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The space weak \(H^ 1\). (English) Zbl 0626.42013

Let \(\psi\) be a function in \(C_ c^{\infty}({\mathbb{R}}^ n)\) with non- zero integral and let for \(t>0\), \(\psi_ t(x)=t^{-n}\psi (x/t).\) For a distribution in \({\mathbb{R}}^ n\), define the maximal operator \(f^*(x)=\sup_{t>0}| f*\psi_ t(x)|.\) The authors study in detail the space weak \(H^ 1=\{f/f^*\in weak L^ 1\}\). In particular they show how to obtain atomic decompositions of distributions in weak \(H^ 1\). They also provide several interesting applications to harmonic analysis.
Reviewer: M.Milman

MSC:

42B30 \(H^p\)-spaces
46E30 Spaces of measurable functions (\(L^p\)-spaces, Orlicz spaces, Köthe function spaces, Lorentz spaces, rearrangement invariant spaces, ideal spaces, etc.)
46M35 Abstract interpolation of topological vector spaces
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