OAI.RieszRectifiability.quantitative_higher_codimension_riesz_rectifiability

The theorem states that, for all natural numbers d and n with d ≥ 4, n ≥ 2 and n+2 ≤ d, every constant C_AD ≥ 1 and every nonnegative real C_R, there exist a real θ > 0 and a nonnegative real M such that the following holds for every regular Borel measure μ on Euclidean space ℝ^d satisfying two hypotheses. First, μ is n-dimensional Ahlfors–David regular with constant C_AD: for every x in the support of μ and every admissible radius r (meaning 0 < r ≤ the diameter of the support of μ), r^n/C_AD ≤ μ(B(x,r)) ≤ C_AD·r^n. Second, the truncated Riesz transforms with kernel K(x,y) = (x−y)/|x−y|^(n+1) are bounded on L²(μ) with norm at most C_R: for every ε > 0 and every f in L²(μ), the truncation T_ε f(x) = ∫_{|x−y|>ε} f(y)K(x,y) dμ(y) is in L²(μ) and ‖T_ε f‖₂ ≤ C_R‖f‖₂. For such μ, the conclusion is that for every x in the support of μ and every admissible radius r there is an M-Lipschitz map g from the open ball of radius r about 0 in ℝ^n into ℝ^d such that μ(B(x,r) ∩ range g) ≥ θ r^n. The constants θ and M depend only on d, n, C_AD and C_R, not on μ.