Pointwise Domination and Weak \(L^1\) Boundedness of Littlewood-Paley Operators via Sparse Operators

This contribution is based on joint work with Hormozi et al. (J. Fourier Anal. Appl. 29:49, 2023). We simplify and shorten the proofs of the main results of Shi et al. (J. Math. Pures Appl. 101:394–413, 2014) significantly. In particular, the new proof for Shi et al. (J. Math. Pures Appl. 101:394–413, 2014; Theorem 1.1) is quite short and, unlike the original proof, does not rely on the properties of the “Marcinkiewicz function”. This allows us to get a precise linear dependence on Dini constants with a subsequent application to Littlewood–Paley operators by the well-known techniques.