Littlewood-Paley operators and Sobolev spaces
Let \(B(x,t)\) denote a ball in \(\mathbb R^n\) centered at \(x\) with radius \(t\) and let \(|B(x,t)|\) denote its Lebesgue measure. Let \(\alpha\in(0,n)\). Define an operator \[ T_\alpha(f)(x)=\Big(\int_0^\infty \Big|I_\alpha(f)(x)-\frac1{|B(x,t)|} \int_{B(x,t)}I_\alpha(f)(y)\,dy\Big|^2\,\frac{dt}{t^{1+2\alpha}}\Big)^{1/2}, \] where \(I_\alpha\) is the Riesz potential operator. The aim of this note is to generalize a result of \textit{R. Alabern} et al. [Math. Ann. 354, No. 2, 589--626 (2012; Zbl 1267.46048)], where the equivalence \[ C^{-1}\|f\|_p\leq\|T_\alpha(f)\|_p\leq C\|f\|_p \tag{\(*\)} \] (\(\|\cdot\|_p\) denotes the Lebesgue \(L^p\) norm and \(C\geq1\) is a constant independent of \(f\)) is derived for each \(\alpha\in(0,2)\) and \(p\in(1,\infty)\) and used to give a new characterization of the Sobolev space \(W^{\alpha,p}(\mathbb R^n)\). (Note that for a noninteger \(\alpha\) this space is a space of Triebel-Lizorkin type.) The author proves a weighted analogue of \((*)\); moreover, he derives a similar result for the operator \(S_\alpha\), \(0<\alpha<2\) (\(1/2\leq\alpha<2\) for the weighted case), \[ S_\alpha(f)(x)=\Big(\int_0^\infty \Big|I_\alpha(f)(x)- \int_{\mathcal S^{n-1}}I_\alpha(f)(x-ty)\,d\sigma(y)\Big|^2\,\frac{dt}{t^{1+2\alpha}}\Big)^{1/2}, \] (\(\mathcal S^{n-1}\) is the unit sphere with an \((n-1)\)-dimensional Hausdorff measure \(\sigma\) normalized by \(\int_{\mathcal S^{n-1}}d\sigma=1\)). He also shows boundedness of the operator \(S_{1/2}\) from the weighted Hardy space \(H^1_w\) (\(w\) being a Muckenhoupt \(A_1\)-weight) into the weighted weak-\(L^1_w\) space.
scientific article; zbMATH DE number 6510960
scientific article