On almost-everywhere differentiability of functions in Besov spaces
Given a linear topological space \(T\) of functions defined on the ball \(B=\{x\in {\mathbb R}^n:\;|x|<1\}\) and a function \(f\) defined on an open set \(G\subset {\mathbb R}^n\), a polynomial \(P_{x_0}(h)\) in \(h\) (\(h\in {\mathbb R}^n\)) is called a differential of \(f\) of order \(r\) at \(x_0\) whenever, for sufficiently small \(\varepsilon>0\), the function \[ \varphi_{\varepsilon}\: h \to \varepsilon^{-r}[f(x_0+\varepsilon h)- P_{x_0}(\varepsilon h)] \] belongs to \(T\) and tends to zero as \(\varepsilon \to 0\) in \(T\). A function \(f\:G \to \overline{{\mathbb R}}\) having the differential of order \(r\) at a point \(x_0\) is called an \(r\)-multiply differentiable function at \(x_0\). The main result of the article is the following theorem: Theorem. Assume that \(G\) is an open set in \({\mathbb R}^n\), \(1\leq q\leq p <\infty\), \(\alpha>0\) is not integer, and the space \(T\) coincides with the Besov space \(B_{p,q}^{\alpha}(B)\). Then every function \(f\in B_{p,q}^{\alpha}(G)\) is \(l\)-multiply differentiable \((l=[\alpha])\) almost everywhere in \(G\). In this case, the differential of \(f\) of order \(l\) at \(x\) is the polynomial \(P_{x}(h)= \sum_{|\alpha|\leq l}D^{\alpha}f(x)\frac{h^{\alpha}}{\alpha!}\) and \[ \|f(x+\varepsilon h)-P_{x}(\varepsilon h)\|_{B_{p,q}^{\alpha}(G)}\leq \varepsilon^l c(x,\varepsilon), \] where the function \(c(x,\varepsilon)\) tends to \(0\) as \(\varepsilon \to 0\) almost everywhere in \(G\).
scientific article; zbMATH DE number 1396676