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

where ɛ>0 is a c**on**stant depending **on** the domain Ω. To see that this generalizati**on** is the natural**on**e, think of (0.5) as a functi**on** of x ∈ R d and replace ω with Lebesgue measure. It’s importantto note that, **on**tains essentially **on**ly **on**e 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 definiti**on**s still to follow, the rephrased versi**on** 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 Secti**on** 3, isTheorem 3.1. Let Ω ⊂ R d+1 be a bounded Lipschitz domain, and let ω be harm**on**ic measure**on** ∂Ω **on**-negative functi**on** andµ is a positive Borel measure **on** Ω. Define σ ≡ v 1−p′ , and suppose that σdω ∈ A ∞ (ω) **on** ∂Ω. If1