2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/119468We establish simple pointwise characterizations of functions in the Hardy-Sobolev spaces within the range n/(n+1)<p <=1. In addition, classical Hardy inequalities are extended to the case p <= 1.17 pagesFunctional AnalysisClassical Analysis and ODEs46E35; 42B30 (Primary), 26D15; 42B25 (Secondary)text