Basis of wavelets and atomic decompositions of $H^ 1(\textbf {R}^ n)$ and $H^ 1(\textbf {R}^ n\times \textbf {R}^ n)$

Abstract:

It is shown that, as in the case of ${H^1}({\mathbb {R}^n})$ and ${\text {BMO(}}{\mathbb {R}^n})$, wavelets provide unconditional basis of ${H^1}({\mathbb {R}^n} \times \mathbb {R}n)$ and ${\text {BMO(}}{\mathbb {R}^n} \times \mathbb {R}n)$. Moreover, we show how wavelets can be used to obtain the usual atomic decompositions in ${H^1}{\text {(}}{\mathbb {R}^n})$ and ${H^1}{\text {(}}{\mathbb {R}^n} \times \mathbb {R}n)$.