L2 -Boundedness of Gradients of Single Layer Potentials for Elliptic Operators with Coefficients of Dini Mean Oscillation-Type

We consider a uniformly elliptic operator L in divergence form associated with an (n+ 1) × (n+ 1) -matrix A with real, merely bounded, and possibly non-symmetric coefficients. If [Equation not available: see fulltext.]then, under suitable Dini-type assumptions on ω, we prove the following: if μ is a compactly supported Radon measure in R , n≥ 2, and Tμf(x)=∫∇xΓA(x,y)f(y)dμ(y) denotes the gradient of the single layer potential associated with L, then 1+‖Tμ‖L2(μ)→L2(μ)≈1+‖Rμ‖L2(μ)→L2(μ),where R indicates the n-dimensional Riesz transform. This allows us to provide a direct generalization of some deep geometric results, initially obtained for R, which were recently extended to T associated with L with Hölder continuous coefficients. In particular, we show the following: (1)If μ is an n-Ahlfors-David-regular measure on R with compact support, then T is bounded on L(μ) if and only if μ is uniformly n-rectifiable.(2)Let E⊂ R be compact and H(E) < ∞. If THn|E is bounded on L(H| ), then E is n-rectifiable.(3)If μ is a non-zero measure on R such that lim supr→0μ(B(x,r))(2r)n is positive and finite for μ-a.e. x∈ R and lim infr→0μ(B(x,r))(2r)n vanishes for μ-a.e. x∈ R , then the operator T is not bounded on L(μ).(4)Finally, we prove that if μ is a Radon measure on R with compact support which satisfies a proper set of local conditions at the level of a ball B= B(x, r) ⊂ R such that μ(B) ≈ r and r is small enough, then a significant portion of the support of μ| can be covered by a uniformly n-rectifiable set. These assumptions include a flatness condition, the L(μ) -boundedness of T on a large enough dilation of B, and the smallness of the mean oscillation of T at the level of B.