SQUARE FUNCTIONS WITH GENERAL MEASURES

SQUARE FUNCTIONS WITH GENERAL MEASURES

arXiv:1212.3684v1 [math.CA] 15 Dec 2012

HENRI MARTIKAINEN AND MIHALIS MOURGOGLOU

A BSTRACT. We characterize the boundedness of square functions in the upper half-space with general measures. The short proof is based on an averaging identity over good Whitney regions.

1. I NTRODUCTION Let µ be a Borel measure on Rn . We assume that µ(B(x, r)) ≤ λ(x, r) for some λ : Rn × (0, ∞) → (0, ∞) satisfying that r 7→ λ(x, r) is non-decreasing and λ(x, 2r) ≤ Cλ λ(x, r) for all x ∈ Rn and r > 0. Let ˆ θt f (x) = st (x, y)f (y) dµ(y), x ∈ Rn , t > 0, Rn

where st is a kernel satisfying for some α > 0 that |st (x, y)| .

tα tα λ(x, t) + |x − y|αλ(x, |x − y|)

and

|y − z|α tα λ(x, t) + |x − y|αλ(x, |x − y|) whenever |y − z| < t/2. We use the ℓ∞ metric on Rn . If Q ⊂ Rn is a cube with sidelength ℓ(Q), we define the associated Carleson b = Q × (0, ℓ(Q)). In this note we will prove the following theorem: box Q |st (x, y) − st (x, z)| .

1.1. Theorem. Assume that there exists a function b ∈ L∞ (µ) such that ˆ b(x) dµ(x) & µ(Q) Q

and

(1.2)

¨

b Q

|θt b(x)|2 dµ(x)

dt . µ(3Q) t

2010 Mathematics Subject Classification. 42B20. Key words and phrases. Square function, non-homogeneous analysis. H.M. is supported by the Emil Aaltonen Foundation. M.M. is supported by Fondation de Mathématiques Jacques Hadamard (FMJH). The authors wish to thank Université Paris-Sud 11, Orsay, for its hospitality. 1

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HENRI MARTIKAINEN AND MIHALIS MOURGOGLOU

for every cube Q ⊂ Rn . Then there holds that ¨ dt (1.3) |θt f (x)|2 dµ(x) . kf k2L2 (µ) , t Rn+1 +

f ∈ L2 (µ).

1.4. Corollary. If ¨

b Q

|θt χQ (x)|2 dµ(x)

dt . µ(Q) t

for every cube Q ⊂ R , then (1.3) holds. n

To the best of our knowledge such a boundedness result was previously known only in the Lebesgue case (see [1], [8], [3], [2] for such results). Our framework covers, as is well-known, the doubling measures, the power bounded measures (µ(B(x, r)) . r m for some m), and some other additional cases of interest (see Chapter 12 of [5] for an example in the context of Calderón–Zygmund operators). The proof of our result follows by first establishing an averaging equality over good dyadic Whitney regions. Such an identity is inspired by Hytönen’s proof of the A2 conjecture [4], which uses a very nice refinement of the Nazarov–Treil– Volberg method of random dyadic systems. After this the probabilistic part of the proof ends, and we may study just one grid establishing a uniform (in the averaging parameter) bound for these good Whitney averages. Then we expand a function f in the same grid using the standard b-adapted martingale differences. It is not necessary to restrict this expansion into good cubes. The rest of the proof is a non-homogeneous T b type summing argument (see e.g. [7] and [5]), which, in this setting, we manage to perform in a delightfully clear way. Indeed, it only takes a few pages. We find that the proof is of interest, since it is, in particular, a very accessible application of the most recent non-homogeneous methods. 1.5. Remark. The property λ(x, |x − y|) ∼ λ(y, |x − y|) can be assumed without loss of generality. Indeed, in Proposition 1.1 of [6] it is shown that Λ(x, r) := inf z∈Rn λ(z, r + |x − z|) satisfies that r 7→ Λ(x, r) is non-decreasing, Λ(x, 2r) ≤ Cλ Λ(x, r), µ(B(x, r)) ≤ Λ(x, r), Λ(x, r) ≤ λ(x, r) and Λ(x, r) ≤ Cλ Λ(y, r) if |x−y| ≤ r. Therefore, we may (and do) assume that the dominating function λ satisfies the additional symmetry property λ(x, r) ≤ Cλ(y, r) if |x − y| ≤ r. 1.6. Remark. The condition (1.2) is necessary for (1.3) to hold. Indeed, one writes b = bχ3Q + bχ(3Q)c and notices that in (1.2) the part with bχ3Q is dominated by kbχ3Q k2L2 (µ) . µ(3Q), if one assumes (1.3). For the other part, we note that for every x ∈ Q there holds that ˆ tα |θt (bχ(3Q)c )(x)| . dµ(y) α (3Q)c |x − y| λ(x, |x − y|) ˆ |y − cQ |−α α .t dµ(y) . tα ℓ(Q)−α . λ(c , |y − c |) c Q Q Q

