Essential Norms of Hardy Operators into $L^\infty$ on Irregular Supports, with Bilinear Applications

We study the Hardy operator $T_hf(x)=\int_a^x h(r)f(r) dr$ from scalar $L^1(a,c)$ into $L^\infty(S)$, where $S$ is an arbitrary measurable output set, $c=\operatorname{ess\,sup} S$, and $h$ is nonnegative and essentially bounded. We identify the essential norm, the distance to finite-rank operators, and the external Hausdorff measure of noncompactness of the unit-ball image. All three quantities equal one half of a support invariant whose closed form is the maximum of the input amplitude on the essential closure of $S$ and the persistent amplitudes of its internal gaps. Finitely many exceptional gaps do not contribute to this defect. For weighted bilinear Hardy products, we construct a scalar reduction modulo an approximable compact remainder when exactly one input is $L^1$, and an exact scalar quotient when both inputs are $L^1$. These reductions give the corresponding essential defects and a target-endpoint dichotomy: bounded maps without an $L^1$ input are compact, whereas nonzero bounded maps with an $L^1$ input are noncompact.