An α -number characterization of L p spaces on uniformly rectifiable sets
aa r X i v : . [ m a t h . C A ] S e p AN α -NUMBER CHARACTERIZATION OF L p SPACES ON UNIFORMLYRECTIFIABLE SETS
JONAS AZZAM AND DAMIAN D ˛ABROWSKIA
BSTRACT . We give a characterization of L p ( σ ) for uniformly rectifiable measures σ using Tolsa’s α -numbers, by showing, for < p < ∞ and f ∈ L p ( σ ) , that k f k L p ( σ ) ∼ (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:18)Z ∞ (cid:0) α fσ ( x, r ) + | f | x,r α σ ( x, r ) (cid:1) drr (cid:19) (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) L p ( σ ) . C ONTENTS
1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1Sharpness of the result . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3Related work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3Organization of the article . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4Acknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42. Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42.1. Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42.2. Adjacent systems of cubes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52.3. α -numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63. k J f k . k f k . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94. k J f k p . k f k p for < p < ∞ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114.1. Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124.2. Calderón-Zygmund decomposition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134.3. Definition of S . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134.4. Estimating J f . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145. k f k p . k J f k p for < p < ∞ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 165.1. Littlewood-Paley theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 165.2. Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 185.3. Angles between planes approximating f σ and σ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 221. I NTRODUCTION
We say a measure µ in R n is d -rectifiable if µ is absolutely continuous with respect to d -dimensional Hausdorff measure and we may exhaust µ -almost all of R n by countably many Lips-chitz graphs. It is a classical result that, for µ -a.e. x ∈ R n , the densities µ ( B ( x, r )) /r d stabilize as r → in the sense that they converge to a nonzero constant and so on small scales the measure µ scales like d -dimensional Lebesgue measure. What’s more is that the shape of the measure µ alsostabilizes: as we zoom in on x , if we set µ x,r ( A ) = µ ( rA + x ) , then µ x,r r − d converges weakly to d -dimensional Lebesgue measure on some d -dimensional plane. In [Tol09], Tolsa quantified how much a uniformly rectifiable measure can deviate from resem-bling planar Lebesgue measure. Recall that a measure µ is uniformly rectifiable (UR) if firstly, it is Ahlfors d -regular , meaning there is A > so that A − r d ≤ σ ( B ( x, r )) ≤ Ar d for all x ∈ supp σ, < r < diam(supp σ ) , and σ has big pieces of Lipschitz images (BPLI), meaning there are constants L, c > so that foreach x ∈ supp σ and < r < diam(supp σ ) there is an L -Lipschitz mapping f : B d (0 , r ) → B ( x, r ) so that µ ( f ( B d (0 , r )) . We say a set E ⊆ R n is UR if H d | E is UR.Before stating Tolsa’s result, we will describe how he measures the planarity of a measure. First,we define a distance between measures. For two measures µ and ν and a ball B we define F B ( σ, ν ) := sup (cid:26)(cid:12)(cid:12)(cid:12)(cid:12)R φ dσ − R φ dν (cid:12)(cid:12)(cid:12)(cid:12) : φ ∈ Lip ( B ) (cid:27) , where Lip ( B ) is the set of -Lipschitz functions supported in B . This is a variant of the Wasserstein -distance from mass transport theory. See [Mat95, Chapter 14] for a discussion about this distance.For a (possibly real-valued) measure µ and d ∈ N , if B = B ( x, r ) we define α dµ ( x, r ) = α dµ ( B ) := 1 r d +1 inf c ∈ R ,L F B ( µ, c H d | L ) , (1.1)We will often omit the superscript d , as it will be fixed throughout. Theorem 1.1 ([Tol09, Theorem 1.2]) . An Ahlfors d -regular measure σ is UR if and only if themeasure α dσ ( x, r ) dσ ( x ) drr is a Carleson measure, meaning that for all balls B centered on supp σ with < r B < diam(supp σ ) , Z r B Z B α dσ ( x, r ) dσ ( x ) drr ≤ Cσ ( B ) for some fixed C > . Estimates on α -type numbers are particularly useful in studying rectifiability. From a geometricview point, they give quite a lot of information about the shape of a measure. David and Semmes[DS91] gave an earlier characterization of UR sets in terms of a Carleson meausure condition on β -numbers, which are quantities like α -numbers except they only measure the average distance ofa measure to a plane, so while a measure could be very close to lying on a plane, its mass could bevery unevenly distributed resulting in a large α -number. The additional information provided by the α -numbers was crucial for the main result of [Tol09], where Tolsa improved on the work in [DS91]by expanding the class of Calderon-Zygmund operators on UR sets that were known to be bounded.See also [Tol08] where α -numbers are used to characterize rectifiability of sets of finite measure interms of existence of principal values for the Riesz transform, and [DEM18, Fen20, DM20] wherethey are used to study higher co-dimensional analogues of harmonic measure.The purpose of this note is to extend Tolsa’s result to measures that are not Ahlfors regular, butare given by L p functions defined on UR sets.Given a Radon measure σ , f ∈ L loc ( σ ) , and a ball B = B ( x, r ) with σ ( B ( x, r )) > set f B = f x,r = R B f dσσ ( B ) . Main Theorem.
Let σ be a UR measure and f ∈ L p ( σ ) where < p < ∞ . Then k f k L p ( σ ) ∼ (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) Z ∞ (cid:16) α fσ ( x, r ) + | f | x,r α σ ( x, r ) (cid:17) drr ! (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) L p ( σ ) , (1.2) with the implicit constant depending on p and σ . N α -NUMBER CHARACTERIZATION OF L p SPACES 3
Sharpness of the result.
An interesting aspect of our result is the presence of two terms that com-prise our square function. We don’t know whether the result holds for general UR sets without thesecond term. Neither of the terms bounds the other in the pointwise sense: one could be zero whilethe other is nonzero. On the other hand, we don’t know whether the norm of the square functioninvolving only α fσ dominates the one involving only | f | x,r α σ . The reverse inequality is certainlynot true, as the latter square function vanishes if σ is the Lebesgue measure on Σ = R d . Question.
Let σ be a UR measure and f ∈ L p ( σ ) where < p < ∞ . Do we have k f k L p ( σ ) . (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) Z ∞ α fσ ( x, r ) drr ! (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) L p ( σ ) ? (1.3) Equivalently, is it true that (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) Z ∞ ( | f | x,r α σ ( x, r )) drr ! (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) L p ( σ ) . (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) Z ∞ α fσ ( x, r ) drr ! (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) L p ( σ ) ? The answer to the question above is obviously affirmative in the flat case, i.e. σ = H d L for L a d -dimensional plane. It is also positive if σ is an Ahlfors d -regular measure on a d -dimensionalplane L , i.e. σ = g H d L for some g satisfying A − ≤ g ≤ A . Indeed, let ˜ σ = H d L , so that f σ = f g ˜ σ . In that case, by the Main Theorem k f k L p ( σ ) ∼ A k f g k L p (˜ σ ) ∼ p (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) Z ∞ α fg ˜ σ ( x, r ) drr ! (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) L p (˜ σ ) ∼ A (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) Z ∞ α fσ ( x, r ) drr ! (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) L p ( σ ) . Finally, one could show that (1.3) is true for “sufficiently flat” UR measures σ . What we mean bythis is that if the constant C from Theorem 1.1 is sufficiently small, then some variant of Carleson’sembedding theorem can be used to show that (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) Z ∞ (cid:0) | f | x,r α σ ( x, r ) (cid:1) drr ! (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) L p ( σ ) . p C k f k L p ( σ ) ≪ k f k L p ( σ ) . This means that the second term from the square function in (1.2) can essentially be absorbed bythe left hand side. To make this more rigorous, one should perhaps track the dependence of theimplicit constants in (1.2) on the UR constants of σ with more diligence than we did. However, theimplicit constants can only get better as σ becomes flatter, and they certainly cannot blow-up as theCarleson constant C goes to : if σ satisfies the Carleson condition of Theorem 1.1 with some C ,then it also satisfies it with constant C ′ for every C ′ ≥ C . Related work.
While our focus has been in the Ahlfors regular setting, α -numbers have also beenused to study measures in more general general settings. In [ATT18], it was shown that pointwisedoubling measures µ were d -rectifiable on the set where the square function R ∞ α µ ( x, r ) drr wasfinite, resolving a question left open in [ADT16]. This paper also exposed some limitations withworking with α -numbers, as a counterexample showed that the same result is not true for generalmeasures. However, the second author of this paper obtained such a generalization in [D ˛ab19]using a different α -number, which measures distance between a measure and planar measure usingthe Wasserstein -metric, which Tolsa had earlier used to give a characterization of UR measures in For p = 2 use e.g. [Tol14, Theorem 5.8], for p = 2 one can show a corresponding statement by proving anappropriate good-lambda inequality, in the spirit of what we do in Section 4 (but simpler). AZZAM AND D ˛ABROWSKI [Tol12]. So while using the Wasserstein -distance allows one to get a more complete picture, the α -number in Theorem 1.1 has a more transparent definition and thus is easier to work with.In [Orp18] Orponen uses a similar square function to characterize when two measures on thereal line (one being doubling) are absolutely continuous, however among a few of the differencesbetween the α -numbers he uses and ours, while we compare distance between a measure and aplane, his numbers compare the distance between the two measures, which is another interestingdirection. Organization of the article.
