Littlewood-Paley characterizations of second-order Sobolev spaces via averages on balls (Q2788793)

The main goal is to characterize the second-order Sobolev space \(W^{2,p}(\mathbb{R}^n)\) by means of the Lusin area function \(\mathcal{S}(f,g)\) NEWLINE\[NEWLINE\mathcal{S}(f,g)(x)=\left(\int_0^{\infty}\int_{B(x,t)}\left|\frac{B_tf(y)-f(y)}{t^2}-B_tg(y)\right|^2dy \frac{dt}{t^{n+1}}\right)^{1/2},NEWLINE\]NEWLINE and the Littlewood-Paley \(g^*_{\lambda}\)-function \(\mathcal{G}^*_{\lambda}(f,g)\), NEWLINE\[NEWLINE\mathcal{G}^*_{\lambda}(f,g)(x)=\left(\int_0^{\infty}\int_{\mathbb{R}^n}\left|\frac{B_tf(y)-f(y)}{t^2}-B_tg(y)\right|^2\left(\frac{t}{t+|x-y|}\right)^{\lambda n}dy\frac{dt}{t^{n+1}}\right)^{1/2},NEWLINE\]NEWLINE where \(\lambda\in (1,\infty)\) and \(B_tf(x)\) is the integral average of \(f\in L_{loc}^1(\mathbb{R}^n)\) on the open ball \(B(x,t)\).NEWLINENEWLINE\textit{R. Alabern} et al. [Math. Ann. 354, No. 2, 589--626 (2012; Zbl 1267.46048)] have presented a new characterization of the second-order Sobolev space \(W^{2,p}(\mathbb{R}^n)\) with \(1<p<\infty\). To be precise, for \(1<p<\infty\), the following are equivalent: (i) \(f\in W^{2,p}(\mathbb{R}^n)\); (ii) \(f\in L^p(\mathbb{R}^n)\) and there exists a function \(g\in L^p(\mathbb{R}^n)\) such that \(\mathcal{G}(f,g)\in L^p(\mathbb{R}^n)\), where \(\mathcal{G}(f,g)\) is NEWLINE\[NEWLINE\mathcal{G}(f,g)(x)=\left(\int_0^{\infty}\left|\frac{B_tf(x)-f(x)}{t^2}-B_tg(x)\right|^2\frac{dt}{t}\right)^{1/2}.NEWLINE\]NEWLINE In fact, the operator \(\mathcal{G}(f,g)\) can be regarded as the Littlewood-Paley \(g\)-function of \(\frac{B_tf-f}{t^2}-B_tg\). In the paper under review the authors prove that the following statements are equivalent: (i) \(f\in W^{2,p}(\mathbb{R}^n)\); (ii) there exists \(g\in L^p(\mathbb{R}^n)\) such that \(\mathcal{S}(f,g)\in L^p(\mathbb{R}^n)\); (iii) there exists \(g\in L^p(\mathbb{R}^n)\) such that \(\mathcal{G}^*_{\lambda}(f,g)\in L^p(\mathbb{R}^n)\), provided that \(p\in [2,\infty)\), \(n\in \mathbb{N}\), and \(\lambda\in (1,\infty)\) or \(p\in (1,2)\), \(n\in\{1,2,3\}\), and \(\lambda\in (2/p,\infty)\).