Compactness and Essential Distances of Weighted Bilinear Hardy-Steklov Operators

We study weighted bilinear Hardy-Steklov operators on the half-line for $1<p_1,p_2<\mathrm{\infty }$ and $0<q<\mathrm{\infty }$. The operator-theoretic core is an exact output-localisation principle. For a bounded bilinear map into $L^q$, compact middle truncations identify compactness with vanishing of a directed output-tail invariant and, with constant one, identify that invariant with the essential distance and a radius-based Hausdorff measure of noncompactness. Under the additional hypothesis of local finite-rank approximability, the same invariant also equals the finite-rank distance and governs the rank-approximation conclusions. We verify these hypotheses for the Hardy-Steklov map and then translate the tail invariant through the five weighted boundedness regions. In Regions $I,{\mathrm{I}\mathrm{I}}_1,{\mathrm{I}\mathrm{I}}_2,{\mathrm{I}\mathrm{I}\mathrm{I}}_1$, boundedness must be supplemented by lower- and upper-endpoint vanishing, and the essential distance is comparable to the larger endpoint defect. In the strict subcritical Region ${\mathrm{I}\mathrm{I}\mathrm{I}}_2$, boundedness alone implies compactness. This yields a sharp structural transition between the equality boundary $P=Q$ and the summable regime $P>Q$. The quasi-Banach range $0<q<1$ is treated throughout with the powered metric $d_q(f,g)=\mathrm{\parallel }f-g{\mathrm{\parallel }}_q^q$.