
in the last inequality. It suffices to verify
Z
sup
y,t
|〈φ y,t , φ y 0 ,t 0 〉|d y 0
dt 0
< ∞.
t0
By Lemma 2.5 and using bounded support of the φ y,t ’s we have
Z
Z
Z
Z
Z
0
0
dt 0
0 −d
0
0 dt
0 dt
(t ) ω(t/t )d y 0 +
t −d ω(t 0 /t)d y 0 0
|〈φ y,t , φ y 0 ,t 0 〉|d y 0 ®
t
t
t
t≤t 0 k y− y 0 k≤t+t 0
t>t 0 k y− y 0 k≤t+t 0
Z
Z
dt 0
dt 0
®
ω(t/t 0 ) 0 +
ω(t 0 /t) 0 ® kωkDini .
t
t
t≤t 0
t>t 0
This finishes the proof of Lemma 2.6.
SQUARE FUNCTIONS FOR DIRECTIONAL OPERATORS ON THE PLANE
5
Proof of Theorem 2.4. We may assume that the superlevel sets {M f > λ}, where M is the uncentered
Hardy–Littlewood maximal function, have finite measure for all λ > 0, since otherwise the right-hand
side of the conclusion is infinite.
Let {Q i }i be a Whitney decomposition of the superlevel set {M f > λ}. Let x i denote the center and
ri the diameter of Q i . Let
p
[
(2.8)
E :=
T (x i , 3 d ri )
i
and note that
µ(E) ® |{M f > λ}|.
The claim of the theorem will therefore follow from the more precise results
(2.9)
kAc f 1 E c k L ∞ (S ∞ ) ® λ,
(2.10)
kDc f 1 E c k L ∞ (S 2 ) ® λ.
p
Let (x, t) ∈ E c . Then no ball B( y, t/ d) with kx − yk ≤ t is contained in a Whitney cube. It follows
that, for some constant C that depends only on the dimension, the ball B(x, C t) is not contained in
{M f > λ}. Hence
Z
Ac f (x, t) ≤ t −d
| f | ® λ.
B(x,C t)
This completes
P the proof of (2.9). Now we show (2.10). The Calderón–Zygmund decomposition f =
g + b, b = i bi associated to the Whitney decomposition {Q i }i has the properties
(1)
(2)
(3)
(4)
kgk∞ ® λ,
supp
bi ⊂ Q i ,
R
bi = 0,
R
|Q i |−1 |bi | ® λ.
Using the bounded support condition on the wave packets and Lemma 2.6 we obtain
‚ Z
Œ1/2
1
dt
S 2 (Dc g)(T (x, s)) = d
|Dc g( y, t)|2 d y
s T (x,s)
t
‚ Z
Œ1/2
1
dt
= d
|Dc (g1B(x,2s) )( y, t)|2 d y
s T (x,s)
t
® s−d/2 kg1B(x,2s) k2 ® kgk∞ .
Hence (2.10) holds with f replaced by g. By sublinearity of the embedding map D and subadditivity
of the outer L ∞ (S 2 ) norm it remains to show (2.10) holds with f replaced by b. More explicitly, for
every tent T = T (x, r) we want to show
S 2 (Dc b1 E c )(T ) ® λ.
We know
S ∞ (Dc b1 E c )(T ) ® S ∞ (Dc f 1 E c )(T ) + S ∞ (Dc g1 E c )(T ) ® S ∞ (Ac f 1 E c )(T ) + λ ® λ.
By logarithmic convexity of S p sizes it therefore suffices to show
S 1 (Dc b1 E c )(T ) ® λ.
(2.11)
Claim 2.12.
R
t>ri
Dc bi (x, t)dx dtt ® λrid .
Proof of Claim 2.12. Notice that, due to support constraints, Dc bi (x, t) can only be non-zero if kx −
x i k ≤ ri /2 + t. Moreover, under this condition and choosing φ x,t that almost extremizes Dc bi (x, t) we
6
FRANCESCO DI PLINIO, SHAOMING GUO, CHRISTOPH THIELE, AND PAVEL ZORIN-KRANICH
obtain
Z
Z
b (z)φ (z)dz = i
x,t
bi (z)(φ x,t (z) − φ x,t (x i ))dz kz−x i k≤ri /2
≤t
−d
®t
−d
ω(ri /(2t))
Z
|bi (z)|dz
kz−x i k≤ri /2
Hence
Z
t>ri
dt
Dc bi (x, t)dx
≤
t
Z
ω(ri /(2t))λrid .
dt
Dc bi (x, t)dx
®
t
t>r ,kx−x k≤t+r /2
i
i
i
Z
ω(ri /(2t))λrid
t>ri
dt
® λrid kωkDini .
t
This finishes the proof of Claim 2.12.
In order to show (2.11) notice that only the Whitney cubes Q i ⊂ B(x, 10r) contribute to Dc b1 T \E .
Z
dt
1
−d
Dc b(z, t)dz
S (Dc b1 E c )(T ) = r
t
T \E
Z
X
dt
Dc bi (z, t)dz
≤ r −d
t
i:Q i ⊂B(x,10r) T \E
Z
X
dt
≤ r −d
Dc bi (z, t)dz
t
i:Q ⊂B(x,10r) t>r
i
i
using Claim 2.12
® r −d λ
X
|Q i |
i:Q i ⊂B(x,10r)
by disjointness of Whitney cubes
® r −d λ|B(x, 10r)| ® λ.
This finishes the proof of Theorem 2.4.
2.2. Carleson embeddings with tails. It is possible to adapt the proofs in Section 2.1 to embeddings
defined using test functions with tails. Since we do not need testing functions with a sharp decay rates
for tails, we will instead estimate such embeddings by averaging the results in Section 2.1.
In this section we work in dimension d = 1 and consider the following embedding maps:
Z
(2.13)
(2.14)
A f (x, t) :=
t −1 (1 + |x − y|/t)−5 | f ( y)|d y,
Z
D f (x, t) := sup t −1 φ((x − y)/t) f ( y)d y ,
φ∈Φ
where
Φ = {φ : R → C,
Z
φ = 0, |φ(x)| ≤ (1 + |x|)−10 , |φ 0 (x)| ≤ (1 + |x|)−10 }.
The smoothness and decay conditions in these embeddings are not optimal, but they suffice for our
purposes. Decomposing the testing functions (1 + |x|)−5 and φ ∈ Φ into series of compactly supported
bump functions as in [Mus+06, Lemma 3.1], see also Lemma 3.4 in this article, we can deduce the
embeddings
(2.15)
kA f k L p (S ∞ ) ® k f k p ,
(2.16)
kD f k L p (S 2 ) ® k f k p
for 1 < p ≤ ∞ from Theorem 2.4.
