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The Muckenhoupt class \(A_ 1(\mathbb{R})\). (English) Zbl 0808.42010

Let \(\psi\) be a Schwartz function, \(\psi\in S(\mathbb{R})\), \(\int_ \mathbb{R}\psi(x)dx\neq 0\). Set \(\psi_ t(x)=t^{-1}\psi(x/t)\), \(t>0\), \(x\in\mathbb{R}\). For each distribution \(f\in{\mathcal S}'(\mathbb{R})\), define \(f^*(x)=\sup_{t>0}|(f*\psi_ t)(x)|\), \(x\in\mathbb{R}\).
It is shown that the Muckenhoupt structure constants for \(f\) and \(f^*\) on the real line are the same.

MSC:

42B25 Maximal functions, Littlewood-Paley theory
26D15 Inequalities for sums, series and integrals
42A50 Conjugate functions, conjugate series, singular integrals
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