Singular sets of Sobolev functions
Let \(u:\mathbb R^N\rightarrow \overline{\mathbb R}\) be a measurable function. If \(x_0\in \mathbb R^N\) and \(\alpha>0\), say that \(u\) has a singularity at least of order \(\alpha\) at \(x_0\) if there exist \(R>0\) and \(C>0\) such that \(u(x)\geq C\left|x-x_0\right|^{-\alpha}\) for all \(x\in B(x_0,R)\). If \(A\) is a nonempty subset of \(\mathbb R^N\), say that \(u\) has a singularity of order at least \(\alpha\) on \(A\) if there exist \(R>0\) and \(C>0\) such that \(u(x)\geq Cd(x,A)^{-\alpha}\) for all \(x\in A_R\), the \(R\)-neighborhood of \(A\). The singular set of \(u\) is the set of all points in \(\mathbb R^N\) for which there exists \(\alpha>0\) such that \(u\) has a singularity at \(x_0\) at least of order \(\alpha\). It will be denoted by Sing \(u\). If \(X\) is a Banach space of measurable functions \(u\), the lower singular dimension of \(X\) is \[ s-\underline{\text{dim}}X=\sup\left\{\text{dim}_H(\text{Sing }u);\;u\in X\right\} \] where \(\text{dim}_H\) denotes the Hausdorff dimension. The upper singular dimension of \(X\) is \[ s-\overline{\text{dim}}X=\sup\left\{\text{dim}_H(e-\text{Sing }u);\;u\in X\right\} \] where \[ e-\text{Sing }u=\left\{x_0\in \mathbb R^N;\;\overline{\lim}_{r\rightarrow 0} \frac 1{r^N}\int_{B(x_0,r)}u(x)dx=+\infty\right\}. \] For \(A\subset \mathbb R^N\), denote also by \(\overline{\text{dim}}_BA\) the upper box dimension of \(A\) [see \textit{K. Falconer}, ``Fractal geometry: Mathematical foundations and applications'', New-York (1990; Zbl 0689.28003)]. The main result of the present paper is as follows: if \(1<p<+\infty\), \(k\in \mathbb N\), \(kp<N\) and \(A\) is a compact subset of \(\mathbb R^N\) such that \(\overline{\text{dim}}_BA<N-kp\), then there exists a function \(u\in W^{k,p}(\mathbb R^N)\) which is singular exactly on \(A\). Moreover, the lower and upper singular dimensions of \(W^{k,p}(\mathbb R^N)\) coincide and are equal to \(N-kp\). If \(kp=N\), the upper singular dimension of \(W^{k,p}(\mathbb R^N)\) is \(0\). If \(1\leq p<+\infty\), then the lower singular dimension of \(L^p(\mathbb R^N)\) is \(N\). If \(X=\cap_{1\leq p<+\infty} L^p(\mathbb R^N)\), then the lower singular dimension of \(X\) is \(0\), while its upper singular dimension is \(N\). This result is related, in particular, to a previous theorem by \textit{B. Fuglede} [Acta Math 98, 171-219 (1957; Zbl 0079.27703)].
scientific article; zbMATH DE number 1754814
scientific article