SQUARE FUNCTIONS FOR DIRECTIONAL OPERATORS ON THE PLANE
7
2.3. Jones beta numbers. Let A : R → C be a Lipschitz function and let a be its distributional derivative, so that kak∞ = kAkLip . Let ψ be a compactly supported bump function with
Z
Z
ψ(x)dx =
(2.17)
and
∞
Z
ψ̂(ξξ0 )
0
xψ(x)dx = 0
dξ
=1
ξ
ξ0 6= 0.
for
Let ψ t = t −1 ψ(t −1 ·) be an L 1 normalized mean zero bump function at scale t. Let
Z∞
ds
a ∗ ψs (x)
(2.18)
α(x, t) :=
s
t
be the average slope of A near x at scale t and let
(2.19)
βn (x, t) :=
sup
x 0 ,x 1 ,x 2 ∈B(x,2n ·3t),2−n t≤ t̃≤2n t
t −1 |A(x 2 ) − A(x 1 ) − α(x 0 , t̃)(x 2 − x 1 )|.
This definition includes the supremum over the range of uncertainty around (x, t), which seems convenient.
Lemma 2.20. With the notation (2.19) we have
Z 2n t Z
Z
€
Š1/2
βn (x, t) ® t −1
Da( y, s)2 s−1 d y
ds + 22n
0
| y−x|®2n t
∞
Da(x, s)
2n t
tds
s2
Proof. Let x 0 , x 1 , x 2 ∈ B(x, 2n · 3t), 2−n t ≤ t̃ ≤ 2n t. By the fundamental theorem of calculus and
Calderón’s reproducing formula for a we can write
Z x2
a( y)d y − t −1 α(x 0 , t̃)(x 2 − x 1 )
t −1 (A(x 2 ) − A(x 1 ) − α(x 0 , t̃)(x 2 − x 1 )) = t −1
x1
=t
−1
Z
x2
x1
Z
0
∞
ds
a ∗ ψs ( y) d y − t −1
s
x2
Z
x1
∞
Z
a ∗ ψs (x 0 )
t̃
ds
d y.
s
Splitting the integral in s in the former term at t̃ we further obtain
Z x2 Z ∞
Z x 2 Z t̃
ds
ds
−1
−1
=t
a ∗ ψs ( y) d y + t
(a ∗ ψs ( y) − a ∗ ψs (x 0 )) d y
s
s
x
t̃
x
0
1
1
=: I + I I.
We estimate the two terms on the right-hand side separately. In the first term we note ψs = s(ψ̃s )0 , where
ψ̃s is also an L 1 normalized mean zero bump function at scale s, by assumption (2.17). Therefore
Z 2n t Z x 2
ds
−1
I≤t
a ∗ ψs ( y) d y s
0
x1
Z 2n t
= t −1
|a ∗ ψ̃s (x 2 ) − a ∗ ψ̃s (x 1 )|ds
0
®t
−1
Z
2n t
sup
0
Da( y, s)ds.
| y−x|®2n t
Since Da(·, s) is almost constant at scale s, this can be further estimated by
Z 2n t Z
€
Š1/2
−1
I®t
Da( y, s)2 s−1 d y
ds.
0
| y−x|®2n t
8
FRANCESCO DI PLINIO, SHAOMING GUO, CHRISTOPH THIELE, AND PAVEL ZORIN-KRANICH
We split the second term I I ≤ I I a + I I b via
∞
Z
Z
2n t
+
≤
(2.21)
2−n t
t̃
Z
∞
.
2n t
Then
I Ia ≤ t
−1
2n t
Z
sup |a ∗ ψs ( y)|
n
2−n t | y−x|®2 t
ds
,
s
and this can be absorbed into the estimate for I. The latter term from (2.21) is bounded by
Z x2 Z ∞
−1
a ∗ [ψ (· − x + y) − ψ (· − x + x )](x) ds d y.
I Ib ® t
s
s
0
s
x
2n t
1
Since |x − y|, |x − x 0 | ® 2n t ® s, the function in the square brackets is a mean zero L 1 normalized bump
function at scale s with constant ® 2n t/s by the fundamental theorem of calculus, so
Z∞
Z x2 Z ∞
2n tds
tds
−1
2n
Da(x, s) 2 d y ® 2
I Ib ® t
Da(x, s) 2 .
s
s
x
2n t
2n t
1
This finishes the proof of Lemma 2.20.
Lemma 2.22 (cf. [Jon89, Lemma 3]). kβn k L ∞ (S2 ) ® 23n/2 kak∞ .
Proof. We have to show
1
t0
Z
Z
βn (x, t)2 dx
|x−x 0 |<t 0
t<t 0
dt
® 23n kak2∞
t
with the implicit constant independent of (x 0 , t 0 ) ∈ R × R+ .
We estimate the S 2 size on the tent centered at x 0 with height t 0 separately for the two terms in the
conclusion of Lemma 2.20. For the first term we consider the square of the S 2 size:
1
t0
Z
Z
t<t 0
€
t
|x−x 0 |<t 0
−1
Z
2n t
€
Z
2 −1
Da( y, s) s d y
Š1/2
Š2
ds dx
| y−x|®2n t
0
dt
t
Apply Hölder’s inequality in the s-variable
1
≤
t0
Z
Z
t<t 0
Z
|x−x 0 |<t 0
2n t
€
0
Z
Š ds
Da( y, s) d y 1/2 ·
s
| y−x|®2n t
2
Z
0
2n t
ds
dt
dx 3
1/2
s
t
Change the order of integration
Z
Z
Z
Z
dt ds
2n/2
®
dxDa( y, s)2 d y 5/2 1/2
t 0 s≤2n t 2−n s<t<t | y−x |®2n t |x− y|®2n t
t s
0
0
0
0
Z
Z
22n
ds
®
Da( y, s)2 d y ® 23n kDak2L ∞ (S2 ) .
t 0 s≤2n t | y−x |®2n t
s
0
0
0
For the second term we consider the S 2 size
Z
Z
Z
€
€ ∞
tds Š2 dt Š1/2
2n 1
2
Da(x, s) 2 dx
t 0 t<t |x−x |<t
s
t
2n t
0
0
0
SQUARE FUNCTIONS FOR DIRECTIONAL OPERATORS ON THE PLANE
9
By applying a change of variable s → tτ and Minkowski’s integral inequality:
Z
Z
Z∞
€1
dt Š1/2 dτ
2n
Da(x, tτ)2 dx
≤2
t 0 t<t |x−x |<t
t
τ2
2n
0
0
0
Z∞
Z
Z
€ 1
ds Š1/2 dτ
= 22n
Da(x, s)2 dx
τt 0 s<τt |x−x |<t
s
τ3/2
2n
0
0
0
Z∞
dτ
2n
®2
kDak L ∞ (S2 ) 3/2 ® 23n/2 kDak L ∞ (S2 ) .
τ
2n
The conclusion follows from (2.16).
Corollary 2.23 (cf. [Jon89, Lemma 4]). Let ε > 0 and
β(x, t) =
sup
(1 + max(|x i − x|)/t + t̃/t + t/ t̃)−3/2−ε t −1 |A(x 2 ) − A(x 1 ) − α(x 0 , t̃)(x 2 − x 1 )|.
i
x 0 ,x 1 ,x 2 ∈R, t̃∈(0,∞)
Then
kβk L ∞ (S 2 ) ® kak∞ .
The difference from the original formulation of Jones’s beta number estimate is that we take a supremum over an uncertainty region in all available parameters.
