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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

Approximate identities and $ H\sp{1}({\bf R})$


Authors: Akihito Uchiyama and J. Michael Wilson
Journal: Proc. Amer. Math. Soc. 88 (1983), 53-58
MSC: Primary 42B30; Secondary 30D55, 46J15
MathSciNet review: 691278
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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

$\displaystyle f_\varphi ^*(x) = \mathop {\sup }\limits_{y > 0,t \in {\mathbf{R}}:\vert x - t\vert < y} \vert f * {\varphi _y}(t)\vert.$

We investigate the space $ H_\varphi ^1 = \{ f \in {L^1}({\mathbf{R}}):f_\varphi ^* \in {L^1}({\mathbf{R}})\} $.

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Additional Information

DOI: http://dx.doi.org/10.1090/S0002-9939-1983-0691278-8
PII: S 0002-9939(1983)0691278-8
Keywords: $ {H^1}$, BMO, maximal functions
Article copyright: © Copyright 1983 American Mathematical Society