× ˆ
Bσ(x)(x)
X
k∈Z
|ak(z)|r01Θ(θ−tck(z))1/r0
Wt(y−z) dz dt t dydx .
ˆ
x∈R
Ws(x) ˆ
y∈Bs
ˆ (1x−2/L)σ(x) t=σ(y)
t
2σ(x)Wt(y−x)h(y, t)M(y, θt−1, t)dt t dydx Since the inmost integral vanishes unless|y−x|>2σθ(x), we have that
ˆ (1−2/L)σθ(x) t=σθ(y)
t
2σ(x)Wt(y−x)
2dt t
1/2
.Wσ(x)(y−x) it follows that
II .λ ˆ
x∈R
Ws(x) ˆ
y∈Bs
Wσ(x)(y−x)ˆ s t=0
|h(y, t)|2dt t
1/2
dydx
.λ ˆ
x∈R
Ws(x)M ˆ s
t=0
|h(·, t)|2dt t
1/2 (x)dx
. λkhkL2
(dydt/t)
s1/2 This concludes the proof.
We used the super level set ofMpf instead of the super level set of Mp M f
to define Kf,λ,p. As mentioned in section 7.3.1 of [DPO15], the inner maximal function appears only in the reduction from the case with fbcompactly supported to the case with a general f ∈S(R). By our assumptions we can effectively ignore this complication.
Proposition 2.29 (Proposition 3.2 + equations (2.6) and (2.7) of [DPO15]). The estimate kF1D(x,s)kLq(Se).N,q
1 + dist(sptf;Bs(x) s
−N
kfkLq
holds for allN >0 andq∈(2,∞].
Lemma 2.30 (Equation (7.3) of [DPO15]). The estimate kF1D(x,s)kL∞(Se).N 1 + dist sptf;Bs(x)
s
!−N
inf
z∈Bs(x)M f(z) holds for any N >0.
Corollary 2.31. Suppose that sptf ∩B2s(x) =∅then
kF1D(x,s)kLq(Se)s−1/q.N,p 1 +dist sptf;Bs(x) s
!−N
s−1/pkfkLp
for allp∈[1,2),q > p0, andN >0.
Proof. Ifsptf∩B2s(x) =∅theninfz∈Bs(x)M f(z).s−1kfkL1. Using this fact and interpolating between the bounds from Proposition 2.29 and Lemma 2.30 we obtain the required inequality.
Fixp∈(1,∞]andq∈ max(p0; 2),∞
and letp∈ 1,min(p; 2)
such thatq > p0. We will now show that
kF1X\Kf,λ,pkL-q(Se).λ. (2.95)
Sinceν(Kf,λ,p).λ−pkfkpLp this would prove (2.93).
Let us consider a stripD(x, s)∈D and suppose thatD(x, s)6⊂Kf,λ,p, otherwise the estimate is trivial. We haveB5s(x)6⊂ Kf,λ,p. For anN >1large enough to be chosen later let us set
f(x) =f0(x) +
∞
X
k=1
fk(x) =f(x)υx−x0 5s
+
∞
X
k=1
f(x)γ
x−x0 5s2N k
whereγ(·) =υ(·/2N)−υ(·)with
υ∈Cc∞(B2) υ≥0 υ= 1onB1.
LetFk be associated tofk via the embedding (2.6) and letKfk,λ,p be as in (2.94).
SinceKf0,λ,p ⊂Kf,λ,pwe have thatkf0kLps−1/p.λand
kF01X\Kf,λ,p1D(x,s)kLq(Se).λ1−p/qkf0kp/qLp .λs1/q (2.96) by Theorem 2.28
Since Kfk,λ,p ⊂Kf,λ,p 6⊃B5s(x)one has kfkkLp .λν(D(x, s))1/p2N k/p and by Corollary 2.31 we have that
kFk1X\Kf,λ,p1D(x,s)kLq(Se)s−1/q.2−2N k2N k/pλ.2−N kλ. (2.97) By quasi-subadditivity we can add up (2.96) and (2.97) to obtain
kF1X\Kf,λ,p1D(x,s)kLq(Se)
ν(D(x, s))1/q .λ.