2.4. Littlewood–Paley diagonalization of Lipschitz change of variables.
Theorem 2.24. Let A : R → R be a Lipschitz function with kAkLip ≤ 1/100 and consider the change of
variable TA f (x) := f (x +A(x)). Let ψ, Ψ be as in Theorem 1.6 and let Pt f := ψ t ∗ f be the one-dimensional
Littlewood–Paley operators associated to ψ. Then
X
|(1 − Pt )TA(Ψ t ∗ f )| ® kAkLip k f k p , 1 < p < ∞.
p
t∈2Z
Note that the linear dependence on the Lipschitz norm breaks down when the Lipschitz norm becomes
too large.
Proof. Since the Lipschitz norm of A is strictly smaller than 1, the change of variable x 7→ x + A(x) is
invertible and bi-Lipschitz. Denote its inverse function by b, so that z = b(z) + A(b(z)).
Write
Z
∞
TA(Ψ t ∗ f )(x) = TA(Ψ t ∗ Pt f )(x) =
Pt f (z)Ψ t (x + A(x) − z)dz.
−∞
This integral is a linear combination of the functions x 7→ Ψ t (x + A(x) − z) that we view as non-linear
deformations of wave packets centered at b(z). The main idea is to replace the non-linear change of
variable x 7→ x +A(x)−z in the argument of Ψ t by the linear change of variable x 7→ (1+α(b(z), t))(x −
b(z)), where α is the average slope of the function A in the sense of (2.18). Since |α| ≤ kAkLip , the
function
Z
∞
Pt f (z)Ψ t ((1 + α(b(z), t))(x − b(z)))dz
x 7→
−∞
has Fourier support inside t −1 [99/100, 103/100], so it is annihilated by I − Pt .
It remains to estimate the error that has been made in approximating the non-linear change of coordinates in the argument of Ψ t by a linear one. To this end we compute the difference of the arguments:
(2.25)
|x + A(x) − z − (1 + α(b(z), t))(x − b(z))| = |(A(x) − A(b(z)) − α(b(z), t)(x − b(z)))|
By the Lipschitz property of A and since |α| ≤ kAkLip we have
1
(2.25) ≤ |x − b(z)|,
2
10
FRANCESCO DI PLINIO, SHAOMING GUO, CHRISTOPH THIELE, AND PAVEL ZORIN-KRANICH
and it follows that both x + A(x) − z and (1 + α(b(z), t))(x − b(z)) have (signed) distance of the order
≈ x − b(z) from zero. Therefore
|Ψ t (x + A(x) − z) − Ψ t ((1 + α(b(z), t))(x − b(z)))|
® t −2 (1 + |x − b(z)|/t)−20 · (2.25)
by decay of Ψ t0
® t −1 β(b(z), t)(1 + |x − b(z)|/t)−10
by definition of β numbers.
It follows that
X
|(1 − Pt )TA(Ψ t ∗ f )|
t∈2Z
Z
X
(1 − P ) P f (z)(Ψ (x + A(x) − z) − Ψ ((1 + α(b(z), t))(x − b(z))))dz =
t
t
t
t
t∈2Z
®
X
(δ0 + t −1 (1 + |·|/t)−10 ) ∗
Z
|Pt f (z)|t −1 β(b(z), t)(1 + |x − b(z)|/t)−10 dz
t∈2Z
®
XZ
|Pt f (z)|t −1 β(b(z), t)(1 + |x − b(z)|/t)−5 .
t∈2Z
0
Multiplying this with a function g ∈ L p (R) and integrating in x we obtain the estimate
XZ
D f (z, t)β(b(z), t)A g(b(z), t)dz.
t∈2Z
R∞
The sum over t can be dominated by 0 dtt since all functions D, β, A are almost (up to a multiplicative
factor) constant on Carleson boxes B(x, t) × [t, 2t]. By [DT15, Proposition 3.6] and outer Hölder
inequality [DT15, Proposition 3.4] this is bounded by
kD f k L p (S2 ) kβ(b(·), ·)k L ∞ (S 2 ) kA g(b(·), ·)k L p0 (S ∞ ) .
Since the function b is bi-Lipschitz, it does not affect outer norms up to a multiplicative constant. To
see this note that
kF 1(∪i T (x i ,si ))c k L ∞ (S q ) ≤ λ =⇒ kF (b(·), ·)1(∪i T (b−1 (x i ),2si ))c k L ∞ (S q ) ≤ Cλ
for a sufficiently large constant C.
Thus we obtain the estimate
kD f k L p (S2 ) kβk L ∞ (S 2 ) kA gk L p0 (S ∞ ) .
Estimating the first term using (2.16), the middle term using Corollary 2.23, and the last term using
(2.15) we obtain the claim.
2.5. Application to truncated directional Hilbert transforms.
Proof of Theorem 1.6. By Minkowski’s integral inequality we obtain
X
|(1 − Pt )Tu (Ψ t ∗2 f )|
L p (R2 )
t∈2Z
Z
1
€
≤
r=−1
Z X
p
|(1 − Pt ) Ψ t ∗ f (x 1 + r, · + ru(x 1 , ·)) |
p
L (R)
R t∈2Z
dx 1
Š1/p dr
|r|
By Theorem 2.24, we further obtain
Z1
Z
€
Š1/p dr
p
p
®
kru(x 1 , ·)kLip k f (x 1 + r, ·)k L p (R) dx 1
|r|
r=−1
R
Z1
Z
Z1
€
Š1/p
p
®
k f (x 1 + r, ·)k L p (R) dx 1
dr ®
k f k L p (R2 ) dr ® k f k L p (R2 ) .
r=−1
R
This finishes the proof of Theorem 1.6.
r=−1
SQUARE FUNCTIONS FOR DIRECTIONAL OPERATORS ON THE PLANE
11
Proof of Theorem 1.7. By [LL10, Theorem 1.18] the operators f 7→ Tu (Ψ t ∗ f ) are bounded on L 2 (R2 )
uniformly in 0 < t < kukLip . They are also trivially bounded for all t ≥ kukLip . To see this split
Z1
Z1
dt
dt
( f (x + t, y + u(x, y)t) − f (x + t, y)) .
f (x + t, y) +
Tu f (x, y) =
t
t
−1
−1
The first term is a one-dimensional truncated Hilbert transform on each horizontal line, and therefore
bounded on any L p , 1 < p < ∞. The second term can be written as
Z 1 Z u(x, y)t
dt
∂2 f (x + t, y + s)
t
−1 s=0
This is in turn bounded by
1
Z
M2 ∂2 f (x + t, y)dt ≤ M1 M2 ∂2 f (x, y),
−1
where Mi denote the Hardy–Littlewood maximal function in the i-th variable. The differential operator
∂2 is L p bounded on the subspace of functions with fˆ(ξ, η) = 0 for |η| > C and therefore we obtain L p
estimates for this term.
Remark 2.26. The same argument can be used to estimate Tu on functions with small horizontal frequencies, thus simplifying an argument in [GT16, Section 3].
