Using our bound **on** λ Q , the last quantity is less than or equal toC ∑ Q b ∈G(|g(Q b )|µ(T (Q b ))l(Q b ) √ ω(Q b )) p′⎛∫⎜⎝∂Ω⎡⎤∞∑⎣2 −j(2α−τ)ω(2 j Q b ) χ R j (Q b )(x) ⎦j=0p ′ /2⎞⎟σdω⎠ ;which, since p ′ ≥ q ′ , is bounded by⎛(∑C ⎜|g(Q b )|µ(T (Q b ))⎝l(Q b ) √ ω(Q b )Q b ∈G) q′⎛∫⎜⎝∂Ω⎡⎤∞∑⎣2 −j(2α−τ)ω(2 j Q b ) χ R j (Q b )(x) ⎦j=0p ′ /2⎞⎟σdω⎠q ′ /p ′ ⎞⎟⎠p ′ /q ′ .In order ong>forong> our dual inequality to hold, the last quantity must be less than or equal to⎛⎞C ⎝ ∑ |g(Q b )| q′ µ(T (Q b ))Q b ∈G⎠p ′ /q ′ ,which will be true ong>forong> all g’s if(µ(T (Q b ))l(Q b ) √ ω(Q b )) q′⎛∫⎜⎝∂Ω⎡⎤∞∑⎣2 −j(2α−τ)ω(2 j Q b ) χ R j (Q)(x) ⎦j=0p ′ /2⎞⎟σdω⎠q ′ /p ′ ≤ Cµ(T (Q b ))ong>forong> all Q b ∈G; i.e., after some transpositi**on**,⎛∫µ(T (Q b )) 1/q ⎜⎝∂Ω⎡⎤∞∑⎣2 −j(2α−τ)ω(2 j Q b ) χ R j (Q b )(x) ⎦j=0p ′ /2⎞⎟σdω⎠1/p ′ ≤ Cl(Q b ) √ ω(Q b ),which is our c**on**diti**on** from Theorem 3.1. QED.We shall now state the fairly straightong>forong>ward corollary of Theorem 3.1 to soluti**on**s of uniong>forong>mlyelliptic equati**on**s in divergence ong>forong>m.Recall that such an equati**on** has the ong>forong>mLu ≡ div(A(x)∇u) =0, (3.7)where A(x) =[a i,j (x)] is a real, bounded, measurable, (d +1)× (d + 1) symmetric matrix satisfyingΛ −1 |ξ| 2 ≤ ∑ i,ja i,j ξ i ξ j ≤ Λ|ξ| 2ong>forong> all x ∈ Ω and all ξ ∈ R d+1 , where Λ is a fixed positive c**on**stant. A u that satisfies (3.7) iscalled elliptic.30

The Dirichlet problem ong>forong> the operator L is solvable **on** bounded Lipschitz **domains** Ω, andthe soluti**on** is written in terms of elliptic measure ω L . Unong>forong>tunately, this measure need not beA ∞ with respect to surface measure **on** ∂Ω—but it is doubling. It also has a Green’s functi**on**, G L ,which satisfies estimates (relative to ω L ) analogous to those that hold ong>forong> G and ω. Indeed, theproofs we have given c**on**cerning weighted norm ong>inequalitiesong> of the ong>forong>m(∫ ) 1/q (∫|∇u| q dµ ≤ |f| p vdωΩ δ ∂Ω(where u is f’s harm**on**ic extensi**on**, and ω is harm**on**ic measure ong>forong> the Laplacian), will go throughalmost verbatim ong>forong> weighted norm ong>inequalitiesong> of the ong>forong>m) 1/p(∫ ) 1/q (∫) 1/p|∇u| q dµ ≤ |f| p vdω LΩ δ ∂Ωwhere u is now f’s elliptic extensi**on**. The **on**e difficulty is that ∇u need not be defined pointwise.This requires us to rephrase our weighted norm inequality somewhat, and to look carefully at theproof of **on**e of the auxiliary results used in the proof of Theorem 3.1.Let us assume that we have our family of boundary cubes G, as used in the proof of Theorem3.1. In that proof we used the fact that, ong>forong> every Q b ∈G,sup T (Qb )) |∇u| was “morally equivalent”tosup l(Q b ) −1 |u(X) − u(Y )| (3.8)X,Y ∈T (Q b )when u was harm**on**ic. In our rephrased ong>forong>m of Theorem 3.1, we will essentially replace |∇u| withthe right-hand side of (3.8).Let G be a family of boundary cubes as described above. Let {X Qb } and {Y Qb } be two familiesof points in Ω, both indexed over G, such that, ong>forong> all Q b ∈G, X Qb ∈ T (Q b )andY Qb ∈ T (Q b ). Weshall call such a double sequence hyperbolically close.Theorem 3.2. Let Ω ⊂ R d+1 be a bounded Lipschitz domain, and let ω L be elliptic measure **on**∂Ω ong>forong> some fixed point X 0 ∈ Ω. LetG be a familyof boundarycubes as described above. Supposethat v ∈ L 1 (∂Ω,dω L ) is a n**on**-negative functi**on** and µ is a positive Borel measure **on** Ω. Defineσ ≡ v 1−p′ , and suppose that σdω L ∈ A ∞ (ω L ) **on** ∂Ω. If1