Essential Norms of Bilinear Hardy–Steklov Operators from L¹×Lᵖ to L∞

We study weighted complex-bilinear Hardy–Steklov maps from L¹×Lᵖ, 1 ≤ p ≤ ∞, and the factor-swapped spaces to literal L∞((0,∞),dx). The moving intervals have differentiable strictly increasing boundaries with the usual endpoint limits. The local dual-exponent admissibility of the positive input weights makes every L¹-side weight locally essentially bounded. An L¹ input permits strict-middle noncompactness even when both endpoint amplitude defects vanish, and the kernel rows need not be strongly measurable in norm. We develop the scalar and vector kernel methods needed to retain this obstruction. For double L¹, the exact essential norm is 𝒪₁₁/2, where 𝒪₁₁ is the positive rectangle oscillation over finite measurable output partitions. For 1 < p ≤ ∞, the exact value is 𝒲ₚ. This characteristic fixes the number of centres, tests arbitrarily long finite output tuples through product-essential suprema, and then takes the infimum over centre numbers. Every finite test is evaluated by finite strip programs using window membership, essential weight suprema, weighted masses and finite-dimensional cluster radii. At p = ∞, restriction to C₀, Radon representation and absolutely continuous extraction contract arbitrary bilinear centres to density centres without increasing their residual distances to the operator rows. Weak-star finite consistency supplies common centres for the complete hierarchy. Finite-piece bulk and tail assembly yields the compactness classification. Throughout the boundary sector, essential norm equals finite-output-rank distance. The characterization asserts no universal finite tuple cutoff, convergence rate or finite-time algorithm for arbitrary measurable data.