P
With a suitable choice of Ψ and a lacunary set of t’s we will have f = t Ψ t ∗2 f . Then the operators
f 7→ Pt Tu (Ψ t ∗2 f ) are uniformly bounded, almost orthogonal, and their sum over a suitable set of t’s
differs from Tu by an L 2 bounded operator by Theorem 1.6.
2.6. Application to Hilbert transforms along Lipschitz variable parabolas. In this section we prove
(2)
Theorem 1.9. For each k ∈ Z, let Pk be a Littlewood-Paley projection in the second variable. By
Theorem 1.6, it suffices to show
X
1/2 (2)
® kf k .
(2.27)
|Tu(α) Pk f |2
p
p
k∈Z
We will reduce this square function estimate to an estimate in [Guo+16]. In the following, we will
assume that u > 0 almost everywhere. The region that u ≤ 0 can be handled similarly.
We begin with splitting off pieces of our operator that can be reduced to one-dimensional objects.
The small scale operator
Z
X
dt 1/2 (2)
p
|
Pk f (x − t, y − u(x, y)t α ) |2
L (x, y)
t
|t|≤(2k u(x, y))−1/α
k∈Z
can be estimated using the fundamental theorem of calculus by
Z
X
|
k∈Z
(2)
(Pk f
Z
(x − t, y) +
|t|≤(2k u(x, y))−1/α
−u(x, y)t α
(2)
∂2 Pk f (x − t, y + s)ds)
s=0
Z
X
≤
|
k∈Z
(2)
Pk f (x − t, y)
|t|≤(2k u(x, y))−1/α
Z
X
+
k∈Z
|t|≤(2k u(x, y))−1/α
dt 2 1/2 p
|
L (x, y)
t
dt 2 1/2 p
|
L (x, y)
t
(2)
|u(x, y)t α |MV ∂2 Pk f (x − t, y)
dt 2 1/2 p
.
L (x, y)
|t|
The former term can be estimated using the vector-valued estimate for the maximally truncated Hilbert
transform. Using integrability of |t|α−1 near zero we estimate the latter term by
X
1/2 (2)
p
|MH 2−k MV ∂2 Pk f (x, y)|2
.
L (x, y)
k∈Z
12
FRANCESCO DI PLINIO, SHAOMING GUO, CHRISTOPH THIELE, AND PAVEL ZORIN-KRANICH
(2)
Using the Fourier support assumption on Pk f this is estimated by
X
1/2 (2)
p
|MH MV MV Pk f (x, y)|2
,
L (x, y)
k∈Z
and this can be estimated using the Fefferman–Stein maximal inequality.
The remaining part of the kernel with (2k u(x, y))−1/α < |t| < 1 is decomposed dyadically. Let
v(x, y) := blog2 u(x, y)c. For the sake of convenience, u(x, y) and v(x, y) will be abbreviated as uz
and vz correspondingly, with z standing for (x, y). Denote
Z
vz
dt
f (x − t, y − uz t α )ψl (2 a (2−vz uz )β t) ,
(2.28)
Auz ,l f (x, y) =
t
R
1
−l
with β := P
α−1 and ψl (t) = ψ0 (2 t), where ψ0 is a non-negative even function supported on ±[1/2, 2]
such that l∈Z ψl (t) = 1 for all t 6= 0 (as in [Guo+16, (1.10)]). Notice that
1 ≤ 2−vz uz < 2 for every z ∈ R2 .
(2.29)
This factor is introduced for a scaling purpose. We refer to Lemma 4.1 in [Guo+16]. With this factor,
one can apply a local smoothing estimate (Theorem A.1 [Guo+16]).
Our operator can be written as
€ XX
2 Š1/2
(2)
.
1l−k/α≤v/α Auz ,l−k/α Pk f (x, y)
(2.30)
k≥0 l>0
Pulling the summation in l out of the `2 norm and performing a further Littlewood-Paley decomposition in the first variable we obtain the estimate
X € X
2 Š1/2
(1)
(2)
1
A
P
(2.31)
P
f
(x,
y)
l−k/α≤v/α uz ,l−k/α <k/α−10l+v /α k
z
k∈Z
l>0
(2.32)
+
X € X
X
2 Š1/2
(2)
1l−k/α≤v/α Auz ,l−k/α Pm(1) Pk f (x, y)
,
k∈Z m≥k/α−10l+vz /α
l>0
P
(1)
where P<k/α−10l+v /α = m<k/α−10l+vz /α Pm(1) . In both terms we will obtain estimates that decay exponenz
tially in l. The term (2.31) can be estimated similarly to the small scale term, writing
(2)
(2)
(1)
(1)
P<...
Pk f (x − t, y − uz t α ) − P<...
Pk f (x, y − uz t α )
(2)
(1)
as an integral of ∂1 P<...
Pk f . In the term involving the derivative we use compact Fourier support to
replace ∂1 by a convolution with a test function, while in the remaining term we find a one-dimensional
(2)
integral operator in the vertical direction at scale 2k−αl , which leads to additional cancellation with Pk .
It remains to estimate the term (2.32). To this end it suffices to find some γ > 0 such that
€ X X
2 Š1/2 (2)
® 2−γl k f k .
(2.33)
1l−k/α≤vz /α Auz ,− k +l Pm(1) Pk f (x, y)
p
p
k∈Z m≥ k −10l+ vz
α
α
α
Here we point out that this estimate has essentially been established in [Guo+16]. We will only give
a sketch of the proof. First of all, we recognise that the left hand side of (2.33) is essentially the term
(5.13) in [Guo+16]. By the local smoothing estimates and certain interpolation argument, the L p
bounds of (2.33) for all 1 < p ≤ 2 have been established in Subsection 5.3 in [Guo+16]. To prove L p
bounds for all p > 2, we cite the pointwise estimate (3.19) in [Guo+16], which further implies that
€ X X
2 Š1/2 (2)
® l 4k f k .
(2.34)
1l−k/α≤vz /α Auz ,− k +l Pm(1) Pk f (x, y)
p
p
k∈Z m≥ k −10l+ vz
α
α
α
A further interpolation gives the desired estimate (2.33) for all 1 < p < ∞. This finishes the proof of
the square function estimate (2.27).
SQUARE FUNCTIONS FOR DIRECTIONAL OPERATORS ON THE PLANE
3. SINGLE
13
SCALE OPERATOR
In this section we prove Theorem 1.2. The strategy is to use duality and outer Hölder inequality to reduce the estimate to two estimates of Carleson embedding flavor, the “energy embedding” in Section 3.2
and the “mass embedding” in Section 3.3.
3.1. Tiles and the outer measure space. We subdivide the parameter space into tiles. Each tile can
be represented in three equivalent ways:
(1) by a shearing matrix
 k1
‹
2
0
A=
, k1 , k2 , l ∈ Z
l2k1 2k2
and the spatial location (2−k1 n1 , 2−k2 n2 ), n1 , n2 ∈ Z.
(2) by the corresponding spatial parallelogram P = A−1 ([0, 1] × [0, 1]) + (2−k1 n1 , 2−k2 n2 ), or
(3) by the corresponding frequency parallelogram A∗ ([0, 1]×[1, 2]) and the spatial location (2−k1 n1 , 2−k2 n2 ).
