2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/103656We describe the spaces $H^1(R)$ and BMO$(R)$ in terms of their closely related, simpler dyadic and two-sided counterparts. As a result of these characterizations we establish when a bounded linear operator defined on dyadic or two-sided $H^1(R)$ into a Banach space has a continuous extension to $H^1(R)$ and when a bounded linear operator that maps a Banach space into dyadic or two-sided BMO$(R)$ actually maps continuously into BMO$(R)$.33 pagesFunctional AnalysisClassical Analysis and ODEs42B20; 42B99text