In Section 2 we introduce the necessary tools and make some initialreductions. We define also
J f , a dyadic variant of the square function from the Main Theorem, see(2.4).We show that k J f k . k f k in Section 3. The proof uses martingale difference operators,and it is inspired by how Theorem 1.1 was originally proved, see [Tol09, Section 4]. In Section4 we use the estimate k J f k . k f k and an appropriate good-lambda inequality to conclude that k J f k p . k f k p for general < p < ∞ .Finally, in Section 5 we prove k f k p . k J f k p . To do that we use the Littlewood-Paley theory ofDavid, Journé and Semmes [DJS85]. Acknowledgments.
We would like to thank Xavier Tolsa for helpful discussions about the paper.Most of work on the article has been done during second author’s visit to the University ofEdinburgh. We are very grateful to the university staff for their kindness and hospitality.The second author received support from the Spanish Ministry of Economy and Competitiveness,through the María de Maeztu Programme for Units of Excellence in R&D (MDM-2014-0445), andalso partial support from the Catalan Agency for Management of University and Research Grants(2017-SGR-0395), and from the Spanish Ministry of Science, Innovation and Universities (MTM-2016-77635-P). 2. P
RELIMINARIES
Notation.
In our estimates we will write f . g to denote f ≤ Cg for some constant C (the so-called “implicit constant”). If the implicit constant depends on a parameter t , i.e. C = C ( t ) , we willwrite f . t g . The notation f ∼ g and f ∼ t g stands for g . f . g and g . t f . t f , respectively.To make the notation lighter, we will usually not track the dependence of C on dimensions n, d, onthe Ahlfors regularity constant of σ , or the parameter < p < ∞ .Given x ∈ R n and r > we denote by B ( x, r ) the open ball centered at x with radius r .Conversely, given a ball B (either open or closed, and either n or d -dimensional), r B and z B denotethe radius and the center of B , respectively.For simplicity, we will sometimes write k f k p := k f k L p ( σ ) . In the introduction we introduced the notation f B to signify the average of f over a ball B withrespect to σ . For general Borel sets E ⊂ R d with σ ( E ) > and f ∈ L loc ( σ ) we will write h f i E = R E f dσσ ( E ) . For a finite set I we will write I to denote the cardinality of I .If v, w ∈ R n , then v · w denotes their scalar product.Given E, F ⊂ R d , dist H ( E, F ) stands for the Hausdorff distance between E and F . N α -NUMBER CHARACTERIZATION OF L p SPACES 5
Adjacent systems of cubes.
As usual, we will work with a family of subsets of supp σ =: Σ that in many ways resemble the family of dyadic cubes on R d . For this reason we will call these sets“cubes”. Many different systems of cubes have been constructed throughout the years, beginningwith the work of David [Dav88] and Christ [Chr90]. In our proof it will be convenient to useadjacent systems of cubes constructed by Hytönen and Tapiola [HT14]. One should think of themas a generalization of the translated dyadic grids in R d , widely used to perform the “ / trick”.First, we will say that a family D of Borel subsets of Σ satisfies the usual properties of David-Christ cubes if D = S k ∈ Z D k , and for each k ∈ Z :(a) for P, Q ∈ D k , P = Q, we have σ ( P ∩ Q ) = ∅ ,(b) the sets in D k cover Σ : Σ = [ Q ∈D k Q, (c) for each Q ∈ D k and each l ≥ kQ = [ P ∈D l : P ⊂ Q P, (d) there exists < δ < (independent of k ) such that each Q ∈ D k has a center z Q ∈ Q satisfying B (cid:16) z Q , δ k (cid:17) ∩ Σ ⊂ Q ⊂ B ( z Q , δ k ) ∩ Σ . (2.1)Consequently, as long as δ k . diam(Σ) , we have σ ( Q ) ∼ δ kd . Set ℓ ( Q ) := δ k .(e) the cubes Q ∈ D k have thin boundaries, that is, there exists γ ∈ (0 , such that for η ∈ (0 , . we have σ ( { x ∈ Σ : dist( x, Q ) + dist( x, Σ \ Q ) < ηℓ ( Q ) } ) ≤ η γ σ ( Q ) . (2.2) Remark 2.1.
Note that in the above we assume D k to be defined for all k ∈ Z . In the case ofunbounded Σ , this translates to having arbitrarily large cubes as k → −∞ . In the case of compact Σ , there exists some k such that for all k ≤ k we have D k = { Σ } . However, in our proof we willassume that Σ is unbounded, see Lemma 2.5.In our setting, the results [HT14, Theorem 2.9, Theorem 5.9] can be summarized as follows. Lemma 2.2.
Let σ be a d -Ahlfors regular measure on R n . Then, there exist ≤ N < ∞ and asmall constant < δ < . , depending only on the Ahlfors regularity constants of σ , such that thefollowing holds. Let Ω = { , . . . , N } . For each ω ∈ Ω we have a system of cubes D ( ω ) satisfyingthe usual properties of David-Christ cubes, and additionally, for all x ∈ supp σ and r > thereare ω ∈ Ω , k ∈ Z and Q ∈ D k ( ω ) with B ( x, r ) ∩ supp σ ⊂ Q and ℓ ( Q ) = δ k ∼ δ r. Remark 2.3.
The construction in [HT14] is valid for general (geometrically) doubling metricspaces, possibly with no underlying measure space structure. The constants N and δ from Lemma 2.2depend on the doubling constant of the metric space. Hytönen and Tapiola construct two differentkinds of cubes, which they call open and closed cubes, see [HT14, Theorem 2.9]. In the above weconsider closed cubes, so that properties (b), (c) and (d) follow immediately from [HT14, Theorem2.9]. To get the property (a) one uses the fact that interiors of P and Q are disjoint by [HT14,(2.11)], and then σ ( ∂P ) = σ ( ∂Q ) = 0 follows from (e). To prove the thin boundaries property (e)one may adapt the proof of Christ [Chr90, pp. 610–612] together with Ahlfors regularity of σ . Weomit the details. AZZAM AND D ˛ABROWSKI
From now on, let us fix a uniformly rectifiable measure σ , with Σ = supp σ . Let Ω , δ and D ( ω ) be as in Lemma 2.2. For simplicity, in our estimates we will not track the dependence of implicitconstants on δ .For all ω ∈ Ω and Q ∈ D k ( ω ) we will write D ( Q ) := { P ∈ D ( ω ) : P ⊂ Q } , Ch ( Q ) := D ( Q ) ∩ D k +1 ( ω ) . The elements of Ch ( Q ) will be called children of Q , and Q will be called their parent.Set B Q := B ( z Q , ℓ ( Q )) , so that Q ⊂ B Q ∩ Σ , and whenever P ∈ D ( Q ) we also have B P ⊂ B Q .Fix some ω ∈ Ω , and set D := D ( ω ) . This will be our system of reference. Given Q ∈ D we define ω ( Q ) ∈ Ω to be the index such thatthere exists R ( Q ) ∈ D ( ω ( Q )) satisfying B Q ∩ supp σ ⊂ R ( Q ) and ℓ ( R ( Q )) ∼ ℓ ( Q ) . If there ismore than one such ω , we simply choose one. We define also G ( ω ) ⊂ D as the family of cubes Q ∈ D such that ω ( Q ) = ω . Clearly, [ ω ∈ Ω G ( ω ) = D . α -numbers. In proving the main theorem, it will be more convenient to work with dyadicversions of the α -numbers. Below we will introduce the notation needed for this framework. Givena Radon measure µ we denote by L µx,r a minimizing d -plane for α µ ( x, r ) , and by c µx,r the corre-sponding constant. They may be non-unique, in which case we just choose one of the minimizers.Set P µx,r = H d L µx,r and L µx,r = c µx,r P µx,r . If B = B ( x, r ) we will also write L µB , c µB etc.For Q ∈ D and a Radon measure µ we set α µ ( Q ) := α µ ( B Q ) . We will write L µQ := L µB Q , c µQ := c µB Q etc.Observe that whenever B ⊂ B are balls, we have Lip ( B ) ⊂ Lip ( B ) , and so if r B ≥ Cr B , then α µ ( B ) = 1 r d +1 B inf c ∈ R ,L F B ( µ, c H d L ) ≤ r d +1 B F B ( µ, L µB ) ≤ r d +1 B F B ( µ, L µB ) ∼ C α µ ( B ) . (2.3)Consider the following square function: J ( x ) = X x ∈ Q ∈D α fσ ( Q ) + | f | B Q α σ ( Q ) ! / . (2.4)The Main Theorem will follow from the following dyadic version: Theorem 2.4.
Let σ be a uniformly rectifiable measure with unbounded support, and let f ∈ L p ( σ ) for some < p < ∞ . Then k J f k L p ( σ ) ∼ k f k L p ( σ ) . First, let us show why we may assume that supp σ is unbounded. Lemma 2.5.
It suffices to only prove Main Theorem in the case that supp σ is unbounded. N α -NUMBER CHARACTERIZATION OF L p SPACES 7
Proof.
