Littlewood-Paley functions and Sobolev spaces (Q2312548)

The authors follow the research of \textit{R. Alabern} et al. [Math. Ann. 354, No. 2, 589--626 (2012; Zbl 1267.46048)], where a new Littlewood-Paley characterization of the Sobolev space \(W^{\alpha,p}(\mathbb R^n)\), \(\alpha\in(0,4)\), \(p\in(1,\infty)\), was obtained. They show that this characterization is actually a consequence of a theorem from [\textit{Y. Ding} et al., Hokkaido Math. J. 29, No. 3, 537--552 (2000; Zbl 0973.42009)] about the \(L^p\)-boundedness of the square function \[ S_\Phi(f)(x)=\Big(\int_{0}^{\infty}|(\Phi_t*f)(x)|^2\frac{\text{d}t}{t}\Big)^{1/2}, \] where \(\Phi\in L^1(\mathbb R^n)\) is a suitable function and \(\Phi_t(x)=t^{-n}\Phi(x/t)\), \(t\in(0,\infty)\), \(x\in\mathbb R^n\). Moreover, their approach enables them to extend and improve the main results of [Alabern et al., loc.\,cit.] (in particular, they enlarge the range for \(\alpha\)) and the results of [\textit{S. Sato} et al., Commun. Contemp. Math. 20, No. 7, Article ID 1750077, 48 p. (2018; Zbl 1410.46024)].