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 
such that \({\left\| {R\mu } \right\|_{{L^\infty }({m_2})}} < + \infty \), where Rµ = µ*x/|x|s+1 and m
2 is the Lebesgue measure in ℝ2. Combined with known results of Prat and Vihtilä, this shows that for any s ∈ (0, 1) ∪ (1, 2) and any finite positive Borel measure in ℝ2 with 
, we have \({\left\| {R\mu } \right\|_{{L^\infty }({m_2})}} = \infty \).
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Work of F. Nazarov and A. Volberg is supported by the National Science Foundation under the grant DMS-0758552.
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Eiderman, V., Nazarov, F. & Volberg, A. The s-Riesz transform of an s-dimensional measure in ℝ2 is unbounded for 1 < s < 2. JAMA 122, 1–23 (2014). https://doi.org/10.1007/s11854-014-0001-1
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DOI: https://doi.org/10.1007/s11854-014-0001-1