Estimates from below for the functions with bounded specific oscillation (Q1920691)

Let \(U\) be a domain in \(\mathbb{R}^n\). A function \(f:U\to \mathbb{R}^d\) is said to be of \textit{bounded specific oscillation} if there exists a positive \(\varepsilon< 1\) such that for every cube \(Q\subset U\) one can indicate a vector \(C_Q\in\mathbb{R}^d\) for which \[ \int_Q| f(x)- C_Q| dx\leq\varepsilon| C_Q| | Q|, \] \(| Q|\) denotes the Lebesgue measure of \(Q\). The term ``functions with bounded specific oscillation'' (BSO-functions) was suggested by \textit{Yu. G. Reshetnyak} for the function class introduced in our joint article [1] [Sibir. Mat. Zh. 17, 540-546 (1976; Zbl 0341.26006)]. The above is a simplified version of a more general definition of [1]. In [1] the measure was estimated of the set on which a BSO-function takes ``large'' values. In the present article, we estimate the measure of the set on which a BSO-function \(| f|\) takes values close to zero. Our estimate implies that the function \(1/| f|\) is locally integrable to the power \(p<\text{const}/\varepsilon\). As a result we prove that every BSO-function belongs to the Muckenhoupt class \(A_p\) for \(p>1+\text{const}\cdot\varepsilon\).