Weak-type (1,1) estimates for strongly singular operators (Q2322516)

The authors use stationary phase methods and explicit estimates on the difference of some suitable integrals to obtain some estimates for some kind of convolution operators. \par The paper is inspired by \textit{C.-H. Cho} and \textit{C. W. Yang} [J. Math. Anal. Appl. 362, No. 2, 523--533 (2010; Zbl 1181.42013)] who studied such operators. \par Moreover, the proof uses the Lebesgue differentiation theorem and the extension of a standard Whitney decomposition, meshing dyadic intervals of length \(2{-k}\) in \(\mathbb{R}\).\par The first version of the theorem was proved by \textit{I. I. Hirschman jun.} [Duke Math. J. 26, 221--242 (1959; Zbl 0085.09201)] and by \textit{C. L. Fefferman} and \textit{E. M. Stein} [Acta Math. 129, 137--193 (1972; Zbl 0257.46078)]. Now, are assumed certain regularity and growth conditions that characterize this study. Preliminary Lemmas enriched the paper and can be useful independently on the main Theorem.