Figure 1 shows the spatial and the frequency parallelograms of a tile (with n1 = n2 = 0). The frequency
picture also includes the symmetric parallelograms A∗ ([0, 1] × [−2, −1]) (in a lighter shade of gray),
because the Fourier transforms of the wave packets associated to tiles will concentrate on both these
parallelograms. However, for combinatorial purposes it suffices to consider only the upper parallelogram. The slope of a tile is the number −l2−k2 +k1 . It is the slope of the lower and the upper side
of the corresponding spatial parallelogram. The spatial parallelogram seems to be the most concise
description of a tile, so we denote tiles by the letter P (for “parallelogram”).
The fact that we are dealing with a single scale operator in Section 3 is reflected in that we define an
outer measure on a finite set X of tiles with k1 = 0, that is, tiles with the fixed horizontal scale 1. (The
restriction to finite sets of tiles avoids technicalities associated with infinite sums. All estimates will be
independent of the specific finite set, so we can pass to the set of all tiles at the end of the argument.)
The outer measure is generated by a function σ whose domain E is the collection of all non-empty
subsets of X . We denote by C P the parallelogram with the same slope and center as P but side lengths
multiplied by C. For R ∈ E set
(3.1)
σ(R) := sup L −C ∪
LR,
R∈R
L≥1
where C is a large number to be chosen later. The three sizes that we need are
X
−1
S1 (F )(R) := σ(R)
|R||F (R)|,
R∈R
S2 (F )(R) := σ(R)
−1
X
|R||F (R)|2
1/2
= S1 (F 2 )(R)1/2 ,
R∈R
S∞ (G)(R) := sup|G(R)|.
R∈R
3.2. Wave packets and the energy embedding. Let Φ = ΦC be the set of functions on R2 that satisfy
|∂ α φ(x)| ≤ (1 + |x|)−C ,
for some sufficiently large C that will be chosen later and
Z
x 2n φ(x 1 , x 2 )dx 2 = 0,
x 1 ∈ R,
kαk`1 ≤ C,
n = 0, . . . , C − 2.
R
We think of φ as morally supported on [0, 1]2 and of φ̂ as morally supported on [0, 1]×[1, 2] for φ ∈ Φ.
The L ∞ normalized wave packets associated to a tile P = (A, n1 , n2 ) are the functions of the form
(∞)
φP
(x) = φ(A(x 1 − 2−k1 n1 , x 2 − 2−k2 n2 )),
φ ∈ Φ.
(p)
(∞)
The L p normalized wave packets, 1 ≤ p < ∞, are the functions φ P = det(A)1/p φ P . Note that
×
φ(A·)(ξ)
= (det A)−1 φ̂(A−∗ ξ). The spatial and the frequency parallelograms of a tile correspond to the
moral space/frequency support of the wave packets associated to this tile.
14
FRANCESCO DI PLINIO, SHAOMING GUO, CHRISTOPH THIELE, AND PAVEL ZORIN-KRANICH
x2
1

1 x1
0
x2
(−l + 1)2−k2
−l2−k2
2−k2

2−k1
0 
A−1 =
−l2−k2 2−k2
−−−−−−−−−−−−−−→
0
ξ2
2
2
1
2−k1x 1
ξ2
k2 +1
2 k2
0
1 ξ1
ξ1
k
l2k1 (l + 1)2 1 0


2k1 l2k1 
0 2 k2
−−−−−−−−−−→
A∗ =
FIGURE 1. Spatial and frequency parallelograms of a tile
3.2.1. Almost orthogonality. The fundamental property of the wave packets is their almost orthogonality for tiles with different scales or slopes.
Lemma 3.2.
(2)
(2)
0
0
0
|〈φ P , φ P 0 〉| ® min(1, (2max(k2 ,k2 ) |2−k2 l − 2−k2 l 0 |)−C , 2−C|k2 −k2 | ),
where C can be made arbitrarily large provided that the order of decay in the definition of Φ is sufficiently
large.
Proof. Without loss of generality suppose k2 ≥ k20 . We will estimate
Z
× φ
Ù
0 (A0 ·)|
|φ(A·)||
R2
for φ, φ ∈ Φ. This is sufficient because the spatial location of the tiles only affects the phase of the
Fourier transforms of the associated wave packets, but not their magnitude.
0
Correlation decay due to shearing. Let 0 < ε 1 and SN = {−N , N } × R be a vertical strip of width
N ≥ 1. The critical intersection A∗ SN ∩ (A0 )∗ SN is a parallelogram centered at zero of width ∼ N and
0
height ∼ N /|2−k2 l − 2−k2 l 0 |. By the vanishing moments assumption we have
0
× ® 2−k2 (2−k2 N /|2−k2 l − 2−k2 l 0 |)C
|φ(A·)|
× and φ
Ù
0 (A0 ·) are L 1 normalon the critical intersection. Using the fact that the Fourier transforms φ(A·)
ized functions and the decay of these Fourier transforms at infinity we obtain
Z
Z
Z
Z
× φ
Ù
0 (A0 ·)| ≤
|φ(A·)||
+
+
R2 \(A0 )∗ SN
R2 \A∗ SN
R2
× +
≤ sup |φ(A·)|
R2 \A∗ SN
A∗ SN ∩(A0 )∗ SN
Ù
0 (A0 ·)| +
sup |φ
R2 \(A0 )∗ SN
sup
A∗ SN ∩(A0 )∗ SN
×
|φ(A·)|
0
0
® 2−k2 N −C(1/ε−1) + 2−k2 N −C(1/ε−1) + 2−k2 (2−k2 N /|2−k2 l − 2−k2 l 0 |)C .
0
0
0
Choosing N = 2ε(k2 −k2 )/C (2k2 |2−k2 l − 2−k2 l 0 |)ε as we may provided that |l − 2k2 −k2 l 0 | ≥ 1, we obtain
Z
0
0
× φ
Ù
0 (A0 ·)| ® 2−k2 +ε(k2 −k2 ) (2k2 |2−k2 l − 2−k2 l 0 |)−(1−ε)C ,
|φ(A·)||
R2
and this gives the second estimate in the conclusion of the lemma.
SQUARE FUNCTIONS FOR DIRECTIONAL OPERATORS ON THE PLANE
15
0
Correlation decay for separated scales. Let 2k2 N 2k2 . Using again the fact that the Fourier trans× and φ
Ù
0 (A0 ·) are L 1 normalized functions and the decay of Fourier transforms near ξ = 0
forms φ(A·)
2
and at infinity we obtain
Z
Z
Z
× φ
Ù
0 (A0 ·)| ≤
+
|φ(A·)||
R2
|ξ2 |≤N
|ξ2 |≥N
× + sup |φ
Ù
0 (A0 ·)|
≤ sup |φ(A·)|
|ξ2 |≤N
−k2
®2
0
|ξ2 |≥N
0
0
(N /2 ) + 2−k2 (N /2k2 )−(C+1)/ε .
k2 C
0
Choosing N ∼ 2k2 +ε(k2 −k2 ) we obtain
Z
0
0
0
0
× φ
Ù
0 (A0 ·)| ® 2−k2 −C(1−ε)(k2 −k2 ) = 2−k2 /2−k2 /2−(C+1/2−ε )(k2 −k2 ) ,
|φ(A·)||
R2
and this gives the third estimate in the conclusion of the lemma.
