Incidence Geometry and L∞ Essential Norms of Bilinear Hardy–Steklov Operators

We study weighted bilinear Hardy–Steklov operators whose moving intervals have differentiable strictly increasing endpoints, with both endpoints tending to zero and infinity at the corresponding ends of the half-line. For complex inputs in the exact sector 1 < p₁, p₂ ≤ ∞ and literal L∞((0,∞), dx) output, we determine the distance from arbitrary compact complex-bilinear maps. The normalized row amplitude has endpoint defects Aⱼ, but these defects alone do not determine the distance. For finite input exponents, cumulative weight coordinates assign dyadic addresses to the intervals and define a graph through positive-measure joint occurrence. Writing σ = 1/p₁ + 1/p₂, we prove that the endpoint radius is Aⱼ when σ < 1 and max{Aⱼ/2, Bⱼ} when σ ≥ 1. Here Bⱼ is the largest amplitude level above which unbounded incidence persists in every endpoint tail. When σ ≥ 1, the normalized geometry has exactly three endpoint radii, zero for eventual inactivity, one half for persistent activity with bounded incidence, and one for persistent unbounded incidence. Finite amplitude filtration then gives the weighted formula with the endpoint limits taken in a specified order. If either input exponent is infinity, the essential norm is max{A₀, A∞}. Throughout the stated sector this norm equals the finite-output-rank distance, and compactness is equivalent to bounded amplitude with both endpoint defects zero. The L¹ input faces require different kernel-width methods and lie outside this paper.