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

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

We claim that⎛H i (x) ≤ C ⎝ ∑ jω(P j )[ ∞∑k=0] ⎞2⎠1/2−kτω(2 k P j ) χ R k (P j )(x) . (1.6)Note that the right-hand side of (1.6) does not depend on i. Oncewehave(1.6), we will obtain∑∫iC(P i )\D⎛∫ ∑|F 2 (P i ,x)| 2 dω(x) ≤ Cγ 2 ⎝j≤ Cγ 2 ∑ jω(P j )[ ∞∑k=0] ⎞2 −kτω(2 k P j ) χ R k (P j )(x) ⎠ dω(x)∫ [ ∑ ∞2 −kτω(P j )ω(2 k P j ) χ R k (P j )(x)k=0]dω(x)≤ Cγ 2 ∑ jω(P j )≤ Cγ 2 ω(Q 0 ),and the bound ong>forong> (II) will follow from Chebyshev’s inequality.Let us now fix a j ≠ i and consider the sum∑Q: Q⊂P jω(Q)[ ∞∑k=0]2 −kτω(2 k Q) χ R k (Q)(x)ong>forong> x ∈ C(P i ) \ D. We rewrite the sum as:∞∑l=0∑Q: Q⊂P jl(Q)=2 −l l(P j )ω(Q)[ ∞∑k=0]2 −kτω(2 k Q) χ R k (Q)(x) .Now let’s fix l. Let k ′ be such that x Pi ∈ R k ′(P j ). If Q ⊂ P j , l(Q) = 2 −l l(P j ), andx ∈ C(P i ) \ D, then x ∈ R k (Q) ong>forong> some k such that |k ′ + l − k| ≤C, where C only depends onthe dimension d. Conversely, if x ∈ C(P i ) \ D and belongs to R k (Q), then x ∈ R˜k(P j ) ong>forong> some ˜ksatisfying |˜k + l − k| ≤C ′ . The reason ong>forong> these ong>inequalitiesong> is that the distance between x Q andx is comparable to the distance between x Pi and x Pj , with “comparability constants” dependingonly on d and κ. For all such Q, the ω-measures of 2 k Q will be comparable to ω(2 k−l P j ), andthus comparable to ω(2 k′ P j ), because ω is doubling. Thereong>forong>e, ong>forong> each fixed l ≥ 0, and all16

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