SinceD(x, s)is arbitrary this implies (2.95).
2.6.2 Non-iterated bounds
We conclude by explaining that forr∈(2,∞]andp∈(2, r)and simpler embedding bounds on the maps f 7→ F and a 7→ A are sufficient to prove boundedness onLp(R) of the Variational Carleson Operator (2.2) and thus also (2.1).
Hereafter we work with the non-iterated outer measure space(X, µ). The energy embedding map satisfies theLp bounds
kFkLp(Se).kfkLp p∈(2,∞]. (2.98)
This follows directly from Proposition 2.29 by takingsarbitrarily large.
Similarly, in Proposition 2.20 we have shown that the auxiliary embedding satisfies
kMkLp0(Sm).kakLp0(lr0) p0∈(r0,∞]. (2.99) and thus, by Proposition 2.23 we have that the variational mass embedding also satisfies such bounds:
kAkLp0(Sm).kakLp0(lr0) p0 ∈(r0,∞]. (2.100) It follows by the outer Hölder inequality 2.7 that
˚
X
F(y, η, t)A(y, η, t)dydηdt
.kFkLp(Se)kAkLp0
(Sm).
Using (2.98) and (2.100) and the wave-packet domination (2.13) it follows that (2.2) is bounded onLp(R).
In conclusion we remark that the iterated outer-measureLp spaces that were introduced provide an effective way of capturing the spatial locality property of the embedding maps. Both the proof of Theorem 2.3 and of Theorem 2.2 rely on first obtaining non-iterated bounds (see Propositions 2.20 and 2.29) and then using a locality lemma (see Lemmata 2.21 and 2.30 ) and a projection lemma (see Lemma 2.22 and Lemma 7.8 of [DPO15]) to bootstrap the full result.
Positive sparse domination of variational Carleson operators
Life, as we know it, is based onL2. –F.D.P.
This Chapter contains the paper [DPDU16] Due to its nonlocal nature, the r-variation norm Carleson operatorCrdoes not yield to the sparse domination techniques of Lerner [Ler16; Ler13], Di Plinio and Lerner [DPL14], Lacey [Lac17]. We overcome this difficulty and prove that the dual form toCr can be dominated by a positive sparse form involvingLp averages. Our result strengthens theLp-estimates by Oberlin et. al. [Obe+12]. As a corollary, we obtain quantitative weighted norm inequalities improving on [DL12a] by Do and Lacey. Our proof relies on the localized outerLp-embeddings of Di Plinio and Ou [DPO15] and Uraltsev [Ura16].
3.1 Introduction and main results
The technique of controlling Calderón-Zygmund singular integrals, which are a-priorinon-local, bylocalized positive sparse operators has recently emerged as a leading trend in Euclidean Har-monic Analysis. We briefly review the advancements which are most relevant for the present article and postpone further references to the body of the introduction. The original domination in norm result of [Ler13] for Calderón-Zygmund operators has since been upgraded to a point-wise positive sparse domination by Conde and Rey [CAR16] and Lerner and Nazarov [LN15], and later by Lacey [Lac17] by means of an inspiring stopping time argument forgoing local mean oscillation. Lacey’s approach was further clarified in [Ler16], resulting in the following principle:
ifT is a sub-linear operator of weak-type(p, p)and in addition the maximal operator f 7→ sup
Q⊂Rinterval
T(f1R\3Q)
L∞(Q)1Q (3.1)
embodying the non-locality of T, is of weak-type (s, s), for some 1 ≤ p ≤ s < ∞, then T is pointwise dominated by a positive sparse operator involvingLsaverages off.
