The classical Chang–Wilson–Wolff theorem [3] in Euclidean spaces characterizes the uniformly locally exponential square integrability of a function when its Lusin area function is bounded. This result can be derived by exploiting martingale differences and intrinsically relies on the conservation law of the Laplacian operator. In this article, applying Hoeffding’s inequality for the sum of atomic functions in almost orthogonal settings (with martingale differences being a special case) established by J. Li, J. Pipher and the author in [12], we go beyond the conservation law and provide a unified methodology for Chang–Wilson–Wolff type theorem associated to square functions in spaces of homogeneous type.
