Bojarski, B.; Sbordone, C.; Wik, I. The Muckenhoupt class \(A_ 1(\mathbb{R})\). (English) Zbl 0808.42010 Stud. Math. 101, No. 2, 155-163 (1992). 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. Cited in 6 ReviewsCited in 32 Documents MSC: 42B25 Maximal functions, Littlewood-Paley theory 26D15 Inequalities for sums, series and integrals 42A50 Conjugate functions, conjugate series, singular integrals Keywords:Muckenhoupt class; distribution; Schwartz function; structure constants PDFBibTeX XMLCite \textit{B. Bojarski} et al., Stud. Math. 101, No. 2, 155--163 (1992; Zbl 0808.42010) Full Text: DOI EuDML