or, more succinctly:|φ (Q) (x)| ≤ √ ω(Q)∞∑( 2−jαj=0ω(2 j Q))χ Rj (Q)(x).2) Smoothness. For any x and y in R d ,|φ (Q) (x) − φ (Q) (y)| ≤( |x − y|l(Q)) β√ ∑∞ ( 2−jαω(Q)j=0ω(2 j Q))(χ Rj (Q)(x)+χ Rj (Q)(y)).Note that, given the size c**on**diti**on**, the **smooth**ness c**on**diti**on** is **on**ly meaningful when |x−y| ≤l(Q).3) Cancellati**on**. For every finite linear combinati**on** ∑ Q γ Qφ (Q) ,∫R d | ∑ Qγ Q φ (Q) | 2 dω ≤ ∑ Q|γ Q | 2 .All of our results depend **on** the next theorem.Theorem 1.1. Let {φ (Q) } Q∈D be a standard familyof functi**on**s, and let ν ∈ A ∞ (ω). If0

The key to our argument lies in defining the right maximal functi**on**. Let us assume that wehave a fixed finite linear combinati**on** f = ∑ Q λ Qφ (Q) .IfI ∈D,wedefineS(I) ≡{Q ∈D: Q ⊄ I}.It is useful to think of S(I) as the family of dyadic cubes that “surround” I. Ifx ∈ I, wedefineF (I,x) ≡∑Q:Q∈S(I)λ Q φ (Q) (x),and we do not define F (I,x)ong>forong>x/∈ I. Ifx I is the center of I, then we set F (I) ≡ F (I,x I ). Theright maximal functi**on** ong>forong> the Littlewood-Paley functi**on** g ∗ (f) turns out to beF ∗ (x) ≡ sup |F (I)|.I:x∈ICorresp**on**ding to F ∗ (x) is a “maximal square functi**on**” adapted to g ∗ (f). For x ∈ I, wedefine⎛G(I,x) ≡ ⎝ ∑Q∈S(I)⎡∞∑|λ Q | 2 ⎣j=02 −j(2α−τ)⎤⎞ω(2 j Q) χ R j (Q)(x) ⎦⎠and we do not define G(I,x)ong>forong>x/∈ I. We similarly define G(I) ≡ G(I,x I )andG ∗ (x) ≡ sup G(I).I:x∈IIn order to prove Theorem 1.1, we shall prove seven fairly elementary lemmas, followed by adifficult lemma, which is really the heart of the proof of Theorem 1.1. These lemmas are directlyanalogous to, respectively, Lemmas 1–7 and Lemma 1.9 in [Wi]. Our more general ong>forong>mulati**on** ofthe φ (Q) ’s requires us to surmount some n**on**-trivial technical obstacles.Lemma 1.2. For ω-a. e. x, |f(x)| ≤F ∗ (x).Proof. The inequality is obviously true Lebesgue almost everywhere. However, the **on**lyexcepti**on**al points lie **on** the faces of dyadic cubes, and these have ω-measure 0, because ω isdoubling. QED.Lemma 1.3. There is a c**on**stant C such that G ∗ (x) ≤ Cg ∗ (f)(x) almost everywhere.Proof. Let I ∈Dand x ∈ I. We need to show that G(I) ≤ Cg ∗ (f)(x), ong>forong> which it is clearlysufficient to show that⎡⎤(G(I)) 2 ≤ C∑∞∑|λ Q | 2 ⎣2 −j(2α−τ)ω(2 j Q) χ R j (Q)(x) ⎦ ,where (recall the definiti**on** above)(G(I)) 2 =Q:Q∈S(I)∑Q:Q∈S(I)j=0⎡∞∑|λ Q | 2 ⎣j=072 −j(2α−τ)1/2⎤ω(2 j Q) χ R j (Q)(x I ) ⎦ .,