Suppose σ did have compact support. Without loss of generality, we may assume diam(supp σ ) =1 , supp σ ⊆ B = B (0 , , and L σ B = R d . Let µ = σ + P σ B ( R d \ B ) . It is not hard to show that µ is also UR. If Main Theorem holds for UR measures of unboundedsupport, then it holds for µ . Let f ∈ L p ( σ ) ⊆ L p ( µ ) and let θ fσ ( x, r ) := α fσ ( x, r ) + | f | x,r α σ ( x, r ) , so that, by the Main Theorem, (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) Z ∞ θ fµ ( x, r ) drr ! / (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) L p ( µ ) ∼ k f k L p ( µ ) = k f k L p ( σ ) . (2.5)Observe that θ fσ ( x, r ) = θ fµ ( x, r ) for x ∈ supp σ and < r < . Thus, (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) Z θ fσ ( x, r ) drr ! / (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) L p ( σ ) = (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) Z θ fµ ( x, r ) drr ! / (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) L p ( σ ) ≤ (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) Z ∞ θ fµ ( x, r ) drr ! / (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) L p ( µ ) (2.5) . k f k L p ( σ ) . Furthermore, we claim that for any x ∈ supp σ and r > we have θ fσ ( x, r ) . r − d | f | B . (2.6)Indeed, since supp f ⊂ supp σ ⊂ B , α fσ ( x, r ) ≤ r d +1 F B ( x,r ) ( f σ, ≤ r d Z B | f | dσ ∼ r d | f | B , (2.7)and also | f | x,r α σ ( x, r ) . r d Z B | f | dσ ∼ r d | f | B . It follows from (2.6) that Z ∞ θ fσ ( x, r ) drr . Z ∞ | f | B drr d +1 . | f | B , and so (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) Z ∞ θ fσ ( x, r ) drr ! / (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) L p ( σ ) . k f k L p ( σ ) + Z B | f | dσ . k f k L p ( σ ) . To finish the proof we now need to show the reverse inequality. Notice that since f is supportedon supp σ , α fµ ( x, r ) = α fσ ( x, r ) for all x ∈ supp σ and r > . We can argue just as in (2.7) toget that for x ∈ supp µ and r ≥ , α fµ ( x, r ) = α fσ ( x, r ) ≤ r d +1 F B ( x,r ) ( f σ, . | f | B r d . | f | B r d α σ (2 B ) , where we also used α σ (2 B ) ∼ . AZZAM AND D ˛ABROWSKI
Hence, Z B Z ∞ α fµ ( x, r ) drr ! p dµ ( x ) . Z B Z α fµ ( x, r ) drr ! p dµ ( x ) + Z B Z ∞ α fµ ( x, r ) drr ! p dµ ( x ) . Z B Z α fσ ( x, r ) drr ! p dµ ( x ) + Z B Z ∞ ( | f | B α σ (2 B )) drr d +1 ! p dµ ( x ) . Z B Z α fσ ( x, r ) drr ! p dµ ( x ) + ( | f | B α σ (2 B )) p . Z B Z ∞ θ fσ ( x, r ) drr ! p dµ ( x ) where we used (2.3) in the final inequality.Furthermore, for x ∈ R d \ B , if α fµ ( x, r ) = 0 , then r ≥ | x | / and so Z R d \ B Z ∞ α fµ ( x, r ) drr ! p dµ ( x ) = ∞ X j =2 Z R d ∩ (2 j +1 B \ j B ) Z ∞| x | / ( | f | B α σ (2 B )) drr d +1 ! p dµ ( x ) . ( | f | B α σ (2 B )) p ∞ X j =2 Z R d ∩ (2 j +1 B \ j B ) | x | − pd dµ ( x ) . ( | f | B α σ (2 B )) p . Z B Z ∞ | f | x,r α σ ( x, r ) drr ! p dµ ( x ) again using (2.3). These two estimates imply (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) Z ∞ α fµ ( x, r ) drr ! / (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) L p ( µ ) . (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) Z ∞ θ fσ ( x, r ) drr ! / (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) L p ( σ ) . (2.8)Note that for x ∈ supp σ and r < , we have P σx,r = P µx,r . For r ≥ , notice that α σ (2 B ) ∼ ,and so α µ ( x, r ) ≤ r d +1 F B ( x,r ) ( µ, P σ B ) = 1 r d +1 F B ( x,r ) ( σ, P σ B B ) . r − d . α σ (2 B ) r d , hence | f | µx,r α µ ( x, r ) ≤ | f | B α σ (2 B ) r d , where | f | µx,r = R B ( x,r ) f dµ/µ ( B ( x, r )) . Thus, just as how we proved (2.8), we can show (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)Z ∞ ( | f | µx,r ) α µ ( x, r ) drr (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) L p ( µ ) . (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)Z ∞ | f | x,r α σ ( x, r ) drr (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) L p ( σ ) . This, (2.8) and (2.5) imply the desired estimate: k f k L p ( σ ) . (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) Z ∞ θ fσ ( x, r ) drr ! / (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) L p ( σ ) . (cid:3) Proof of the Main Theorem using Theorem 2.4.
By Lemma 2.5, we may assume that supp σ = Σ is unbounded, so that Theorem 2.4 holds. N α -NUMBER CHARACTERIZATION OF L p SPACES 9
Let x ∈ Σ , r > . Let k ∈ Z be such that δ k +1 < r ≤ δ k , and let Q be a cube in D k containing x . Recall that Q ⊂ B ( z Q , ℓ ( Q )) . Since r ≤ ℓ ( Q ) , we have B ( x, r ) ⊂ B ( z Q , ℓ ( Q ) + r ) ⊂ B ( z Q , ℓ ( Q )) = B Q . Hence, by (2.3), α fσ ( x, r ) . α fσ ( Q ) . We also have | f | x,r . | f | B Q , and so | f | x,r α σ ( x, r ) . | f | B Q α σ ( Q ) . Consequently, Z δ k δ k +1 ( α fσ ( x, r ) + | f | x,r α σ ( x, r )) drr . α fσ ( Q ) + | f | B Q α σ ( Q )) . Summing over k ∈ Z yields Z ∞ ( α fσ ( x, r ) + | f | x,r α σ ( x, r )) drr . X x ∈ Q ∈D α fσ ( Q ) + | f | B Q α σ ( Q )) . Similarly, for x ∈ Σ , r > , δ k +1 < r ≤ δ k , we may consider a cube Q ∈ D k +2 such that x ∈ Q ⊂ B Q ⊂ B ( x, r ) . Mimicking the estimates above, one gets X x ∈ Q ∈D α fσ ( Q ) + | f | B Q α σ ( Q )) . Z ∞ ( α fσ ( x, r ) + | f | x,r α σ ( x, r )) drr . Putting the two estimates together, we get the comparability of the dyadic and continuous variantsof the square function:
J f ( x ) = X x ∈ Q ∈D α fσ ( Q ) + | f | B Q α σ ( Q )) ∼ Z ∞ ( α fσ ( x, r ) + | f | x,r α σ ( x, r )) drr . (cid:3) Theorem 2.4 will follow from the results from the next three sections. From now on we assumethat σ is a uniformly rectifiable measure with unbounded support.3. k J f k . k f k First, we prove the estimate k J f k p . k f k p in the case p = 2 . Proposition 3.1.
Let f ∈ L ( σ ) . Then X Q ∈D ( α fσ ( Q ) + | f | B Q α σ ( Q ) ) ℓ ( Q ) d . k f k L ( σ ) . Our main tool in the proof of Proposition 3.1 are the martingale difference operators associatedto systems of cubes D ( ω ).Given ω ∈ Ω , Q ∈ D ( ω ) , and f ∈ L loc ( σ ) we set ∆ Q f = X P ∈ Ch ( Q ) h f i P P − h f i Q Q . Observe that all ∆ Q f have zero mean, i.e. R ∆ Q f dσ = 0 . It is well known (see e.g. [Gra14, Chapter 6.4]) that given f ∈ L ( σ ) and some system of cubes D ( ω ) we have f = X Q ∈D ( ω ) ∆ Q f with the convergence understood in the L sense. It is crucial that σ (Σ) = ∞ , so that f + C ∈ L ( σ ) if and only if C = 0 (in the case σ (Σ) < ∞ one would have to subtract from the left hand sideabove the average of f ).Note that ∆ Q f are mutually orthogonal in L ( σ ) , so that k f k L ( σ ) = X Q ∈D ( ω ) (cid:13)(cid:13) ∆ Q f (cid:13)(cid:13) L ( σ ) (3.1)Moreover, if Q ∈ D ( ω ) , then for σ -a.e. x ∈ Qf ( x ) = h f i Q + X P ∈D ( Q ) ∆ P f ( x ) . (3.2) Lemma 3.2.
Suppose Q ∈ D , and let R = R ( Q ) ∈ D ( ω ( Q )) be as in Section 2.1. Then, for f ∈ L ( σ ) we have α fσ ( Q ) . |h f i R | α σ ( R ) + X P ∈D ( R ) ℓ ( P ) d/ ℓ ( Q ) d k ∆ P f k . (3.3) Proof.
Let ϕ ∈ Lip ( B Q ) and consider a candidate for L fσQ of the form h f i R L σQ . For all x ∈ B Q ∩ supp σ we have x ∈ R , so that using (3.2) (cid:12)(cid:12)(cid:12)(cid:12)Z ϕ ( x ) f ( x ) dσ ( x ) − h f i R Z ϕ ( x ) d L σQ ( x ) (cid:12)(cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12)(cid:12)Z ϕ ( x ) h f i R + X P ∈D ( R ) ϕ ( x )∆ P f ( x ) dσ ( x ) − Z ϕ ( x ) h f i R d L σQ ( x ) (cid:12)(cid:12)(cid:12)(cid:12) ≤ (cid:12)(cid:12) h f i R (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)Z ϕ ( x ) dσ ( x ) − Z ϕ ( x ) d L σQ ( x ) (cid:12)(cid:12)(cid:12)(cid:12) + X P ∈D ( R ) (cid:12)(cid:12)(cid:12)(cid:12)Z ϕ ( x )∆ P f ( x ) dσ ( x ) (cid:12)(cid:12)(cid:12)(cid:12) =: I + I . It is clear that I ≤ (cid:12)(cid:12) h f i R (cid:12)(cid:12) α σ ( Q ) ℓ ( Q ) d +1 (2.3) . (cid:12)(cid:12) h f i R (cid:12)(cid:12) α σ ( R ) ℓ ( Q ) d +1 , which gives rise to the first term on the right hand side of (3.3).Concerning I , we use the zero mean property of martingale difference operators, and the factthat ϕ ∈ Lip ( B Q ) , to get I = X P ∈D ( R ) (cid:12)(cid:12)(cid:12)(cid:12)Z (cid:16) ϕ ( x ) − ϕ ( z P ) (cid:17) ∆ P f ( x ) dσ ( x ) (cid:12)(cid:12)(cid:12)(cid:12) ≤ X P ∈D ( R ) Z (cid:12)(cid:12) ϕ ( x ) − ϕ ( z P ) (cid:12)(cid:12)(cid:12)(cid:12) ∆ P f ( x ) (cid:12)(cid:12) dσ ( x ) . X P ∈D ( R ) ℓ ( P ) k ∆ P f k Hölder . X P ∈D ( R ) ℓ ( P ) d/ k ∆ P f k . Dividing by ℓ ( Q ) d +1 and taking supremum over ϕ ∈ Lip ( B Q ) yields (3.3). (cid:3) Proof of Proposition 3.1.
We begin by noting that, since σ is uniformly rectifiable, α σ ( Q ) ℓ ( Q ) d isa Carleson measure by the results from [Tol09], see Theorem 1.1. Therefore, the estimate X Q ∈D | f | B Q α σ ( Q ) ℓ ( Q ) d . k f k L ( σ ) follows immediately from Carleson’s embedding theorem, see e.g. [Tol14, Theorem 5.8], and weonly need to estimate the sum involving α fσ ( Q ) .Observe that for each ω ∈ Ω and R ∈ D ( ω ) there is at most a bounded number of cubes Q ∈ D such that R ( Q ) = R . N α -NUMBER CHARACTERIZATION OF L p SPACES 11
Fix some ω ∈ Ω . Recall that G ( ω ) is the family of cubes Q ∈ D such that ω ( Q ) = ω . We apply(3.3) and the observation above to get X Q ∈G ( ω ) α fσ ( Q ) ℓ ( Q ) d . X R ∈D ( ω ) |h f i R | α σ ( R ) ℓ ( R ) d + X R ∈D ( ω ) X P ∈D ( R ) ℓ ( P ) d/ ℓ ( R ) d/ k ∆ P f k ! =: S + S . Concerning S , we may use Carleson’s embedding theorem again to estimate S . k f k .Moving on to S , we apply Cauchy-Schwarz inequality to get S ≤ X R ∈D ( ω ) X P ∈D ( R ) ℓ ( P ) ℓ ( R ) k ∆ P f k ! X P ∈D ( R ) ℓ ( P ) d +1 ℓ ( R ) d +1 ! . It is easy to see that, due to Ahlfors regularity of σ , P P ∈D ( R ) ℓ ( P ) d +1 ℓ ( R ) d +1 . . Thus, S ≤ X R ∈D ( ω ) X P ∈D ( R ) ℓ ( P ) ℓ ( R ) k ∆ P f k = X P ∈D ( ω ) k ∆ P f k X R ∈D ( ω ) R ⊃ P ℓ ( P ) ℓ ( R ) . X P ∈D ( ω ) k ∆ P f k (3.1) . k f k . Putting the estimates above together we arrive at X Q ∈G ( ω ) α fσ ( Q ) ℓ ( Q ) d . k f k . Summing over all ω ∈ Ω (recall that is bounded) we get the desired estimate. (cid:3) k J f k p . k f k p FOR < p < ∞ In this section we use the estimate || J f || . k f k to prove || J f || p . k f k p for general
Let < p < ∞ and f ∈ L p ( σ ) . Then, k J f k L p ( Q ) . p k f k L p ( B Q ) . The proposition follows easily from a good-lambda inequality stated below. Let M denote thenon-centered maximal Hardy-Littlewood operator with respect to σ , i.e. M f ( x ) = sup {| f | B : x ∈ B, B is a ball } . Since σ is Ahlfors regular, the operator M is bounded on L p ( σ ) for p > , see e.g. [Tol14, Theorem2.6, Remark 2.7]. Lemma 4.2.
Let f ∈ L loc ( σ ) . For any α > there exists ε = ε ( α ) > such that for all λ > σ ( { x ∈ Q : J f ( x ) > αλ, M f ( x ) ≤ ελ } ) ≤ σ ( { x ∈ Q : J f ( x ) > λ } ) . (4.1)Let us show how to use the above to prove Proposition 4.1. Proof of Proposition 4.1.
Note that J f = J ( f B Q ) , so without loss of generality we may as-sume that supp f ⊂ B Q . Let α = α ( p ) > be so close to that . α p < . , and let ε = ε ( α ) be as in Lemma 4.2. We use the layer cake representation to get Z Q J f ( x ) p dσ ( x ) = p Z ∞ λ p − σ ( { x ∈ Q : J f ( x ) > λ } ) dλ = pα p Z ∞ λ p − σ ( { x ∈ Q : J f ( x ) > αλ } ) dλ ≤ pα p Z ∞ λ p − σ ( { x ∈ Q : J f ( x ) > αλ, M f ( x ) ≤ ελ } ) dλ + pα p Z ∞ λ p − σ ( { x ∈ Q : M f ( x ) > ελ } ) dλ (4.1) ≤ pα p Z ∞ λ p − σ ( { x ∈ Q : J f ( x ) > λ } ) dλ + α p ε − p Z Q M f ( x ) p dσ ( x ) ≤ p Z ∞ λ p − σ ( { x ∈ Q : J f ( x ) > λ } ) dλ + α p ε − p Z Q M f ( x ) p dσ ( x )= 1920 Z Q J f ( x ) p dσ ( x ) + α p ε − p Z Q M f ( x ) p dσ ( x ) . Absorbing the first term from the right hand side into the left hand side, we arrive at Z Q J f ( x ) p dσ ( x ) ≤ α p ε − p Z M f ( x ) p dσ ( x ) . We use the L p boundedness of M and the assumption supp f ⊂ B Q to conclude Z Q J f ( x ) p dσ ( x ) . α,ε Z B Q f ( x ) p dσ ( x ) . (cid:3) The remainder of this section is dedicated to proving Lemma 4.2.4.1.
Preliminaries.
Fix α > and λ > . First, we set E λ = { x ∈ Q : J f ( x ) > λ } . Consider the covering of E λ with a family of cubes C λ ⊂ D ( Q ) such that for every S ∈ C λ wehave σ ( S ∩ E λ ) ≥ . σ ( S ) and S is the maximal cube with this property. Since the cubes from C λ are pairwise disjoint, to get(4.1) it is enough to find ε = ε ( α ) such that for each S ∈ C λ we have σ ( { x ∈ S : J f ( x ) > αλ, M f ( x ) ≤ ελ } ) ≤ σ ( S ) . (4.2)Fix S ∈ C λ . Without loss of generality assume that σ ( { x ∈ S : M f ( x ) ≤ ελ } ) > σ ( S ) , (4.3)otherwise there is nothing to prove.Given x ∈ S , we split the sum from the definition of J f ( x ) into two parts: J f ( x ) = X x ∈ Q ∈D ( S ) (cid:16) α fσ ( Q ) + | f | B Q α σ ( Q ) (cid:17) + X S ( Q ∈D ( Q ) (cid:16) α fσ ( Q ) + | f | B Q α σ ( Q ) (cid:17) =: J f ( x ) + J f ( x ) . (4.4) N α -NUMBER CHARACTERIZATION OF L p SPACES 13
Clearly, J f ( x ) ≡ J f is just a constant. By the definition of C λ there exists y ∈ ˆ S (where ˆ S is theparent of S ) such that y E λ . By the definition of E λ , we get that J f ≤ J f ( y ) ≤ λ. We will show the following.
Lemma 4.3.
There exists a set S ⊂ S such that σ ( S ) ≥ . σ ( S ) and Z S J f ( x ) dσ ( x ) . ε λ σ ( S ) The estimate (4.2) follows from the above easily. Indeed, using Chebyshev, we can find S ⊂ S such that for all x ∈ S we have J f ( x ) . ελ and σ ( S ) ≥ . σ ( S ) ≥ . σ ( S ) . Then, choosing ε = ε ( α ) small enough, (4.4) gives J f ( x ) ≤ λ + Cε λ ≤ α λ on S , so that σ ( { x ∈ S : J f ( x ) > αλ, M f ( x ) ≤ ελ } ) ≤ σ ( S \ S ) ≤ σ ( S ) . So our goal is to prove Lemma 4.3.4.2.
Calderón-Zygmund decomposition.
