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 $B^{;p, q};_{;\alpha};(R^N)$ 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 $B^{;p, q};_{;\alpha};(R^N)$ is $\le N-\alpha p$. Similar results are obtained for Lizorkin-Triebel spaces $F^{;p, q};_{;\alpha};(R^N)$ and for the Hardy space $H^1(R^N)$. Some open problems will be listed related to the program of finding singular dimension of various function spaces, and of solutions of PDE-s. The above results will be published in Archiv der athematik, and they are a continuation of author's previous work on finding maximally singular functions in Bessel potential spaces, including Sobolev spaces.

Besov space; maximally singular function; fractal set

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