3.2.2. Bessel inequality.
Lemma 3.3. For each tile P fix an L 2 normalized wave packet φ P adapted to P. Then
X
|〈 f , φ P 〉|2 ® k f k22 .
P
Proof. Schur’s test
X
X
φ P 〈φ P , f 〉
|〈 f , φ P 〉|2 = f ,
P
P
X
≤ k f k2 φ P 〈φ P , f 〉2
P
X
1/2
〈 f , φ P 〉〈φ P , φ P 0 〉〈φ P 0 , f 〉
= k f k2
P,P 0
X
X
1/2
≤ k f k2
|〈 f , φ P 〉|2
|〈φ P , φ P 0 〉|
P0
P
shows that it suffices to prove
sup
P
X
|〈φ P , φ P 0 〉| < ∞.
P0
For a fixed tile P we split the above sum according to the shearing matrix A0 of the tile P 0 . For a given
shearing matrix A0 we distinguish the cases k2 ≤ k20 and k2 > k20 .
In the case k2 ≤ k20 the tile P has larger scale than P 0 , so the tail of the associated wave packet is
more important. For L ∈ 2N let
R̃ L := {P 0 with shearing matrix A0 such that LP ∩ P 0 6= ;}
and let R1 := R̃1 , R L := R̃ L \ R̃ L/2 for L ≥ 2. Then
0
0
|R̃ L | ® L(L2−k2 + |2−k2 l − 2−k2 l 0 |)/2−k2 ,
and
X X
L∈2N P 0 ∈R L
|〈φ P , φ P 0 〉| ®
X
L∈2N
0
0
0
|R̃ L | min(L −C , 2−C(k2 −k2 ) , (2k2 |2−k2 l − 2−k2 l 0 |)−C )),
16
FRANCESCO DI PLINIO, SHAOMING GUO, CHRISTOPH THIELE, AND PAVEL ZORIN-KRANICH
where the first estimate inside the minimum is due to spatial separation and the other two estimates
come from Lemma 3.2. Summing this over k20 ≥ k2 and l 0 we obtain
X
0
0
0
0
0
L(L2−k2 + |2−k2 l − 2−k2 l 0 |)/2−k2 min(L −C , 2−C(k2 −k2 ) , (2k2 |2−k2 l − 2−k2 l 0 |)−C )
L∈2N ,k20 ≥k2 ,l 0 ∈Z
®
X
L(L2k + |2k l − l 0 |) min(L −C , 2−C k , |2k l − l 0 |−C )
L∈2N ,k≥0,l 0 ∈Z
X
®
L(L2k + |2k l − l 0 |)(L + 2k + |2k l − l 0 |)−C ≤ C.
L∈2N ,k≥0,l 0 ∈Z
In the region k2 ≥ k20 we make a similar decomposition with
R̃ L := {P 0 with shearing matrix A0 such that P ∩ LP 0 6= ;}.
The resulting estimate is similar to the above with the roles of k2 and k20 reversed.
3.2.3. Splitting into compactly supported wave packets. In order to obtain a localized Bessel inequality
we decompose wave packets into compactly supported parts as in [Mus+06, Lemma 3.1].
Lemma 3.4. For every C there exists C 0 such that if C 0 φ ∈ ΦC 0 , then there exists a decomposition
X
φ=
2−C k φk , φk ∈ ΦC , supp φk ⊂ B(0, 2k ).
k≥0
Sketch of proof. Let ψ be a smooth function supported on B(0, 1/2) and identically equal to 1 on
B(0, 1/4). Write ψk (x) = ψ(2−k x) for its L ∞ dilates. Let also η(0) , . . . , η(C−2) be smooth functions
supported on [−1/2, 1/2] with
Z
x n η(m) (x)dx = 1n=m .
For k ∈ N and x 1 ∈ R let
(n)
mk (x 1 )
Z
x 2n φ(x 1 , x 2 )ψk (x 1 , x 2 )dx 2 ,
:=
R
then
|∂
α
(n)
mk (x 1 )|
Z
= x 2n ∂1α φ(x 1 , x 2 )(ψk (x 1 , x 2 ) − 1)dx 2 ® 2−C k (1 + |x 1 |)−C ,
|α| ≤ C, n < C,
R
provided that C 0 is sufficiently large. The claimed splitting is given by
PC−2 (n)
(n)
φ(ψk − ψk−1 ) − n=0 (mk − mk−1 ) ⊗ η(n) ,
P
φk :=
C−2
(n)
φψ0 − n=0 m0 ⊗ η(n) ,
k > 0,
k = 0.
3.2.4. Energy embedding. The energy embedding is defined by
(1)
F (R) := sup|〈 f , φR 〉|,
(1)
φR
R ∈ X,
where the supremum is taken over all L 1 normalized wave packets adapted to R with a sufficiently large
order of decay C 0 .
Lemma 3.5. kF k L 2,∞ (S2 ) ® k f k2 .
Proof. Let R be a maximal collection of tiles with S2 (F )(R) ≥ λ. If R 0 ⊂ X \ R also has size ≥ λ, then
using subadditivity of σ it is easy to see that R ∪ R 0 also has size ≥ λ, contradicting maximality. Hence
by maximality we have outsupX \R S2 (F ) ≤ λ. On the other hand,
X
σ(R) ≤ λ−2
|R||F (R)|2 ® λ−2 k f k22
R∈R
by Lemma 3.3.
Lemma 3.6. kF k L ∞ (S2 ) ® k f k∞ .
SQUARE FUNCTIONS FOR DIRECTIONAL OPERATORS ON THE PLANE
17
Proof. Let R ∈ E and let φR , R ∈ R, be wave packets that almost extremizePF (R). Splitting the corresponding members of ΦC 0 using Lemma 3.4 we obtain decompositions φR = k≥0 2−C k φR,k , where each
φR,k is an L 1 normalized wave packet adapted to R (with a lower order of decay C) and supported on
2k R.
By Lemma 3.3 and the support condition we have
X
2
|R||〈 f , φR,k 〉|2 ® 2−2C k f 1∪{2k R:R∈R} 2
R∈R
≤ 2−2C k k f k2∞ ∪R∈R 2k R
≤ 2(C3.1 −2C)k k f k2∞ σ(R),
and summing in k we obtain
X
|R||〈 f , φR 〉|2 ® k f k2∞ σ(R),
R∈R
so that S2 (F )(R) ® k f k∞ as required.
3.3. Covering lemma for parallelograms and the mass embedding. For completeness we include a
slightly streamlined proof of a covering lemma from [BT13] inspired by [CF75]. We consider parallelograms with two vertical edges as shown below:
B
A
C
D
R
I
The height H(R) is the common length of AB and C D. The shadow I(R) is the projection of R onto the
horizontal axis. The slope s(R) is the common slope of the edges BC and AD. The uncertainty interval
U(R) ⊂ R is the interval between the slopes of BD and AC. It is the interval of length 2H(R)/|I(R)|
centered at s(R).
