Littlewood-Paley endpoint scope and the H1-BMO dual pair

The two-sided square-function equivalence of Littlewood-Paley square-function equivalence on Lp for 1<p<infinity is stated for $1<p<\mathrm{\infty }$. Its contract supplies no estimate at $p=1$ or $p=\mathrm{\infty }$, so substituting either endpoint into it or its consumers is not justified by that theorem.

The locally defined real Hardy space $H^1({\mathbb{R}}^n)$ is The real Hardy space $H^p$ defined by a radial maximal function. Under the Axiom of Choice, its continuous dual is isomorphic, with equivalent norms, to BMO modulo constants (BMO seminorm and the quotient by constants, Real H1-BMO duality), with the fixed Hardy kernel and auxiliary order required by that duality theorem. Thus the library supplies the pair $(H^1,\mathrm{B}\mathrm{M}\mathrm{O}\mathrm{/}\mathbb{C})$ as a proved duality of these defined spaces. That duality supplies no square-function endpoint estimate on its own.

Choice. The dual identification inherits the full Axiom of Choice from Real H1-BMO duality, declared through The Axiom of Choice. The strict-range statement is used only within its own hypotheses. The conclusions here use the defined spaces and local duality; no homogeneous or inhomogeneous endpoint square-function characterization is asserted.