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

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

where ɛ>0 is a constant depending on the domain Ω. To see that this generalization is the naturalone, think of (0.5) as a function of x ∈ R d and replace ω with Lebesgue measure. It’s importantto note that, ong>forong> any x, the sum in (0.5) contains essentially only one term.Finally there is the right-hand term l(Q) d+1 . We will replace this with l(Q b )ω(Q b ).With the precise definitions still to follow, the rephrased version of (0.2) is⎛∫µ(T (Q b )) 1/q ⎜⎝∂Ω⎡⎤∞∑ 2⎣ω(Q −jɛb )ω(2 j Q b ) χ R j (Q b )(x) ⎦j=0p ′ /2⎞⎟σ(x) dω(x) ⎠1/p ′ ≤ cl(Q b )ω(Q b ),and our main theorem (Theorem 3.1), which we prove in Section 3, isTheorem 3.1. Let Ω ⊂ R d+1 be a bounded Lipschitz domain, and let ω be harmonic measureon ∂Ω ong>forong> some fixed point X 0 ∈ Ω. Suppose that v ∈ L 1 (∂Ω,dω) is a non-negative function andµ is a positive Borel measure on Ω. Define σ ≡ v 1−p′ , and suppose that σdω ∈ A ∞ (ω) on ∂Ω. If1

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