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Global orthogonality implies local almost-orthogonality.

J. Michael Wilson — 2000

Revista Matemática Iberoamericana

We introduce a new stopping-time argument, adapted to handle linear sums of noncompactly-supported functions that satisfy fairly weak decay, smoothness, and cancellation conditions. We use the argument to obtain a new Littlewood-Paley-type result for such sums.

Weighted inequalities for gradients on non-smooth domains

We prove sufficiency of conditions on pairs of measures μ and ν, defined respectively on the interior and the boundary of a bounded Lipschitz domain Ω in d-dimensional Euclidean space, which ensure that, if u is the solution of the Dirichlet problem. Δu = 0 in Ω, u | Ω = f , with f belonging to a reasonable test class, then ( Ω | u | q d μ ) 1 / q ( Ω | f | p d ν ) 1 / p , where 1 < p ≤ q < ∞ and q ≥ 2. Our sufficiency conditions resemble those found by Wheeden and Wilson for the Dirichlet problem on d + 1 . As in that case we attack the problem by...

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