Let R = R ( S ) be as in Section 2.2, so that B S ∩ supp σ ⊂ R . We consider a variant of the Calderón-Zygmund decomposition of f R with respectto D ( R ) at the level ελ .First, let { Q j } j ⊂ D ( R ) be maximal cubes satisfying | f | B Qj ≥ ελ . Note that for all x ∈ Q j (and recalling that M is the non-centered maximal function) we have M f ( x ) ≥ | f | B Qj ≥ ελ. Hence, S j Q j ⊂ { x ∈ S : M f ( x ) ≥ ελ } , and so σ ( R \ [ j Q j ) ≥ σ ( S \ [ j Q j ) ≥ σ ( { x ∈ S : M f ( x ) ≤ ελ } ) (4.3) ≥ σ ( S ) ∼ ℓ ( S ) d ∼ ℓ ( R ) d . (4.5)In particular, Q j = R for all j . Thus, by the maximality of Q j we get easily | f | B Qj ∼ ελ. (4.6)We define g ∈ L ∞ ( σ ) by g ( x ) = f ( x ) R \ S j Q j ( x ) + X j h f i Q j Q j ( x ) . From the definition of Q j and (4.6) it follows that k g k ∞ . ελ . We define also b ∈ L ( σ ) as b ( x ) = X j ( f ( x ) − h f i Q j ) Q j ( x ) =: X j b j ( x ) . Note that f = g + b and for all j we have R b j dσ = 0 .4.3. Definition of S . We set S = S \ N η , where N η is some small neighbourhood of S j Q j . Tomake this more precise, given a small η > we define N η = S j N η,j , where N η,j = { x ∈ supp σ : dist( x, Q j ) < ηℓ ( Q j ) } . The thin boundaries property of D (2.2) gives σ ( N η,j \ Q j ) ≤ η γ σ ( Q j ) for some γ ∈ (0 , . From (4.5) and the fact that σ ( S ) ∼ σ ( R ) we get σ ( S \ N η ) ≥ σ ( S \ [ j Q j ) − X j σ ( N η,j \ Q j ) (4.5) ≥ σ ( S ) − X j η γ σ ( Q j ) ≥ σ ( S ) − η γ σ ( R ) ≥ σ ( S ) − Cη γ σ ( S ) = (cid:18) − Cη γ (cid:19) σ ( S ) . Here C depends only on the implicit constant in σ ( S ) ∼ σ ( R ) , which in turn depends on the Ahlforsregularity constant of σ and on the parameters from the definition of the system D .Choosing η so small that Cη γ < . , we get that S = S \ N η satisfies σ ( S ) ≥ σ ( S ) . Estimating J f . Now, we will show that Z S J f ( x ) dσ ( x ) . ε λ σ ( S ) (4.7)Recall that J f ( x ) = X x ∈ Q ∈D ( S ) α fσ ( Q ) + X x ∈ Q ∈D ( S ) | f | B Q α σ ( Q ) =: J ′ f ( x ) + J ′′ f ( x ) . First we deal with J ′′ f . Observe that for all Q ∈ D ( S ) intersecting S we have | f | B Q . ελ. (4.8)Indeed, let y ∈ Q ∩ S , and let P ∈ D ( R ) be such that y ∈ P , ℓ ( Q ) ∼ ℓ ( P ) , and B Q ⊂ B P . Bythe maximality of Q j and the fact that P \ S j Q j = ∅ we get | f | B P ≤ ελ . Estimate (4.8) followsfrom the inclusion B Q ⊂ B P .Using (4.8) as well as Theorem 1.1 we get Z S X x ∈ Q ∈D ( S ) | f | B Q α σ ( Q ) dσ ( x ) . ε λ X Q ∈D ( S ) α σ ( Q ) σ ( Q ∩ S ) . ε λ X Q ∈D ( S ) α σ ( Q ) σ ( Q ) . ε λ σ ( S ) ∼ ε λ σ ( S ) . Thus, we are only left with showing Z S J ′ f ( x ) dσ ( x ) = Z S X x ∈ Q ∈D ( S ) α fσ ( Q ) dσ ( x ) . ε λ σ ( S ) . (4.9) Lemma 4.4.
For Q ∈ D ( S ) we have α fσ ( Q ) . α gσ ( Q ) + ελ X j : Q j ∩ B Q = ∅ ℓ ( Q j ) d +1 ℓ ( Q ) d +1 . Proof.
Let ϕ ∈ Lip ( B Q ) . Then, using the decomposition f ( y ) = g ( y ) + b ( y ) valid for all y ∈ R ⊃ B S ∩ supp σ ⊃ B Q ∩ supp σ , (cid:12)(cid:12)(cid:12)(cid:12)Z ϕ ( y ) f ( y ) dσ ( y ) − Z ϕ ( y ) d L gσQ ( y ) (cid:12)(cid:12)(cid:12)(cid:12) ≤ (cid:12)(cid:12)(cid:12)(cid:12)Z ϕ ( y ) g ( y ) dσ ( y ) − Z ϕ ( y ) d L gσQ ( y ) (cid:12)(cid:12)(cid:12)(cid:12) + (cid:12)(cid:12)(cid:12)(cid:12)Z ϕ ( y ) b ( y ) dσ ( y ) (cid:12)(cid:12)(cid:12)(cid:12) . ℓ ( Q ) d +1 α gσ ( Q ) + X j (cid:12)(cid:12)(cid:12)(cid:12)Z ϕ ( y ) b j ( y ) dσ ( y ) (cid:12)(cid:12)(cid:12)(cid:12) . N α -NUMBER CHARACTERIZATION OF L p SPACES 15
Concerning the second term on the right hand side, recall that R b j dσ = 0 and that supp b j ⊂ Q j .Keeping that in mind, denoting by x j the center of Q j , we estimate in the following way: X j (cid:12)(cid:12)(cid:12)(cid:12)Z ϕ ( y ) b j ( y ) dσ ( y ) (cid:12)(cid:12)(cid:12)(cid:12) = X j (cid:12)(cid:12)(cid:12)(cid:12)Z ( ϕ ( y ) − ϕ ( x j )) b j ( y ) dσ ( y ) (cid:12)(cid:12)(cid:12)(cid:12) ≤ X j Z (cid:12)(cid:12) ( ϕ ( y ) − ϕ ( x j )) b j ( y ) (cid:12)(cid:12) dσ ( y ) . X j : Q j ∩ B Q = ∅ ℓ ( Q j ) Z (cid:12)(cid:12) b j ( y ) (cid:12)(cid:12) dσ ( y )= X j : Q j ∩ B Q = ∅ ℓ ( Q j ) Z Q j (cid:12)(cid:12)(cid:12) f ( y ) − h f i Q j (cid:12)(cid:12)(cid:12) dσ ( y ) . X j : Q j ∩ B Q = ∅ ℓ ( Q j ) d +1 h| f |i Q j (4.6) . ελ X j : Q j ∩ B Q = ∅ ℓ ( Q j ) d +1 . Together with the previous string of estimates, taking supremum over all ϕ ∈ Lip ( B Q ) , we get α fσ ( Q ) . α gσ ( Q ) + ελ X j : Q j ∩ B Q = ∅ ℓ ( Q j ) d +1 ℓ ( Q ) d +1 . (cid:3) An immediate consequence of Lemma 4.4 is the estimate Z S J ′ f ( x ) dσ ( x ) . Z S J ′ g ( x ) dσ ( x ) + ε λ Z S X x ∈ Q ∈D ( S ) X j : Q j ∩ B Q = ∅ ℓ ( Q j ) d +1 ℓ ( Q ) d +1 dσ ( x ) . (4.10)Using Proposition 3.1 and the fact that k g k ∞ . ελ, supp g ⊂ R, we get Z S J ′ g ( x ) dσ ( x ) ≤ k J g k . k g k ≤ k g k ∞ σ ( R ) . ε λ σ ( R ) ∼ ε λ σ ( S ) . (4.11)Moving on to the second term from the right hand side of (4.10), denote by Tree ⊂ D ( S ) thefamily of cubes contained in S that intersect S . We have Z S X x ∈ Q ∈D ( S ) X j : Q j ∩ B Q = ∅ ℓ ( Q j ) d +1 ℓ ( Q ) d +1 dσ ( x ) ≤ X Q ∈ Tree σ ( Q ) X j : Q j ∩ B Q = ∅ ℓ ( Q j ) d +1 ℓ ( Q ) d +1 Cauchy-Schwarz . X Q ∈ Tree ℓ ( Q ) − d − X j : Q j ∩ B Q = ∅ ℓ ( Q j ) d +2 X j : Q j ∩ B Q = ∅ ℓ ( Q j ) d (4.12)Note that since Q ∈ Tree , we have Q ∩ S = ∅ . By the definition of S , this implies that for all j such that Q j ∩ B Q = ∅ we have ℓ ( Q ) & η ℓ ( Q j ) . Indeed, if ℓ ( Q ) ≪ ηℓ ( Q j ) , then B Q ∩ Q j = ∅ implies Q ⊂ N η,j , which would contradict Q ∩ S = ∅ .By the observation above, we have some C = C ( η ) such that if B Q ∩ Q j = ∅ , then Q j ⊂ CB Q .Consequently, X j : Q j ∩ B Q = ∅ ℓ ( Q j ) d . X j : Q j ⊂ CB Q σ ( Q j ) ≤ σ ( CB Q ) ∼ η ℓ ( Q ) d . Thus, the right hand side of (4.12) can be estimated by X Q ∈ Tree ℓ ( Q ) − X j : Q j ∩ B Q = ∅ ℓ ( Q j ) d +2 = X j ℓ ( Q j ) d +2 X Q ∈ Tree : Q j ∩ B Q = ∅ ℓ ( Q ) − . (4.13)As noted above, Q j ∩ B Q = ∅ implies ℓ ( Q ) & η ℓ ( Q j ) . Hence, X Q ∈ Tree : Q j ∩ B Q = ∅ ℓ ( Q ) − . η ℓ ( Q j ) − , where we used the fact that the sum above is essentially a geometric series. Putting this togetherwith (4.13) and (4.12), we get Z S X x ∈ Q ∈D ( S ) X j : Q j ∩ B Q = ∅ ℓ ( Q j ) d +1 ℓ ( Q ) d +1 dσ ( x ) . η X j ℓ ( Q j ) d . ℓ ( R ) d ∼ σ ( S ) . Together with (4.10) and (4.11) this gives the desired estimate (4.9): Z S J ′ f ( x ) dσ ( x ) . η ε λ σ ( S ) . This finishes the proof of Lemma 4.3.5. k f k p . k J f k p FOR < p < ∞ In this section we show the second inequality of Theorem 2.4.
