Essential Norms of Hardy Operators into L^∞ on Irregular Supports, with Bilinear Applications

We study the Hardy operator Tₕf(x)=∫ₐˣ h(r)f(r) dr from scalar L¹(a,c) into L^∞(S), where S is an arbitrary measurable output set, c=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 L¹, and an exact scalar quotient when both inputs are L¹. These reductions give the corresponding essential defects and a target-endpoint dichotomy: bounded maps without an L¹ input are compact, whereas nonzero bounded maps with an L¹ input are noncompact.