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

The Dirichlet problem <strong>for</strong> the operator L is solvable on bounded Lipschitz domains Ω, andthe solution is written in terms of elliptic measure ω L . Un<strong>for</strong>tunately, this measure need not beA ∞ with respect to surface measure on ∂Ω—but it is doubling. It also has a Green’s function, G L ,which satisfies estimates (relative to ω L ) analogous to those that hold <strong>for</strong> G and ω. Indeed, theproofs we have given concerning weighted norm <strong>inequalities</strong> of the <strong>for</strong>m(∫ ) 1/q (∫|∇u| q dµ ≤ |f| p vdωΩ δ ∂Ω(where u is f’s harmonic extension, and ω is harmonic measure <strong>for</strong> the Laplacian), will go throughalmost verbatim <strong>for</strong> weighted norm <strong>inequalities</strong> of the <strong>for</strong>m) 1/p(∫ ) 1/q (∫) 1/p|∇u| q dµ ≤ |f| p vdω LΩ δ ∂Ωwhere u is now f’s elliptic extension. The one 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 one 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, <strong>for</strong> 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 harmonic. In our rephrased <strong>for</strong>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, <strong>for</strong> 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 <strong>for</strong> some fixed point X 0 ∈ Ω. LetG be a familyof boundarycubes as described above. Supposethat v ∈ L 1 (∂Ω,dω L ) is a non-negative function and µ is a positive Borel measure on Ω. Defineσ ≡ v 1−p′ , and suppose that σdω L ∈ A ∞ (ω L ) on ∂Ω. If1