A remark on Calderón-Zygmund classes and Sobolev spaces (Q2781334)

In the paper under review, the author investigates how the Sobolev space \(W^{k,p}(\mathbb{R}^n)\) can be characterised in terms of the local behaviour of its members. He uses the classes \(T^{k,p}(x)\), introduced by \textit{A.~P.~Calderón} and \textit{A.~Zygmund} [Proc. Natl. Acad. Sci. USA 46, 1385-1389 (1960; Zbl 0196.40501)] in connection with their study of the regularity of elliptic partial differential equations. For \(x\in \mathbb{R}^n\), \(p\geq 1\), and \(k\geq -n/p\) (not necessarily an integer), the class \(T^{k,p}(x)\) consists of those functions \(f\in L_p(\mathbb{R}^n)\) for which there exists a polynomial \(P_x\) with degree not exceeding \(k-1\) such that NEWLINE\[NEWLINE \sup\limits _{r>0} r^{-k}\left ( \frac{1}{| B(x,r)|}\int _{B(x,r)} | f(y)-P_x(y)| ^p\, dy \right )^{1/p}<\infty. NEWLINE\]NEWLINE Calderón and Zygmund showed that for \(k\geq 1\) and \(1<p<\infty\) a function \(f\) from the Sobolev space \(W^{k,p}(\mathbb{R}^n)\) belongs to \(T^{k,p}(x)\) for almost all \(x\in \mathbb{R}^n\). It is clear that in this general setting a true converse cannot hold (indicator function of the unit ball); the main result in the paper treats this question.