Research proof index

Chebyshev bound for the superlevel set {u > λ} using (u-θ)₊.

Closeout: pass from iterated L^{pₙ}(B_{rₙ}) bounds to an a.e. L^∞ bound on u⁻¹ over B_{1/2}. Since rₙ > 1/2 for all n and pₙ → ∞, the L^{pₙ} norms converge to the L^∞ norm. The uniform bound from the iteration + geometric majorant gives the pointwise bound.

Iteration of the inverse-power one-step bound by induction. At each step, supersolution_preMoser_inverse provides the Lᵖⁿ → Lᵖⁿ⁺¹ gain, and we accumulate the product of step constants.

Each inverse-power step constant is bounded by a simpler expression. The key simplification: pₙ/(1+pₙ) ≤ pₙ (since pₙ > 0), and gap_n = 2^{-(n+2)}, so 1/gap_n² = 4^{n+2} = 16 · 4^n.

Vitali covering lemma (5r-covering): from any family of balls, one can extract a disjoint subfamily such that the 5× enlargements cover the union, and every original ball meets a selected ball of at least half its radius.

For u ∈ W^{1,1}(a,b) with weak derivative g, u agrees a.e. with x ↦ C + ∫_a^x g(t) dt for some constant C. Proof: define F(x) = ∫_a^x g. By AC-IBP, F also has weak derivative g on (a,b). Then u - F has zero weak derivative. By du_bois_reymond, u - F = D a.e.