Proposition 5.1.
Let f ∈ L p ( σ ) for some < p < ∞ . Then k f k L p ( σ ) . k J f k L p ( σ ) . (5.1)5.1. Littlewood-Paley theory.
Our main tool will be the Littlewood-Paley theory for spaces ofhomogeneous type developed by David, Journé and Semmes in [DJS85]. We follow the way it wasparaphrased (in English) in [Tol17, Section 15].For r > , x ∈ Σ , and g ∈ L loc ( σ ) , let D r g ( x ) = φ r ∗ ( gσ )( x ) φ r ∗ σ ( x ) − φ r ∗ ( gσ )( x ) φ r ∗ σ ( x ) where φ r ( y ) = r − d φ ( y/r ) and φ is a radially symmetric smooth nonnegative function supported in B (0 , with R R n φ = 1 .For a function g ∈ L loc ( σ ) and r > , we denote S r g ( x ) = φ r ∗ ( gσ )( x ) φ r ∗ σ ( x ) , so that D r g = S r g − S r g. Let W r be the operator of multiplication by /S ∗ r . We consider the operators ˜ S r = S r W r S ∗ r and ˜ D r = ˜ S r − ˜ S r . Note that ˜ S r , and thus ˜ D r , are self-adjoint and ˜ S r ≡ , so that ˜ D r D ∗ r . (5.2)Let s r ( x, y ) the kernel of S r with respect to σ , that is, so we can write S r g ( x ) = Z s r ( x, y ) g ( y ) dσ ( y ) . N α -NUMBER CHARACTERIZATION OF L p SPACES 17
Observe that s r ( x, y ) = 1 φ r ∗ σ ( x ) φ r ( x − y ) and the kernel of ˜ S r is ˜ s r ( x, y ) = Z s r ( x, z ) 1 S ∗ r z ) s r ( y, z ) dσ ( z ) . We claim that the kernel ˜ d r ( x, · ) for the operator ˜ D r is supported in B ( x, r ) and satisfies theLipschitz bounds | ˜ d r ( x, y ) − ˜ d r ( x, z ) | . | y − z | r − d − . (5.3)Indeed, let x, x ′ ∈ supp σ . Since φ r is Cr − d − -Lipschitz and σ is Ahlfors regular, | φ r ∗ σ ( x ) − φ r ∗ σ ( x ′ ) | = (cid:12)(cid:12)(cid:12)(cid:12)Z ( φ r ( x − y ) − φ r ( x ′ − y )) dσ ( y ) (cid:12)(cid:12)(cid:12)(cid:12) . | x − x ′ | r d +1 σ ( B ( x, r ) ∪ B ( x ′ , r )) . | x − x ′ | r . Thus, for y ∈ supp σ , | s r ( x, y ) − s r ( x ′ , y ) | ≤ | φ r ( x − y ) − φ r ( x ′ − y ) | φ r ∗ σ ( x ) + φ r ( x ′ − y ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) φ r ∗ σ ( x ) − φ r ∗ σ ( x ′ ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) . | x − x ′ | r d +1 + r − d | φ r ∗ σ ( x ) − φ r ∗ σ ( x ′ ) | φ r ∗ σ ( x ) ∼ | x − x ′ | r d +1 . Hence, | ˜ s r ( x, y ) − ˜ s r ( x ′ , y ) = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)Z ( s r ( x, z ) − s r ( x ′ , z )) 1 S ∗ r z ) s r ( y, z ) dσ ( z ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) . | x − x ′ | r d +1 (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)Z S ∗ r z ) s r ( y, z ) dσ ( z ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) . | x − x ′ | r d +1 where in the last line we used the fact that R s r ( y, z ) dσ ( z ) = 1 and S ∗ r z ) = Z φ r ( x − z ) φ r ∗ σ ( x ) dσ ( x ) ≥ Z B ( z,r/ r − d φ r ∗ σ ( x ) dσ ( x ) ∼ . Since ˜ d r = ˜ s r − ˜ s r and is symmetric, this proves (5.3). Moreover, notice that if x ∈ supp σ , supp s r ( x, · ) ⊆ B ( x, r ) , and so the integrand of ˜ s r is nonzero only when z ∈ B ( x, r ) ∩ B ( y, r ) ,meaning | x − y | ≤ r , and so supp ˜ s r ⊆ B ( x, r ) , hence supp ˜ d r ⊆ B ( x, r ) , which proves ourclaim. Theorem 5.2. [DJS85]
Let r k = 2 − k , and g ∈ L p ( σ ) , < p < ∞ , we have k g k L p ( σ ) ∼ (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)X k ∈ Z | ˜ D r k g | (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) L p ( σ ) . (5.4)The original result is stated for p = 2 , but this case implies the other cases (see for example theproof of [Tol01, Corollary 6.1]).Let ˜ D k := ˜ D r k , ˜ d k := ˜ d r k . By (5.4), it is clear that to prove (5.1), it suffices to show that (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)X k ∈ Z | ˜ D k f | (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) L p ( σ ) . k J f k L p ( σ ) . In fact, we will show a stronger, pointwise inequality which immediately implies the one above.
Lemma 5.3.
Let x ∈ Σ , k ∈ Z , and let Q ∈ D be the smallest cube containing x and such that supp ˜ d k ( x, · ) ⊂ . B Q . Then, ℓ ( Q ) ∼ r k and | ˜ D k f ( x ) | . α fσ ( Q ) + | f | B Q α σ ( Q ) . (5.5)The remainder of this section is devoted to the proof of this lemma.5.2. Preliminaries.
Fix x ∈ Σ , k ∈ Z , and let Q be as above. As noted just above (5.3), we have ˜ d k ( x, · ) ⊂ B ( x, r k ) , and so ℓ ( Q ) ∼ r k follows immediately.We make a few simple reductions. Remark 5.4.
Without loss of generality we may assume that α σ ( Q ) ≤ ε for some small ε . Indeed,if we had α σ ( Q ) ≥ ε , then using (5.3) and the fact that supp ˜ d k ( x, · ) ⊂ B Q | ˜ D k f ( x ) | = (cid:12)(cid:12)(cid:12)(cid:12)Z ˜ d k ( x, y ) f ( y ) dσ ( y ) (cid:12)(cid:12)(cid:12)(cid:12) ≤ (cid:13)(cid:13)(cid:13) ˜ d k ( x, · ) (cid:13)(cid:13)(cid:13) ∞ Z B Q | f ( y ) | dσ ( y ) . ℓ ( Q ) − d Z B Q | f ( y ) | dσ ( y ) ∼ | f | B Q . ε | f | B Q α σ ( Q ) , and so in this case (5.5) holds. From now on we assume α σ ( Q ) ≤ ε . Remark 5.5.
Similarly, without loss of generality we may assume that L fσQ ∩ . B Q = ∅ . If wehad L fσQ ∩ . B Q = ∅ , then L fσQ ∩ supp ˜ d r ( x, · ) = ∅ so that Z ˜ d k ( x, y ) d L fσQ ( y ) = 0 . This implies | ˜ D k f ( x ) | = (cid:12)(cid:12)(cid:12)(cid:12)Z ˜ d k ( x, y ) f ( y ) dσ ( y ) (cid:12)(cid:12)(cid:12)(cid:12) . α fσ ( B ) , and so (5.5) is true also in this case.Recall that c fσQ , c σQ are the constants minimizing α fσ ( Q ) , α σ ( Q ) , respectively. Since σ isAhlfors regular and α σ ( Q ) ≤ ε , we get by [ATT18, Lemma 3.3] c σQ ∼ . (5.6)To show (5.5) we begin by using (5.2) and the triangle inequality: | ˜ D k f ( x ) | = (cid:12)(cid:12)(cid:12)(cid:12)Z Σ ˜ d k ( x, y ) f ( y ) dσ ( y ) (cid:12)(cid:12)(cid:12)(cid:12) (5.2) = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)Z Σ ˜ d k ( x, y ) f ( y ) dσ ( y ) − c fσQ c σQ Z Σ ˜ d k ( x, y ) dσ ( y ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)Z Σ ˜ d k ( x, y ) f ( y ) dσ ( y ) − Z L fσQ ˜ d k ( x, y ) d L fσQ ( y ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) + (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)Z L fσQ ˜ d k ( x, y ) d L fσQ ( y ) − c fσQ c σQ Z L σQ ˜ d k ( x, y ) d L σQ ( y ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) + (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) c fσQ c σQ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)Z L σQ ˜ d k ( x, y ) d L σQ ( y ) − Z Σ ˜ d k ( x, y ) dσ ( y ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) =: ( I ) + ( II ) + ( III ) . (5.7)Using the Lipschitz property of ˜ d k (5.3) we immediately get that ( I ) . α fσ ( Q ) , and that ( III ) . (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) c fσQ c σQ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) α σ ( Q ) (5.6) ∼ (cid:12)(cid:12)(cid:12) c fσQ (cid:12)(cid:12)(cid:12) α σ ( Q ) . (5.8) N α -NUMBER CHARACTERIZATION OF L p SPACES 19
Lemma 5.6.
