Products of derivatives of interval functions with continuous functions (Q1308865)

It is known that the family of all derivatives (from \(\mathbb{R}\) into \(\mathbb{R})\) whose product with every continuous function is a derivative is the same as the family of all locally summable derivatives such that \(\limsup_{h\to 0^ +}{1\over 2h}\int^{x+h}_{x-h}| f|<\infty\) for each \(x\in\mathbb{R}\). In this paper, the author proves three versions of this theorem in multidimensional case, by considering, respectively, \(s\)-derivatives, \(o\)-derivatives, and \(w\)-derivatives functions.