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

The integral we need to estimate naturally breaks into two pieces. Let us recall the region wedenoted by R in Section 2:R≡{(x, y) : ψ(x) κ}. By our estimates onthe φ (Q) ’s,|Tg(x)| ≤C κ,Ω∑Q b ∈G|λ Q | √ ω(Q b )∞∑j=02 −jαω(2 j Q b ) χ R j (Q b )(x).2There is a C, independent of x ∈ℵ, such that−jαω(2 j Q b ) χ R j (Q b )(x) can be non-zero ong>forong> at most Cmany j’s. For each of these j’s, 2 −j is essentially equal to l(Q b ), with the comparability constantsdepending on κ and Ω. Also, ong>forong> such j, ω(2 j Q b ) is comparable to ω(∂Ω), which equals 1. Thereong>forong>e,ong>forong> x ∈ℵ,∑|Tg(x)| ≤C κ,Ω |λ Q | √ ω(Q b )l(Q b ) α≤ C ∑ Q b ∈GQ b ∈G|g(Q)|µ(T (Q b ))l(Q b ) −1 l(Q b ) α ;which looks funny—but we have a good reason ong>forong> not combining the exponents in the l(Q b )’s.We assume that we have⎛∫µ(T (Q b )) 1/q ⎜⎝∂Ω⎡⎣ω(Q b )⎤∞∑ 2 −jɛω(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 )ong>forong> every Q b ∈G. We may replace the integral on the left-hand side of this inequality by an integralover the smaller region ℵ. Doing so, we may rewrite the inequality (after a change in c) as:where ɛ>0 is small.Thus, ong>forong> x ∈ℵ,|Tg(x)| ≤c ∑ Q b ∈G(µ(T (Q b )) 1/q ≤ cl(Q b ) −ɛ/2 l(Q b ) √ )ω(Q b )(∫,ℵ σdω) 1/p ′|g(Q)|µ(T (Q b ))l(Q b ) −1 l(Q b ) α= c ∑ |g(Q)|µ(T (Q b )) 1/q′ µ(T (Q b )) 1/q l(Q b ) −1 l(Q b ) αQ b ∈G⎡⎤≤ c ⎣ ∑ |g(Q)|µ(T (Q b )) 1/q′ l(Q b ) α−ɛ/2√ (∫ω(Q b ) ⎦ σdωQ b ∈Gℵ25) −1/p′.

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