Square functions with general measures

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.

If Q ⊂ R n is a cube with sidelength ℓ(Q), we define the associated Carleson box Q = Q × (0, ℓ(Q)). In this note we will prove the following theorem: 1.1. Theorem. Assume that there exists a function b ∈ L ∞ (µ) such that and for every cube Q ⊂ R n . Then there holds that

Corollary. If¨
for every cube Q ⊂ R n , then (1.3) holds.
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 A 2 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.
1.6. Remark. The condition (1.2) is necessary for (1.3) to hold. Indeed, one writes b = bχ 3Q + bχ (3Q) c and notices that in (1.2) the part with bχ 3Q is dominated by bχ 3Q 2 L 2 (µ) µ(3Q), if one assumes (1.3). For the other part, we note that for every x ∈ Q there holds that This implies thaẗ 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).

A random dyadic grid. Let us be given a random dyadic grid
Here D 0 is the standard dyadic grid of R n .
We set γ = α/(2d + 2α), where α > 0 appears in the kernel estimates and Furthermore, it is important to note that for a fixed Q ∈ D 0 the set Q + w depends on w i with 2 −i < ℓ(Q), while the goodness (or badness) of Q+w depends on w i with 2 −i ≥ ℓ(Q). In particular, these notions are independent.
2.2. Averaging over good Whitney regions. Let f ∈ L 2 (µ). For R ∈ D, let W R = 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 π good := P w (R + w is good) = E w χ good (R + w) for any R ∈ D 0 we may now writë R n+1 Notice that we used the independence of χ good (R+w) and˜W for a fixed R ∈ D 0 . 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 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 mar- and We can write in L 2 (µ) that There also holds that We plug this decomposition into (2.1) noting that we need to prove that where we abuse notation by redefining the operator ∆ Q to be 2.4. The case ℓ(Q) < ℓ(R). Here we show that

Let us set
where c Q is the center of the cube Q. Noting that |y −c Q | ≤ ℓ(Q)/2 ≤ ℓ(R)/4 < t/2 for every y ∈ Q, we may estimate The last estimate is seen as follows: In the numerator, simply estimate ℓ(Q) α ≤ ℓ(Q) α/2 ℓ(R) α/2 . In the denominator we split into two cases. If d(Q, R) ≤ ℓ(R) one has D(Q, R) ℓ(R), while in the case d(Q, R) > 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 This yields that 2.5. The case ℓ(Q) ≥ ℓ(R) and d(Q, R) > ℓ(R) γ ℓ(Q) 1−γ . In this subsection we deal with Let (x, t) ∈ W R . The size estimate gives that where we claim that D(Q, R)) .
Noting again that if z ∈ Q ∪ R, then |x − z| D(Q, R), we have shown that This is enough by Proposition 2.2 like in the previous subsection.
We also used the fact that given R there are 1 cubes Q for which Q ∼ R.
Let (x, t) ∈ W R . The size estimate gives that Therefore, we have thaẗ and so 2.7. The case ℓ(Q) > 2 r 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 where gen(R) is determined by ℓ(R) = 2 −gen(R) , and R (k) ∈ D is the unique cube for which ℓ(R (k) ) = 2 k ℓ(R) and R ⊂ R (k) . We decompose Noticing that we have that Let us first deal with the last term. We bound where the last estimate follows from Carleson embedding theorem and the next lemma. for every R ∈ D.
Proof. Fix R ∈ D, and let F (R) denote the maximal Q ∈ D such that ℓ(Q) ≤ 2 −r ℓ(R) and d(Q, R c ) ≥ 3ℓ(Q). Notice that where we used goodness, the fact that 2 r(1−γ) ≥ 3, our assumption about b and the fact that Q∈F (R) χ 3Q χ R .
To complete the proof of our main theorem, it remains to control By the accretivity condition for b, there holds that Let (x, t) ∈ W R . The size estimate gives that where goodness was used to conclude that d(R, R n \ R (k−1) ) ≥ ℓ(R) 1/2 ℓ(R (k−1) ) 1/2 . Let then S ∈ ch(R (k) ), S ⊂ R (k) \ R (k−1) . Notice that d(R, S) ≥ ℓ(R) γ ℓ(S) 1−γ . Therefore, an estimate like in the subsection 2.5 gives that What we need then readily follows from the following estimate: