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

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

or, more succinctly:|φ (Q) (x)| ≤ √ ω(Q)∞∑( 2−jαj=0ω(2 j Q))χ Rj (Q)(x).2) Smoothness. For any x and y in R d ,|φ (Q) (x) − φ (Q) (y)| ≤( |x − y|l(Q)) β√ ∑∞ ( 2−jαω(Q)j=0ω(2 j Q))(χ Rj (Q)(x)+χ Rj (Q)(y)).Note that, given the size condition, the smoothness condition is only meaningful when |x−y| ≤l(Q).3) Cancellation. For every finite linear combination ∑ Q γ Qφ (Q) ,∫R d | ∑ Qγ Q φ (Q) | 2 dω ≤ ∑ Q|γ Q | 2 .All of our results depend on the next theorem.Theorem 1.1. Let {φ (Q) } Q∈D be a standard familyof functions, and let ν ∈ A ∞ (ω). If0

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