Lemma 3.7 ([BT13, Lemma 7]). Let R a finite collection of parallelograms with vertical edges and dyadic
shadow. Let u : R2 → R be a measurable function and
E(R) := {(x, y) ∈ R : u(x, y) ∈ U(R)}.
Then there exists a decomposition R = G t B such that
X
[
(3.8)
R| ®
|R|,
|
R∈R
(3.9)
R∈G
Z X
X
n
(
1 E(R) ) ®n
|R|,
R∈G
n ∈ N.
R∈G
In [BT13] this lemma is stated for one-variable vector fields, but this structural assumption is not
used in the proof.
In the proof of Lemma 3.7 we denote by CR the parallelogram with the same center, slope, and
shadow as R but height C H(R) (this definition of CR is used only here). We need the following geometric
observation:
Lemma 3.10. Let R, R0 be two parallelograms with I(R) = I(R0 ), U(R) ∩ U(R0 ) 6= ;, and R ∩ R0 6= ;. If
7H(R) ≤ H(R0 ), then 7R ⊆ 7R0 .
Let MV denote the Hardy–Littlewood maximal operator in the vertical direction:
Z
(3.11)
MV f (x, y) = sup|J|−1
y∈J
| f (x, z)|dz,
J
where the supremum is taken over all intervals J containing y.
18
FRANCESCO DI PLINIO, SHAOMING GUO, CHRISTOPH THIELE, AND PAVEL ZORIN-KRANICH
Proof of Lemma 3.7. We partition R by the following iterative procedure. Initialize
ST OC K = R
G =;
B = ;.
While S T OC K 6= ;, choose an R ∈ ST OC K with maximal |I(R)| and update ST OC K := ST OC K \ {R}.
Let G (R) be the set of all elements in G (which are in particular chosen prior to R) such that E(R) ∩
E(R0 ) 6= ;. If
X
MV (
17R0 )(x, y) ≥ 10−4
R0 ∈G (R)
for all (x, y) ∈ R then update B := B ∪ {R}, otherwise update G := G ∪ {R}. It is clear that this
procedure yields a partition R = G t B.
By construction
X
[
R ⊂ {x : MV (
(3.12)
17R )(x) ≥ 10−4 },
R∈R
r∈G
and (3.8) follows by the weak (1, 1) inequality for MV .
We prove (3.9) by induction on n. For n = 1 the statement clearly holds. Suppose now that it holds
for a given n and write
Œn+1
Z ‚X
X
=
1 E(R)
|E(R0 ) ∩ · · · ∩ E(R n )|.
R0 ,...,R n ∈G
R∈G
The summands in which one of the R i ’s occurs at least twice can be estimated by the inductive hypothesis. In the remaining summands we can arrange R0 , . . . , R n in the order in which they have been
selected, and omitting some vanishing terms we obtain the estimate
X
|E(R0 ) ∩ · · · ∩ E(R n )|.
(n + 1)!
R0 ∈G ,R1 ∈G (R0 ),...,R n ∈G (R n−1 )
Replacing the exceptional sets by dilates of the corresponding recatangles we estimate this by
X
(3.13)
|7R0 ∩ · · · ∩ 7R n |.
R0 ∈G ,R1 ∈G (R0 ),...,R n ∈G (R n−1 )
Consider R ∈ G and R0 ∈ G (R). Then in particular I(R) ⊂ I(R0 ) and U(R) ∩ U(R0 ) 6= ;. If H(R) ≤ 7H(R0 ),
then Lemma 3.10 shows that R ⊂ 49R0 , so that MV (17R0 ) ≥ 49−1 on R, contradicting R ∈ G . Therefore
H(R) > 7H(R0 ), and Lemma 3.10 shows that
7R0 ∩ (I(R) × R) ⊂ 7R.
It follows that (7H(R))−1
0
R0 ∈G (R) 7H(R )
P
(3.13) =
< 10−4 , since otherwise R would have been put into B. Hence
X
|I(R0 )| · 7|H(R n )|
R0 ∈G ,R1 ∈G (R0 ),...,R n ∈G (R n−1 )
X
®
|I(R0 )| · 7|H(R n−1 )|
R0 ∈G ,R1 ∈G (R0 ),...,R n−1 ∈G (R n−2 )
® ··· ®
X
|I(R0 )| · 7|H(R0 )| ®
R0 ∈G
X
|R0 |.
R0 ∈G
3.3.1. Mass embedding. The mass embedding is given by
Z
G(R) := |R|−1
|g|,
R ∈ X.
ER
Lemma 3.14. Let 1 < q < ∞. If the constant C in the definition of σ is sufficiently large depending on
q, then kGk L q,∞ (S∞ ) ® kgkq .
SQUARE FUNCTIONS FOR DIRECTIONAL OPERATORS ON THE PLANE
19
Recall that C P now again denotes the parallelogram P expanded by the factor C both in the horizontal
and in the vertical direction.
Proof. Let δ > 0, g ∈ L q (R2 ), and let R be a collection of tiles such that G(R) ≥ δ for R ∈ R . We have
to show
(3.15)
sup L −C ∪R∈R LR ®q δ−q kgkqq .
L≥1
Note that the definition of G(R) makes sense for arbitrary parallelograms (not only the dyadic ones that
we call tiles). For the enlarged parallelograms LR we still have G(LR) ≥ δ/L 2 , so it suffices to show
(3.15) with L = 1 and a collection of arbitrary parallelograms R, provided that the constant C3.1 in the
definition of σ is at least 2q.
Enlarging the parallelograms in such a way that their shadows become intervals in adjacent dyadic
grids and the uncertainty intervals stay the same we preserve the hypothesis G(R) ¦ δ up to a multiplicative constant. Hence we may assume that the parallelograms have dyadic shadows.
Let R = G t B be the decomposition provided by Lemma 3.7. In view of (3.8) it suffices to consider
the parallelograms in G . By the density assumption and Hölder’s inequality we have
‚
Œ1/q0
X
X
X
X
X1Z
1
1
1
|g| = 1 E(R) |g|1 ≤ k
1 E(R) kq0 kgkq ®
|R|
kgkq ,
|R| ≤
δ
δ
δ
δ
E(R)
R∈G
R∈G
R∈G
R∈G
R∈G
where in the last passage we have used the estimate (3.9) that also nolds for non-integer values of n
since its left-hand side is monotonic in n. After division by the middle factor of the right hand side we
obtain the claim.
3.4. Estimate for the square function. We finally prove Theorem 1.2. Note that Tu,φ P2,t f (x) is the
integral of f against an L 1 normalized wave packet associated to a tile that contains x and whose
uncertainty interval contains u(x). Hence the left-hand side of (1.3) is bounded by
X
X
1/2
1/2
F (R)2 1 ER
kp = k
F (R)2 1 ER k p/2 .
k
R∈X
Dualizing with a function g ∈ L
R∈X
(p/2)0
we obtain
Z X
X
F (R)2 1 ER g =
|R|F (R)2 G(R).
