engleski

Maximally singular functions in Besov spaces

Assuming that $0<\alpha p<N$, $p, q\in(1, \infty)$, we construct a class of functions in the Besov space $B_\alpha^{; ; ; p, q}; ; ; (\mathbb{; ; ; R}; ; ; ^N)$ such that the Hausdorff dimension of their singular set is equal to $N-\alpha p$. We show that these functions are maximally singular, that is, the Hausdorff dimension of singular set of any other Besov function in $B_{; ; ; \alpha}; ; ; ^{; ; ; p, q}; ; ; (\mathbb{; ; ; R}; ; ; ^N)$ is $\le N-\alpha p$. Similar results are obtained for Lizorkin-Triebel spaces $F_{; ; ; \alpha}; ; ; ^{; ; ; p, q}; ; ; (\mathbb{; ; ; R}; ; ; ^N)$ and for the Hardy space $H^1(\mathbb{; ; ; R}; ; ; ^N)$. Some open problems are listed.

Besov space; Lizorkin-Triebel space; Hardy space; singular set; fractal set

nije evidentirano