Research proof index

A set-family John-Nirenberg tail decay theorem from a base-level one-step decay hypothesis. This is the reusable set-family analogue of john_nirenberg_from_base. It iterates one-step decay starting from the base level A for an arbitrary antitone measurable family E_lam, and is useful when the Calderon-Zygmund / stopping-time work has already been pa...

De Giorgi L∞ bound for subsolutions on the unit ball. The positive part is bounded almost everywhere on the half-ball by an explicit coefficient-dependent constant times the L² size of u₊ on B₁. The result is stated as an a.e. bound rather than a pointwise supremum bound, since the representative/continuity upgrade comes later in the theory.

Height-threshold version of the De Giorgi endpoint. Once the PDE-facing one-step estimate is supplied, a lower bound for lamStar of the form C(d,K) * sqrt(∫_{B₁} (u₊)^2) yields the a.e. half-ball L∞ bound.

Unit-ball smallness packaging for the De Giorgi endpoint. This is the direct successor to the raw hsmall-based theorem: instead of assuming a bound on the abstract initial De Giorgi energy A₀, it is enough to assume the same recurrence threshold for the concrete unit-ball quantity ∫_{B₁} (u₊)^2. The radius/level bookkeeping then supplies the A₀ bo...

Small-data De Giorgi endpoint: if the canonical De Giorgi energy sequence tends to zero, then the final truncation vanishes almost everywhere on the half-ball. This is the formally correct endpoint available from the current Chapter 05 infrastructure before any representative/continuity upgrade.

Existence of a threshold realizing the De Giorgi L∞ estimate.