Maximally singular functions in Besov spaces (Q2501156)

For a measurable function \(u: \mathbb R^n \mapsto \mathbb R\), a point \(a \in \mathbb R^n\) is called singular if \(u(x) \geq C | x-a| ^{-\gamma}\) for some \(C>0\), \(\gamma >0\), near \(a\). Let \[ s- \dim X = \sup \{ \dim_H (\text{Sing\,} u) : \;u \in X \}, \] where \(\text{Sing\,}u\) collects the singular points of \(u\), and \(X\) is a set of functions. Let \[ X = B^s_{pq} (\mathbb R^n) \quad \text{or} \quad X= F^s_{pq} (\mathbb R^n), \quad 1<p,q < \infty, \quad 0< s \leq n/p. \] Then \[ s-\dim (B^s_{pq} (\mathbb R^n)) = s-\dim (F^s_{pq} (\mathbb R^n)) = n-sp \] is the main result of the paper.