Input-Endpoint Compactness and Essential Distances for Weighted Bilinear Hardy-Steklov Operators

We study weighted bilinear Hardy-Steklov operators with finite output exponent and at least one input exponent equal to one or infinity. For continuous increasing moving windows, we prove local finite-rank approximation and identify the distances to compact and finite-rank bilinear maps, together with the ambient-centred Hausdorff radius of noncompactness, with the maximum of two endpoint-tail norms. Boundedness implies compactness below the harmonic input exponent. Exact incidence identities cover the Banach-output endpoint cases, while inputs at infinity admit linear reductions. For finite inputs, a variational framework incorporates factorisation results of Carbery, H"anninen and Valdimarsson. In the common homogeneous window geometry, dual capacities characterize both active quasi-Banach endpoint strips with explicit comparison constants. Their localized versions give compactness criteria through vanishing capacity tails and comparable essential-distance estimates. The capacities retain their auxiliary functions. Examples distinguish accumulation from isolated testing and show the limitations of specified local norm summaries.