Weighted inequalities for gradients on non-smooth domains ...

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Weighted inequalities for gradients on non-smooth domains ...

The key to our argument lies in defining the right maximal function. Let us assume that wehave a fixed finite linear combination 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 function ong>forong> the Littlewood-Paley function g ∗ (f) turns out to beF ∗ (x) ≡ sup |F (I)|.I:x∈ICorresponding to F ∗ (x) is a “maximal square function” 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>mulation ofthe φ (Q) ’s requires us to surmount some non-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 onlyexceptional points lie on the faces of dyadic cubes, and these have ω-measure 0, because ω isdoubling. QED.Lemma 1.3. There is a constant 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 definition 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 ) ⎦ .,

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