The $s$-Riesz transform of an $s$-dimensional measure in $\R^2$ 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 $\R^2$ with\break $\mathcal H^s(\supp\mu)<+\infty$ such that $\|R\mu\|\ci{L^\infty(m_2)}<+\infty$, where $R\mu=\mu\ast\frac{x}{|x|^{s+1}}$ and $m_2$ is the Lebesgue measure in $\R^2$. Combined with known results of Prat and Vihtil\"a, this shows that for any non-integer $s\in(0,2)$ and any finite positive Borel measure in $\R^2$ with $\mathcal H^s(\supp\mu)<+\infty$, we have $\|R\mu\|\ci{L^\infty(m_2)}=\infty$.