The principle (3.1) extends to certain modulated singular integrals. Of interest for us is the maximal partial Fourier transform
Cf(x) = sup
N
ˆ N
∞
fb(ξ) eixξdξ
also known as Carleson’s operator on the real line. The crux of the matter is that (3.1) follows forT =C from its representation as a maximally modulated Hilbert transform, a fact already exploited in the classical weighted norm inequalities for C by Hunt and Young [HY74], and in the more recent work [GMS05]. Together with sharp forms of the Carleson-Hunt theorem near the endpointp= 1 [DP14] this allows, as observed by the first author and Lerner in [DPL14], the domination ofC by sparse operators and thus leads to sharp weighted norm inequalities for C.
In this article we consider ther-variation norm Carleson operator, which is defined for Schwartz functions on the real line as
Crf(x) = sup
N∈N
sup
ξ0<···<ξN
N
X
j=1
ˆ ξj ξj−1
fb(ξ) eixξdξ
r
1/r
.
The importance ofCris revealed by the transference principle, presented in [Obe+12, Appendix B], which shows howr-variational convergence of the Fourier series off ∈Lp(T;w)for a weightw on the torusTfollows fromLp(R;w)-estimates for the sub-linear operatorCr. Values of interest forrare2< r <∞. Indeed the main result of [Obe+12] is that in this range,Crmaps into Lp wheneverp > r0, while noLp-estimates hold for variation exponentsr≤2. Unlike the Carleson operator, its variation norm counterpartCrdoes not have an explicit kernel form and thus fails to yield to Hunt-Young type techniques. The same essential difficulty is encountered in the search forLq-bounds for the nonlocal maximal function (3.1) when T =Cr. Therefore, the approach via (3.1) does not seem to be applicable toCr. In the series [DL12a; DL12b], the second author and Lacey circumvented this issue through a direct proof ofAp-weighted inequalities forCrand its Walsh analogue, based on weighted phase plane analysis.
The main result of the present article is that a sparse domination principle forCrholds in spite of the difficulties described above. More precisely, we sharply dominate the dual form to the r-variational Carleson operatorCrby a single positive sparse form involvingLpaverages, leading to an effortless strengthening of the weighted theory of [DL12a]. Our argument abandons (3.1) in favor of a stopping time construction, relying on the localized Carleson embeddings for suitably modified wave packet transforms of [DPO15] by the first author and Yumeng Ou, and [Ura16] by the third author. In particular, our technique requires noa-priori weak-type information on the operatorT. A similar approach was employed by Culiuc, Ou and the first author in [CDPO16]
in the proof of a sparse domination principle for the family of modulation invariant multi-linear multipliers whose paradigm is the bilinear Hilbert transforms. Interestingly, unlike [CDPO16], our construction of the sparse collection in Section 3.4 seems to be the first in literature which does not make any use of dyadic grids.
We believe that intrinsic sparse domination can prove useful in the study of other classes of multi-linear operators lying way beyond the scope of Calderón-Zygmund theory, such as the iterated Fourier integrals of [DMT17] and the sub-dyadic multipliers of [BB17].
To formulate our main theorem, we recall the notation hfiI,p:= 1
|I|
ˆ
|f|pdx1p
, 1≤p <∞
whereI ⊂Ris any interval, and the notion of asparse collection of intervals. We say that the countable collection of intervalsI ∈ S is η-sparse for some0 < η ≤1 if there exist a choice of measurable sets{EI :I∈S} such that
EI ⊂I, |EI| ≥η|I|, EI∩EJ =∅ ∀I, J∈ S, I 6=J.