SQUARE FUNCTIONS WITH GENERAL MEASURES

3

This implies that ¨ ˆ ℓ(Q) dt 2 −2α |θt (bχ(3Q)c )(x)| dµ(x) . µ(Q) · ℓ(Q) t2α−1 dt . µ(Q). t b Q 0 The assumption of Corollary 1.4 is also necessary. However, even there one may weaken the assumption by replacing on the right-hand side µ(Q) with, say, µ(3Q) (note that Theorem 1.1 is true with µ(3Q) replaced by µ(κQ), κ > 1). 2. P ROOF

OF THE MAIN THEOREM

2.1. A random dyadic grid. Let us be given a random P dyadic grid D = D(w), w = (wi )i∈Z ∈ ({0, 1}n )Z . This means that D = {Q + i: 2−i 0 and 2r(1−γ) ≥ 3. Furthermore, it is important to note that for a fixed Q ∈ D0 the set Q + w depends on wi with 2−i < ℓ(Q), while the goodness (or badness) of Q+w depends on wi with 2−i ≥ ℓ(Q). In particular, these notions are independent. 2.2. Averaging over good Whitney regions. Let f ∈ L2 (µ). For R ∈ D, let WR = R × (ℓ(R)/2, ℓ(R)) be the associated Whitney region. We can assume that w is such that µ(∂R) = 0 for every R ∈ D = D(w) (this is the case for a.e. w). Using that πgood := Pw (R + w is good) = Ew χgood (R + w) for any R ∈ D0 we may now write ¨ X¨ dt dt 2 |θt f (x)|2 dµ(x) |θt f (x)| dµ(x) = Ew t t WR Rn+1 + R∈D ¨ X dt |θt f (x)|2 dµ(x) = Ew t WR+w R∈D0 ¨ 1 X dt πgood Ew = |θt f (x)|2 dµ(x) πgood R∈D t WR+w 0 ¨ 1 X dt Ew [χgood (R + w)]Ew = |θt f (x)|2 dµ(x) πgood R∈D t WR+w 0 ¨ h 1 X dt i Ew χgood (R + w) = |θt f (x)|2 dµ(x) πgood R∈D t WR+w 0 ¨ X dt 1 |θt f (x)|2 dµ(x) . = Ew πgood t WR R∈D good

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HENRI MARTIKAINEN AND MIHALIS MOURGOGLOU

˜ Notice that we used the independence of χgood (R+w) and WR+w |θt f (x)|2 dµ(x) dtt for a fixed R ∈ D0 . In [4], Hytönen used averaging equalities to represent a general Calderón–Zygmund operator as an average of dyadic shifts. These techniques are similar in spirit. We now fix one w. It is enough to show that X ¨ dt |θt f (x)|2 dµ(x) . kf k2L2 (µ) (2.1) t WR R∈D good

ℓ(R)≤2s

with every large s ∈ Z. Let us now fix the s as well. 2.3. Adapted decomposition of f . We now perform the standard´ b-adapted martingale difference decomposition of f . We define hf iQ = µ(Q)−1 Q f dµ, EQ = and

hf iQ χQ b hbiQ

X h hf iQ′ hf iQ i χQ′ b. − hbiQ′ hbiQ ′

∆Q f =

Q ∈ch(Q)

We can write in L (µ) that 2

f=

X

∆Q f +

Q∈D ℓ(Q)≤2s

There also holds that kf k2L2 (µ) ∼

X

X

Q∈D ℓ(Q)=2s

k∆Q f k2L2 (µ) +

Q∈D ℓ(Q)≤2s

EQ f.

X

kEQ f k2L2 (µ) .

Q∈D ℓ(Q)=2s

We plug this decomposition into (2.1) noting that we need to prove that 2 X ¨ X dt θt ∆Q f (x) dµ(x) . kf k2L2 (µ) , t WR Q∈D R∈D good

ℓ(R)≤2s

ℓ(Q)≤2s

where we abuse notation by redefining the operator ∆Q to be ∆Q + EQ if ℓ(Q) = 2s . 2.4. The case ℓ(Q) < ℓ(R). Here we show that 2 X X ¨ dt θt ∆Q f (x) dµ(x) . kf k2L2 (µ) . t WR s R: ℓ(R)≤2

Q: ℓ(Q) ℓ(R) one has D(Q, R) . d(Q, R). It remains to note that if z ∈ Q ∪ R, then |x − z| . D(Q, R). We conclude that ℓ(Q)α/2 ℓ(R)α/2 µ(Q)1/2 k∆Q f kL2 (µ) , (x, t) ∈ WR . |θt ∆Q f (x)| . D(Q, R)α supz∈Q∪R λ(z, D(Q, R)) This yields that X ¨ R: ℓ(R)≤2s