We have (cid:12)(cid:12)(cid:12) c fσQ (cid:12)(cid:12)(cid:12) . | f | B Q .Proof. Indeed, if we had (cid:12)(cid:12)(cid:12) c fσQ (cid:12)(cid:12)(cid:12) ≥ Λ | f | B Q for some big Λ > , then ˜ c fσQ = 0 would be a bettercompetitor for a constant minimizing α fσ ( Q ) . To see that, note that for any ϕ ∈ Lip ( B Q ) (cid:12)(cid:12)(cid:12)(cid:12)Z ϕf dσ − (cid:12)(cid:12)(cid:12)(cid:12) ≤ Cℓ ( Q ) d +1 | f | B Q . That is, F B Q ( f σ, ≤ Cℓ ( Q ) d +1 | f | B Q . On the other hand, taking a positive ψ ∈ Lip ( B Q ) suchthat ψ ( x ) = ℓ ( Q ) for x ∈ . B Q , and using the assumption L fσQ ∩ . B Q = ∅ we get α fσ ( Q ) ℓ ( Q ) d +1 & (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)Z ψf dσ − c fσQ Z L fσQ ψ d H d (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≥ (cid:12)(cid:12)(cid:12) c fσQ (cid:12)(cid:12)(cid:12) ℓ ( Q ) H d (0 . B Q ∩ L fσQ ) − (cid:12)(cid:12)(cid:12)(cid:12)Z ψf dσ (cid:12)(cid:12)(cid:12)(cid:12) ≥ ˜ C Λ | f | B Q ℓ ( Q ) d +1 − Cℓ ( Q ) d +1 | f | B Q ≥ ˜ C Λ2 | f | B Q ℓ ( Q ) d +1 > F B Q ( f σ, , assuming Λ big enough. This contradicts the optimality of c fσQ . (cid:3) Using the lemma above and (5.8) we get ( III ) . | f | B Q α σ ( Q ) . Hence, by (5.7), to finish the proof of (5.5) it remains to show that ( II ) = (cid:12)(cid:12)(cid:12) c fσQ (cid:12)(cid:12)(cid:12) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)Z L fσQ ˜ d k ( x, y ) d H d ( y ) − Z L σQ ˜ d k ( x, y ) d H d ( y ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) . α fσ ( Q ) + | f | B Q α σ ( Q ) . This can be seen as an estimate of how far from each other the planes L fσQ and L σQ are.The inequality above follows immediately from Proposition 5.7 proven in the next subsection,together with the already established estimate (cid:12)(cid:12)(cid:12) c fσQ (cid:12)(cid:12)(cid:12) . | f | B Q .5.3. Angles between planes approximating f σ and σ . In the following proposition we do not useuniform rectifiability in any way, and so we state it for a general Ahlfors regular measure µ . Recallthat given a ball B we defined P µB = H d L µB . Proposition 5.7.
Let µ be an Ahlfors d -regular measure on R n , and let f ∈ L loc ( µ ) . Let x ∈ supp µ, r > , B = B ( x, r ) , and suppose that L fµB ∩ . B = ∅ . Then, | c fµB | r d +1 F B ( P µB , P fµB ) . α fµ ( B ) + | c fµB | α µ ( B ) . (5.9)In the proof of Proposition 5.7 we will use the following lemma. Lemma 5.8.
Let B = B ( x, r ) and let L , L be two d -planes intersecting . B . Set P = H d L , P = H d L . Then, r d F B ( P , P ) . dist H ( L ∩ B, L ∩ B ) . (5.10) Proof.
First, set D = dist H ( L ∩ B, L ∩ B ) r . Note that we always have F B ( P , P ) . r d +1 so that if D & , then (5.10) follows trivially. Hence,without loss of generality we may assume that D ≤ ε for some ε > to be fixed later. We claim that if ε is chosen small enough (depending only on n, d ), then there exists an isometry A : L → L such that for y ∈ B ∩ L we have | y − A ( y ) | . Dr . To see that, let y ∈ L ∩ B bearbitrary. Set y = π L ( y ) . Clearly, | y − y | ≤ Dr ≤ εr. Let v , . . . , v d be an orthonormal basis of the linear plane L ′ := L − y . For i = 1 , . . . , d define w i := π L ( y + v i ) − y ∈ L − y =: L ′ . In fact, since y = π L ( y ) , we have w i = π L ′ ( v i ) . It is easy to see that for all v ∈ L ′ we have | π L ′ ( v ) − v | . D | v | . Hence, | w i − v i | . D ≤ ε and for i = j | w i · w j | = | ( w i − v i ) · ( w j − v j ) + ( w i − v i ) · v j + v i · ( w j − v j ) | . D ≤ ε. Choosing ε small enough (depending only on dimensions), we get easily that { w i } is a basis of L ′ .Moreover, if { ˆ w i } is the orthonormal basis of L ′ constructed from { w i } using the Gram-Schmidtprocess, then it follows from the estimates above that for all i = 1 , . . . , d | ˆ w i − v i | . D. We define the map A : L → L as the unique isometry such that A ( y ) = y and A ( y + v i ) = y +ˆ w i . It follows immediately from basic linear algebra that for y ∈ L ∩ B we have | y − A ( y ) | . Dr .Now, let ϕ ∈ Lip ( B ) . We have (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) Z L ϕ ( y ) d H d ( y ) − Z L ϕ ( y ) d H d ( y ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) Z L ϕ ( y ) d H d ( y ) − Z L ϕ ( A ( y )) d H d ( y ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ Z L | ϕ ( y ) − ϕ ( A ( y )) | d H d ( y ) . Z L ∩ B Dr d H d ( y ) . Dr d +1 . Taking supremum over ϕ ∈ Lip ( B ) finishes the proof. (cid:3) Proof of Proposition 5.7.
For simplicity of notation we will usually omit the subscript B , i.e. wewill write L µ := L µB , c fµ := c fµB , and so on.Without loss of generality we can assume that c fµ ≥ . Indeed, if that was not the case we couldconsider g = − f . Then the plane and constant L gµ = L fµ , c gµ = − c fµ ≥ are minimizing for α gµ ( B ) , and we have α gµ ( B ) = α fµ ( B ) . Thus, proving (5.9) for g is equivalent to proving it for f , and c gµ ≥ .Note that we always have F B ( P µ , P fµ ) . r d +1 so that if α µ ( B ) & , then (5.9) is trivial.Assume that α µ ( B ) ≤ ε for some small ε > (depending on dimensions and Ahlfors regularityconstants), to be fixed later.Note that if ε is small enough, then one can use Ahlfors regularity of µ to conclude that L µ ∩ . B = ∅ (see for example [Tol09, Lemma 3.1]). We use this observation, the assumption L fµ ∩ . B = ∅ and (5.10) to estimate c fµ r d +1 F B ( P µ , P fµ ) . c fµ dist H ( L µ ∩ B, L fµ ∩ B ) r =: c fµ D. Our aim is to show that c fµ D . c fµ α µ ( B ) + α fµ ( B ) . (5.11)Let < η < . be some dimensional constant. Note that, since L fµ ∩ . B = ∅ , the set L fµ ∩ . B is a d -dimensional ball with H d ( L fµ ∩ . B ) ∼ r d . We claim that we can find a d -dimensional ball B contained in L fµ ∩ . B , of radius ηr (in particular r B ∼ η r B ), and suchthat dist( z, L µ ) ≥ ηDr for all z ∈ B . (5.12) N α -NUMBER CHARACTERIZATION OF L p SPACES 21
Indeed, if there was no such ball, i.e. if for all d -dimensional balls B ⊂ L fµ ∩ . B of radius ηr there was some z ∈ B with dist( z, L µ ) ≤ ηDr , then it would follow easily from the definitionof Hausdorff distance, and from the fact that L µ and L fµ are d -planes intersecting . B , that dist H ( L µ ∩ B, L fµ ∩ B ) . ηDr = η dist H ( L µ ∩ B, L fµ ∩ B ) . For η small enough, this is a contradiction. We omit the details, which can be readily filled in e.g.using [AT15, Lemma 6.4].Consider an open neighbourhood of B given by U := { y ∈ R n : dist( y, B ) < ηDr } , and also for λ > set λU := { y ∈ R n : dist( y, B ) < ληDr } . Since D ≤ , one should think of U as an n -dimensional pancake around B of thickness ηDr ,so that the smaller D , the flatter the pancake. Note that by (5.12) for all < λ < we have λU ∩ L µ = ∅ , and also λU ⊂ B because B ⊂ . B .Let ϕ : R d → [0 , ηDr ] be a function satisfying ϕ ≡ ηDr in U , supp ϕ ⊂ U , and Lip( ϕ ) ≤ .Clearly, ϕ ∈ Lip ( B ) , and so (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) Z ϕf dµ − Z ϕ d L fµ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ α fµ ( B ) r d +1 . (5.13)Furthermore, note that ϕ ≡ ηDr on B , so that Z ϕ d L fµ = c fµ Z L fµ ϕ d H d ≥ c fµ ηDr H d ( B ) = C ( d ) c fµ Dη d +1 r d +1 . Together with (5.13) this implies Z ϕf dµ ≥ C ( η, d ) c fµ Dr d +1 − α fµ ( B ) r d +1 . (5.14)Recall that we are trying to prove c fµ D . c fµ α µ ( B ) + α fµ ( B ) . If we had c fµ D ≤ Λ α fµ ( B ) forsome Λ = Λ( η, d ) > , then there is nothing to prove. So without loss of generality assume that c fµ D ≥ Λ α fµ ( B ) . In that case (5.14) gives Z ϕf dµ & η c fµ Dr d +1 . (5.15)Now we define a modified version of ϕ . Recall that supp ϕ ⊂ U . For all y ∈ supp µ ∩ U let B y = B ( y, ηDr/ . We use the r covering theorem to extract from { B y } y ∈ supp µ ∩ U a subfamilyof pairwise disjoint balls { B i } i ∈ I such that supp µ ∩ U ⊂ S i B i . Note that S i B i ⊂ U , andin particular, S i B i ∩ L µ = ∅ . Moreover, the balls B i have bounded intersection. Thus, wemay consider a partition of unity Ψ = X i ∈ I ψ i , such that supp ψ i ⊂ B i for each i ∈ I , Ψ ≡ on S i B i , and Lip Ψ . ( ηDr ) − .Consider Φ = ϕ Ψ . We have k∇ Φ k ∞ ≤ k∇ ϕ k ∞ k Ψ k ∞ + k ϕ k ∞ k∇ Ψ k ∞ . ηDr ( ηDr ) − = 1 . Hence, C Φ ∈ Lip ( B ) for some C ∼ , so that (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) Z Φ f dµ − Z Φ d L fµ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ C − α fµ ( B ) r d +1 . (5.16)On the other hand, observe that Ψ ≡ on supp ϕ ∩ supp µ . By (5.15) Z Φ f dµ = Z ϕf dµ & η c fµ Dr d +1 . Together with (5.16) this gives Z Φ d L fµ ≥ C ( η ) c fµ Dr d +1 − C − α fµ ( B ) r d +1 & η c fµ Dr d +1 , (5.17)where we used once again the additional assumption c fµ D ≥ Λ α fµ ( B ) we made along the way(and choosing Λ large).Now we will show that Z L fµ Φ d H d . η α µ ( B ) r d +1 . (5.18)Since L fµ = c fµ H d L fµ , together with (5.17) this will give c fµ D . η c fµ α µ ( B ) , and so the proofof (5.11) will be finished.Recall that supp Φ ⊂ supp Ψ ⊂ S i B i , and that k Φ k ∞ ≤ k ϕ k ∞ = ηDr . Hence, Z L fµ Φ d H d . η Dr X i ∈ I H d ( L fµ ∩ B i ) . η I ( Dr ) d +1 . To estimate I we will use Ahlfors regularity of µ . Recall that { B i } i ∈ I are pairwise disjoint, theyare centered at points from supp µ ∩ U , and r ( B i ) = ηrD/ . Thus, I ( rD ) d ∼ η X i ∈ I µ ( B i ) = µ (cid:16) [ i ∈ I B i (cid:17) . On the other hand, since the balls { B i } are centered at points from U , we have S i ∈ I B i ⊂ U and µ (cid:16) [ i ∈ I B i (cid:17) ≤ µ (3 U ) . To bound µ (3 U ) consider ˜ ϕ ∈ Lip ( B ) such that ˜ ϕ ≥ , ˜ ϕ ≡ ηrD on U and supp ˜ ϕ ⊂ U .Recalling that U ∩ L µ = ∅ , we arrive at rDµ (3 U ) . η Z ˜ ϕ dµ = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) Z ˜ ϕ dµ − Z ˜ ϕ d L µ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ α µ ( B ) r d +1 . Putting all the estimates above together we get (5.18): Z L fµ Φ d H d . η I ( Dr ) d +1 . η rDµ (3 U ) . η α µ ( B ) r d +1 . (cid:3) R EFERENCES [ADT16] J. Azzam, G. David, and T. Toro. Wasserstein distance and the rectifiability of doubling measures: part I.
Math.Ann. , 364(1-2):151–224, 2016, arXiv:1408.6645. doi:10.1007/s00208-015-1206-z. 3[AT15] J. Azzam and X. Tolsa. Characterization of n -rectifiability in terms of Jones’ square function: Part II. Geom.Funct. Anal. , 25(5):1371–1412, 2015, arXiv:1501.01572. doi:10.1007/s00039-015-0334-7. 21[ATT18] J. Azzam, X. Tolsa, and T. Toro. Characterization of rectifiable measures in terms of α -numbers. Preprint, toappear in Trans. Amer. Math. Soc. , 2018, arXiv:1808.07661. doi:10.1090/tran/8170. 3, 18[Chr90] M. Christ. A T ( b ) theorem with remarks on analytic capacity and the Cauchy integral. Colloq. Math. , 60:601–628, 1990. doi:10.4064/cm-60-61-2-601-628. 5[D ˛ab19] D. D ˛abrowski. Necessary condition for rectifiability involving Wasserstein distance W . Preprint, to appearin Int. Math. Res. Not. IMRN , 2019, arXiv:1904.11000. doi:10.1093/imrn/rnaa012. 3[Dav88] G. David. Morceaux de graphes lipschitziens et intégrales singulieres sur une surface.
Rev. Mat. Iberoam. ,4(1):73–114, 1988. doi:10.4171/RMI/64. 5[DEM18] G. David, M. Engelstein, and S. Mayboroda. Square functions, non-tangential limits and harmonic measure inco-dimensions larger than one.
Preprint, to appear in Duke Math. J. , 2018, arXiv:1808.08882. 2[DJS85] G. David, J.-L. Journé, and S. Semmes. Opérateurs de Calderón-Zygmund, fonctions para-accrétives et inter-polation.
Rev. Mat. Iberoam. , 1(4):1–56, 1985. doi:10.4171/RMI/17. 4, 16, 17[DM20] G. David and S. Mayboroda. Harmonic measure is absolutely continuous with respect to the Hausdorff mea-sure on all low-dimensional uniformly rectifiable sets. arXiv preprint , 2020, arXiv:2006.14661. 2 N α -NUMBER CHARACTERIZATION OF L p SPACES 23 [DS91] G. David and S. Semmes. Singular integrals and rectifiable sets in R n : Au-delà des graphes lipschitziens. Astérisque , 193, 1991. doi:10.24033/ast.68. 2[Fen20] J. Feneuil. Absolute continuity of the harmonic measure on low dimensional rectifiable sets. arXiv preprint ,2020, arXiv:2006.03118. 2[Gra14] L. Grafakos.
Classical Fourier Analysis , volume 249 of
Grad. Texts in Math.
Springer-Verlag New York, 3edition, 2014. doi:10.1007/978-1-4939-1194-3. 9[HT14] T. Hytönen and O. Tapiola. Almost Lipschitz-continuous wavelets in metric spaces via a new randomizationof dyadic cubes.
J. Approx. Theory , 185:12–30, 2014, arXiv:1310.2047. doi:10.1016/j.jat.2014.05.017. 5[Mat95] P. Mattila.
Geometry of sets and measures in Euclidean spaces: fractals and rectifiability , volume 44 of
Cambridge Stud. Adv. Math.
Cambridge Univ. Press, 1995. doi:10.1017/CBO9780511623813. 2[Orp18] T. Orponen. Absolute continuity and α -numbers on the real line. Anal. PDE , 12(4):969–996, 2018,arXiv:1703.02935. doi:10.2140/apde.2019.12.969. 4[Tol01] X. Tolsa. Littlewood–Paley theory and the T (1) theorem with non-doubling measures. Adv. Math. , 164(1):57–116, 2001, arXiv:math/0006039. doi:10.1006/aima.2001.2011. 17[Tol08] X. Tolsa. Principal values for Riesz transforms and rectifiability.
J. Funct. Anal. , 254(7):1811–1863, 2008,arXiv:0708.0109. doi:j.jfa.2007.07.020. 2[Tol09] X. Tolsa. Uniform rectifiability, Calderón-Zygmund operators with odd kernel, and quasiorthogonality.
Proc.Lond. Math. Soc. (3) , 98(2):393–426, 2009, arXiv:0805.1053. doi:10.1112/plms/pdn035. 2, 4, 10, 20[Tol12] X. Tolsa. Mass transport and uniform rectifiability.
Geom. Funct. Anal. , 22(2):478–527, 2012,arXiv:1103.1543. doi:10.1007/s00039-012-0160-0. 4[Tol14] X. Tolsa.
Analytic capacity, the Cauchy transform, and non-homogeneous Calderón-Zygmund theory , volume307 of
Progress in Mathematics . Birkhäuser, 2014. doi:10.1007/978-3-319-00596-6. 3, 10, 11[Tol17] X. Tolsa. Rectifiable measures, square functions involving densities, and the Cauchy transform.
Mem. Amer.Math. Soc. , 245(1158), 2017, arXiv:1408.6979. doi:10.1090/memo/1158. 16J
ONAS A ZZAM , S
CHOOL OF M ATHEMATICS , U
NIVERSITY OF E DINBURGH , JCMB, K
INGS B UILDINGS , M AY - FIELD R OAD , E
DINBURGH , EH9 3JZ, S
COTLAND . E-mail address : j.azzam "at" ed.ac.uk D AMIAN D ˛ABROWSKI , D EPARTAMENT DE M ATEMÀTIQUES , U
NIVERSITAT A UTÒNOMA DE B ARCELONA ; B
ARCELONA G RADUATE S CHOOL OF M ATHEMATICS (BGSM
ATH ), E
DIFICI
C F
ACULTAT DE C IÈNCIES , 08193 B
ELLATERRA (B ARCELONA ), C
ATALONIA , S
PAIN
E-mail address ::