The Measures with 𝐿²-Bounded Riesz Transform and the Painlevé Problem

In this work we provide a geometric characterization of the measures μ μ in R n + 1 Rⁿ⁺¹ with polynomial upper growth of degree n n such that the n n -dimensional Riesz transform R μ ( x ) = ∫ x − y | x − y | n + 1 d μ ( y ) Rμ (x) = ∫ {x-y}{|x-y|ⁿ⁺¹}\,dμ (y) belongs to L 2 ( μ ) L^2(μ ) . More precisely, it is shown that ‖ R μ ‖ L 2 ( μ ) 2 + ‖ μ ‖ ≈