The s-Riesz transform of an s-dimensional measure in ℝ2 is unbounded for 1 < s < 2

Abstract

In this paper, we prove that for s ∈ (1, 2) there exists no totally lower irregular finite positive Borel measure µ in ℝ2 with Open image in new window such that \({\left\| {R\mu } \right\|_{{L^\infty }({m_2})}} < + \infty \), where Rµ = µ*x/|x|s+1 and m2 is the Lebesgue measure in ℝ2. Combined with known results of Prat and Vihtila, this shows that for any s ∈ (0, 1) ∪ (1, 2) and any finite positive Borel measure in ℝ2 with Open image in new window, we have \({\left\| {R\mu } \right\|_{{L^\infty }({m_2})}} = \infty \).