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

TL;DR

This paper proves that the $s$-Riesz transform is unbounded in $L^ty$ for certain measures in  when 1<s<2, extending known results to all non-integer s in that range.

Contribution

It establishes the unboundedness of the $s$-Riesz transform for a broad class of measures in  when 1<s<2, filling a gap in the understanding of singular integral operators.

Findings

The $s$-Riesz transform is unbounded in $L^ty$ for measures with finite  Hausdorff measure support.

No totally lower irregular finite positive Borel measure with finite support measure can have bounded $s$-Riesz transform in this range.

The result extends to all non-integer $s$ in (0,2), confirming the unboundedness in these cases.

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 $\|R\mu\|\ci{L^\infty(m_2)}=\infty$.