Input-Endpoint Compactness and Essential Norms for Weighted Bilinear Hardy Operators

Abstract

We study compactness of the product of two weighted Hardy primitives into $L^q(u)$, $0<q<\infty$, on the input boundary $1\le p_1,p_2\le\infty$, where at least one input exponent is $1$ or $\infty$. For $L^1\times L^p$, $1<p<\infty$, boundedness implies compactness when $q<p/(p+1)$. For $L^1\times L^1$, the corresponding range is $q<1/2$. At and above these boundaries compactness is characterized by additional active-tail decay, and critical power weights give bounded noncompact examples. The endpoint reductions use cumulative Stieltjes measures with their initial atoms and strict Hardy-Copson brackets. An output-mass allocation argument proves local finite-rank density, including quasi-Banach targets. Applying the localization principles developed for the interior-input regime identifies the essential distance, finite-rank distance and Hausdorff noncompactness radius with the coupled active tail, with the appropriate metric power when $q<1$. For a finite left endpoint, ordinary $L^\infty$ inputs admit an exact scalar transfer completed by linear nodal approximation.