WR

2 i2 dt X h X θt ∆Q f (x) dµ(x) . AQR k∆Q f kL2 (µ) t R Q Q: ℓ(Q) ℓ(R)γ ℓ(Q)1−γ . In this subsection we deal with 2 X X ¨ dt θt ∆Q f (x) dµ(x) . t WR s s R: ℓ(R)≤2

Q: ℓ(R)≤ℓ(Q)≤2 d(Q,R)>ℓ(R)γ ℓ(Q)1−γ

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HENRI MARTIKAINEN AND MIHALIS MOURGOGLOU

Let (x, t) ∈ WR . The size estimate gives that ˆ ℓ(R)α |∆Q f (y)| dµ(y), |θt ∆Q f (x)| . α Q d(Q, R) λ(x, d(Q, R)) where we claim that ℓ(R)α ℓ(Q)α/2 ℓ(R)α/2 . . d(Q, R)α λ(x, d(Q, R)) D(Q, R)α λ(x, D(Q, R)) This estimate is trivial if d(Q, R) ≥ ℓ(Q), because in that case D(Q, R) . d(Q, R). So assume that d(Q, R) < ℓ(Q), in which case D(Q, R) . ℓ(Q). This case is more tricky. Note that λ(x, ℓ(Q)) = λ(x, (ℓ(Q)/ℓ(R))γ ℓ(R)γ ℓ(Q)1−γ ) log2 ( ℓ(Q) )γ ℓ(R)

. Cλ

λ(x, ℓ(R)γ ℓ(Q)1−γ ) =

 ℓ(Q) γd ℓ(R)

λ(x, ℓ(R)γ ℓ(Q)1−γ ).

Using the assumption d(Q, R) > ℓ(R)γ ℓ(Q)1−γ and the identity γd + γα = α/2, we conclude that ℓ(Q)α/2 ℓ(R)α/2 ℓ(Q)α/2 ℓ(R)α/2 ℓ(R)α . . . d(Q, R)α λ(x, d(Q, R)) ℓ(Q)α λ(x, ℓ(Q)) D(Q, R)α λ(x, D(Q, R)) Noting again that if z ∈ Q ∪ R, then |x − z| . D(Q, R), we have shown that |θt ∆Q f (x)| .

ℓ(Q)α/2 ℓ(R)α/2 µ(Q)1/2 k∆Q f kL2 (µ) , D(Q, R)α supz∈Q∪R λ(z, D(Q, R))

(x, t) ∈ WR .

This is enough by Proposition 2.2 like in the previous subsection. 2.6. The case ℓ(R) ≤ ℓ(Q) ≤ 2r ℓ(R) and d(Q, R) ≤ ℓ(R)γ ℓ(Q)1−γ . Here we bound 2 X X ¨ dt θt ∆Q f (x) dµ(x) t WR s r s Q: ℓ(R)≤ℓ(Q)≤min(2 ,2 ℓ(R)) d(Q,R)≤ℓ(R)γ ℓ(Q)1−γ

R: ℓ(R)≤2

.

X X ¨ Q R: R∼Q

WR

|θt ∆Q f (x)|2 dµ(x)

dt , t

where we have written Q ∼ R to mean ℓ(Q) ∼ ℓ(R) and d(Q, R) . min(ℓ(Q), ℓ(R)). We also used the fact that given R there are . 1 cubes Q for which Q ∼ R. Let (x, t) ∈ WR . The size estimate gives that 1 µ(Q)1/2 k∆Q f kL2 (µ) 1/2 λ(x, t) λ(x, t)1/2 1 µ(Q)1/2 k∆Q f kL2 (µ) ≤ µ(R)−1/2 k∆Q f kL2 (µ) . . 1/2 λ(cQ , ℓ(Q)) λ(cR , ℓ(R))1/2

|θt ∆Q f (x)| .

SQUARE FUNCTIONS WITH GENERAL MEASURES

Therefore, we have that ¨

|θt ∆Q f (x)|2 dµ(x)

WR

and so X X ¨ Q R: R∼Q

|θt ∆Q f (x)|2 dµ(x)

WR

7

dt . k∆Q f k2L2 (µ) , t

X dt X . k∆Q f k2L2 (µ) 1 . kf k2L2 (µ) . t Q R: R∼Q

2.7. The case ℓ(Q) > 2r ℓ(R) and d(Q, R) ≤ ℓ(R)γ ℓ(Q)1−γ . We finally utilize the goodness of R to conclude that in this case we must actually have that R ⊂ Q. This means that 2 X X ¨ dt θt ∆Q f (x) dµ(x) t WR r s R∈D good

ℓ(R)≤2s

Q: 2 ℓ(R)