By invoking some new ideas, we characterize Sobolev spaces Wα,p(ℝn) with the smoothness order α ∊ (0, 2] and p ∊ (max{1, 2n/(2α + n)},∞), via the Lusin area function and the Littlewood–Paley g*λ-function in terms of centred ball averages. We also show that the assumption p ∊ (max{1, 2n/(2α + n)},∞) is nearly sharp in the sense that these characterizations are no longer true when p ∊ (1, max{1, 2n/(2α + n)}). These characterizations provide a possible new way to introduce Sobolev spaces with smoothness order in (1, 2] on metric measure spaces.

In this paper, the authors characterize second-order Sobolev spaces ${{W}^{2,p}}({{\mathbb{R}}^{n}})$ , with $p\,\in \,[2,\,\infty )$ and $n\,\in \,\mathbb{N}\,\text{or}\,p\,\in \,(1,\,2)\,\text{and}\,n\,\in \,\left\{ 1,\,2,\,3 \right\}$ , via the Lusin area function and the Littlewood–Paley $g_{\text{ }\!\!\lambda\!\!\text{ }}^{*}$ -function in terms of ball means.