Research proof index

Transport subsolutions on B(x₀, R) to the unit ball via affine pullback.

Transport supersolutions on B(x₀, R) to the unit ball via affine pullback.

Simple Iteration Lemma (AKM, Appendix C, Lemma C.6, specialized to ξ = 2). Suppose ρ : ℝ → ℝ satisfies: 1. ρ ≥ 0 on [1/2, 1), 2. sup_{t ∈ [1/2,1)} (1-t)² ρ(t) < ∞ (finiteness of weighted supremum), 3. For all 1/2 ≤ s < t < 1: ρ(s) ≤ (1/2) ρ(t) + A_iter · (t - s)⁻² Then ρ(1/2) ≤ C_iter · A_iter. Proof sketch (following AKM): - Let `M...

Smooth-input interface smoothing for the explicit extension operator. For smooth compactly supported input ψ, the piecewise extension unitBallExtension ψ can itself be approximated globally in full W^{1,p} by smooth compactly supported functions. The gradient side is expressed against some global field Gψ attached to the exact extension. The surro...

The main whole-space Sobolev inequality for functions approximated by smooth compactly supported functions.

Whole-space Sobolev inequality specialized to the MemW₀^{1,p} data already stored in the MemW₀^{1,p} witness predicate.