Approximate identities and $H^{1}(\textbf {R})$

Abstract:

Let $\varphi (x) \in {L^1}({\mathbf {R}}) \cap {L^\infty }({\mathbf {R}})$ be a real-valued function with $\int \limits _{\mathbf {R}} {\varphi dx \ne 0}$. For $y > 0$, let ${\varphi _y}(x) = {y^{ - 1}}\varphi (x/y)$. For $f(x) \in {L^1}({\mathbf {R}})$ define \[ f_\varphi ^*(x) = \sup \limits _{y > 0,t \in {\mathbf {R}}:|x - t| < y} |f * {\varphi _y}(t)|.\] We investigate the space $H_\varphi ^1 = \{ f \in {L^1}({\mathbf {R}}):f_\varphi ^* \in {L^1}({\mathbf {R}})\}$.