The $s$-Riesz transform of an $s$-dimensional measure in ${\mathbb{R}}^2$ is unbounded for $1<s<2$

Abstract

In this paper, we prove that for $s\in (1,2)$ there exists no totally lower irregular finite positive Borel measure $\mu$ in ${\mathbb{R}}^2$ with\break $\mathcal H^s(\suppμ)<+\infty$ such that $\|Rμ\|\ci{L^\infty(m_2)}<+\infty$, where $R\mu =\mu \ast \frac{x}{|x{|}^{s+1}}$ and $m_2$ is the Lebesgue measure in ${\mathbb{R}}^2$. Combined with known results of Prat and Vihtilä, this shows that for any non-integer $s\in (0,2)$ and any finite positive Borel measure in ${\mathbb{R}}^2$ with $\mathcal H^s(\suppμ)<+\infty$, we have $\|Rμ\|\ci{L^\infty(m_2)}=\infty$.

arXiv comment23 pages; here is the final version