Theorem 3.1. Let 2< r <∞ andp > r0. Given f, g∈ C0∞(R) there exists a sparse collection S=S(f, g, p)and an absolute constantK=K(p)such that
|hCrf, gi| ≤K(p)X
I∈S
|I|hfiI,phgiI,1. (3.2) A corollary of Theorem 3.1 is thatCrextends to a bounded sub-linear operator onLq(R) when-everq > r0. As a matter of fact, let us fixq∈(r0,∞], and choosep∈(r0, q). Denoting by
Mpf(x) = sup
I3x
hfiI,p
thep-th Hardy-Littlewood maximal function, the estimate of Theorem 3.1 and the fact thatS is sparse yields
|hCrf, gi|.X
I∈S
|EI|hfiI,phgiI,1≤ hMpf,M1gi.kMpfkqkM1gkq0 .kfkqkgkq0.
Bounds onLq forCr were first proved in [Obe+12], where it is also shown that the restriction q > r0 is necessary, whence no sparse domination of the type occurring in Theorem 3.1 will hold forp < r0. We can thus claim that Theorem 3.1 is sharp, short of the endpointp=r0. In fact, sparse domination as in (3.2) also entailsCr:Lp(R)→Lp,∞(R). Such an estimate is currently unknown forp=r0.
However, Theorem 3.1 yields much more precise information than mere Lq-boundedness. In particular, we obtain precisely quantified weighted norm inequalities forCr. Recall the definition of theAtconstant of a locally integrable nonnegative functionwas
[w]At :=
sup
I⊂R
hwiI,1
w1−t1 t−1
I,1 1< t <∞
inf
A: Mw(x)≤Aw(x)for a.e. x t= 1 Theorem 3.2. Let 2< r <∞ andq > r0 be fixed. Then
(i) there existsK: [1,rq0)→(0,∞)nondecreasing such that
kCrkLq(R;w)→Lq(R;w)≤K(t)[w]max{1,q(t−1)t }
At ;
(ii) there exists a positive increasing functionQ such that fort= rq0
kCrkLq(R;w)→Lq(R;w)≤ Q([w]At). (3.3) We omit the standard deduction of Theorem 3.2 from Theorem 3.1, which follows along lines analogous to the proofs of [CDPO16, Theorem 3] and [LN15, Theorem 17.1]. Estimate (i) of Theorem 3.2 yields in particular that
w∈At =⇒ kCrkLq(R;w)→Lq(R;w)<∞ ∀r >maxn 2,q−tq o
an improvement over [DL12a, Theorem 1.2], where Lq(R;w) boundedness is only shown for variation exponentsr >maxn
2t,q−tqt o
whenw∈At. Fixingrinstead, part (ii) of Theorem 3.2 is sharp in the sense that t = rq0 is the largest exponent such that an estimate of the type of (3.3) is allowed to hold. Indeed, if (3.3) were true for any q = q0 ∈ (r0,∞) and somet = qs0
withs < r0, a version of the Rubio de Francia extrapolation theorem (see for instance [CUMP11, Theorem 3.9]) would yield thatCrmapsLqinto itself for allq∈(s,∞), contradicting the already mentioned counterexample from [Obe+12].
We turn to further comments on the proof and on the structure of the paper. In the upcoming Section 3.2 we reduce the bilinear form estimate (3.2) to an analogous statement for a bilinear form involving integrals over the upper-three space of symmetry parameters for the Carleson operator of a wave packet transforms off and a variational-truncated wave packet transform of g. The natural framework forLp-boundedness of such forms, theLp-theory of outer measures, has been developed by the second author and Thiele in [DT15]. In Section 3.3, we recall the basics of this theory as well as the localized Carleson embeddings of [DPO15] and [Ura16]. These will come to fruition in Section 3.4, where we give the proof of Theorem 3.1. A significant challenge in the course of the proof is the treatment of the nonlocal (tail) components, which are handled via novelad-hocembedding theorems incorporating the fast decay of the wave packet coefficients away from the support of the input functions.
Acknowledgments
This work was initiated and continued during G. Uraltsev’s visit to the Brown University and University of Virginia Mathematics Departments, whose hospitality is gratefully acknowledged.
The authors would like to thank Amalia Culiuc, Michael Lacey, Ben Krause and Yumeng Ou for useful conversations about sparse domination principles.