R∈X
R∈X
For every R ∈ E we have R∈R |R|F (R) = σ(R)S1 (F )(R). Therefore by [DT15, Proposition 3.6] and
outer Hölder inequality [DT15, Proposition 3.4] the above is bounded by
P
kF 2 Gk L 1 (S1 ) ® kF 2 k L p/2 (S1 ) kGk L (p/2)0 (S∞ ) = kF k2L p (S2 ) kGk L (p/2)0 (S∞ ) .
The first term is bounded by k f k2p by Lemmas 3.5 and 3.6 and interpolation [DT15, Proposition 3.5].
The second term is bounded by kgk(p/2)0 by Lemma 3.14 and interpolation [DT15, Proposition 3.5].
3.5. Application to a maximal operator with a restricted set of directions. Although the operator
(1.1) is unbounded for general direction fields u, it is clearly bounded (on any L p , 1 ≤ p ≤ ∞) with
norm p
O(N ) as long as u is allowed to take at most N values. This trivial estimate has been improved
to O( log N ) on L 2 by Katz [Kat99]. Note that we also have the trivial estimate O(1) on L ∞ , and
by interpolation one obtains logarithmic dependence on N of the operator norm of (1.1) on L p also
for all 2 < p < ∞. Demeter [Dem10] gives an alternative proof of Katz’s result and furthermore
hints at an approach to this estimate using a good lambda inequality with sharp constant due to Chang,
Wilson, and Wolff [CWW85]. The suggested argument should be along the lines of similar arguments in
Demeter [Dem10] and in Grafakos, Honzík, and Seeger [GHS06]. We have not been able to reproduce
the endpoint p = 2 using this technique, but observed that our square function inequality allows us
to reproduce the result for p > 2 up to an arbitrarily small loss in the exponent of the logarithm:
Specifically, we prove
20
FRANCESCO DI PLINIO, SHAOMING GUO, CHRISTOPH THIELE, AND PAVEL ZORIN-KRANICH
Corollary 3.16. Assume the measurable function u : R2 → [−1, 1] takes at most N different values. For
2 < p < ∞ there is a constant C depending on φ and p such that
kTu f k p ≤ C log(N + 2)1/2 k f k p .
Proof. For j ∈ Z, define the dyadic martingale averaging operator
X
(3.17)
E j f :=
22 j 〈 f , 1Q 〉1Q ,
where the summation runs over all standard dyadic squares Q in R2 with side length 2− j . Further define
∆ j = E j+1 − E j ,
X
∆ f := ( |∆ j f |2 )1/2 .
j∈Z
Let M denote the non-dyadic Hardy–Littlewood maximal operator. Chang, Wilson, and Wolff [CWW85,
Corollary 3.1] prove that there are universal constants c1 and c2 such that for all λ > 0 and 0 < ε < 1
c1
|{z : | f (z) − E0 f (z)| > 2λ, ∆ f (z) ≤ ελ}| ≤ c2 e− ε2 |{z : M f (z) ≥ λ}|.
(3.18)
Denote the finitely many values of u by ui , 1 ≤ i ≤ N , and write Tui for the operator with the constant
direction field ui . Corollary 3.16 follows by Marcinkiewicz interpolation from the weak type inequality
|{z : sup|Tui f (z)| > 4λ}| ≤ C log(N + 2) p/2 λ−p k f k pp
i
for 2 < p < ∞. Gearing up for Chang, Wilson, and Wolff we estimate
|{z : sup|Tui f (z)| > 4λ}|
i
[
= | {z : |Tui f (z)| > 4λ}|
i
[
≤ | {z : |Tui f (z) − E0 Tui f (z)| > 2λ, ∆Tui f (z) ≤ ελ}|
(3.19)
i
(3.20)
[
+ | {z : |E0 Tui f (z)| > 2λ}|
(3.21)
i
[
+ | {z : ∆Tui f (z) > ελ}|
i
Using (3.18) we estimate
(3.19) ≤
X
|{z : |Tui f (z) − E0 Tui f (z)| > 2λ, ∆Tui f (z) ≤ ελ}|
i
≤C
≤C
X
i
X
c1
e− ε2 |{z : M (Tui f )(z) > 2λ}|
c1
e− ε2 λ−p kTui f k pp
i
c1
≤ C N e− ε2 λ−p k f k pp
≤ Cλ−p k f k pp
1/2
provided ε ≤ c1 log(N + 2)1/2 .
The function E0 Tui f in (3.20) is pointwise dominated by the standard Hardy–Littlewood maximal
operator, because E0 and Tui compose to some averaging operator at scale 0. Therefore
(3.20) ≤ |{z : M f (z) > Cλ}| ® λ−p k f k pp .
REFERENCES
21
To control (3.21) we introduce a suitable Littlewood–Paley decomposition in the second variable,
note that P2k commutes with Tui , and estimate pointwise
X
sup ∆Tui f = sup ∆(
P2k Tui P2k f )
i
i
Xk∈Z
X
= sup( |
∆ j P2k Tui P2k f |2 )1/2
i
j
k
XX
0
® sup( (
2−| j−k|/q M Mq,V Tui P2k f )2 )1/2
i
j
k
XX
0
® sup(
2−| j−k|/q (M Mq,V Tui P2k f )2 )1/2
i
j
k
X
® sup( (M Mq,V Tui P2k f )2 )1/2
i
X t
≤ ( (M Mq,V sup|Tui P2k f |)2 )1/2 ,
k
i
where Mq,V is the q-maximal operator in the vertical direction Mq,V f = (MV ( f q ))1/q for any fixed 1 <
q < 2 with MV as in (3.11), M is the usual two-dimensional Hardy–Littlewood maximal operator, and
0
the pointwise estimate |∆ j P2k f | ® 2| j−k|/q M Mq,V f follows from [GHS06, Sublemma 4.2] applied in the
vertical direction. The Fefferman–Stein maximal inequalities and Theorem 1.2 give
X
X
2 1/2
(M Mq,V sup Tui Pt f ) ) k p ≤ Ck(
(sup Tui Pt f )2 )1/2 k p ≤ Ck f k p .
k(
i
t∈2Z
t∈2Z
i
With Tchebysheff we obtain
(3.21) = |{sup ∆Tui f (z) > ελ}| ≤ C(ελ)−p k f k pp ≤ C log(N + 2) p/2 λ−p k f k pp .
i
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(FDP) DEPARTMENT OF MATHEMATICS, UNIVERSITY OF VIRGINIA, KERCHOF HALL, B OX 400137, CHARLOTTESVILLE, VA
22904-4137, USA
E-mail address: [email protected]
(SG) INDIANA UNIVERSITY BLOOMINGTON, 831 E THIRD ST, BLOOMINGTON, IN 47405, USA
E-mail address: [email protected]
(CT, PZ) MATHEMATICAL INSTITUTE, UNIVERSITY OF B ONN, ENDENICHER ALLEE 60, 53115 B ONN, GERMANY
E-mail address: [email protected]
E-mail address: [email protected]