The Riesz Transform of Codimension Smaller Than One and the Wolff Energy

Fix d ≥ 2, and s ∈ (d-1, d). We characterize the non-negative locally finite non-atomic Borel measures μ in Rdfor which the associated s-Riesz transform is bounded in L2(μ) in terms of the Wolff energy. This extends the range of s in which the Mateu-Prat-Verdera characterization of measures with bounded s-Riesz transform is known. As an application, we give a metric characterization of the removable sets for locally Lipschitz continuous solutions of the fractional Laplacian operator (-Δ)α/2, α ∈ (1, 2), in terms of a well-known capacity from non-linear potential theory. This result contrasts sharply with removability results for Lipschitz harmonic functions.