Research proof index

Second stage of weak Harnack: forward low-power iteration for positive supersolutions. For u > 0 a supersolution on B₁ and 0 < p < q < 1: ‖u‖_{L^{qd/(d-2)}(B_{1/2})}^p ≤ C_fwd(d) (Λp²/(1-q)² + 1)^{dχ/2} ‖u‖_{Lᵖ(B₁)}^p The theorem is stated for arbitrary 0 < p < q < 1, with the nonlinear factor raised to dχ/2. Proof: choose m large enou...

First stage of weak Harnack: inverse-power iteration for positive supersolutions. For u > 0 a supersolution on B₁ and every p₀ > 0: (Λ p₀² + 1)^{-d/2} · (∫_{B₁} |u⁻¹|^{p₀})⁻¹ ≤ C · (inf_{B_{1/2}} u)^{p₀} Proof: the Moser iteration (Steps 1-2 above) gives an a.e. L^∞ bound on u⁻¹ over B_{1/2}. Since u > 0 pointwise, this converts to a...