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

Abstract

We consider a uniformly elliptic operator LA 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 ωA, we prove the following: if μ is a compactly supported Radon measure in Rn+1, n≥ 2 , and Tμf(x)=∫∇xΓA(x,y)f(y)dμ(y) denotes the gradient of the single layer potential associated with LA, then 1+‖Tμ‖L2(μ)→L2(μ)≈1+‖Rμ‖L2(μ)→L2(μ),where Rμ indicates the n-dimensional Riesz transform. This allows us to provide a direct generalization of some deep geometric results, initially obtained for Rμ, which were recently extended to Tμ associated with LA with Hölder continuous coefficients. In particular, we show the following: (1)If μ is an n-Ahlfors-David-regular measure on Rn+1 with compact support, then Tμ is bounded on L2(μ) if and only if μ is uniformly n-rectifiable.(2)Let E⊂ Rn+1 be compact and Hn(E) < ∞. If THn|E is bounded on L2(Hn| E) , then E is n-rectifiable.(3)If μ is a non-zero measure on Rn+1 such that lim supr→0μ(B(x,r))(2r)n is positive and finite for μ-a.e. x∈ Rn+1 and lim infr→0μ(B(x,r))(2r)n vanishes for μ-a.e. x∈ Rn+1, then the operator Tμ is not bounded on L2(μ).(4)Finally, we prove that if μ is a Radon measure on Rn+1 with compact support which satisfies a proper set of local conditions at the level of a ball B= B(x, r) ⊂ Rn+1 such that μ(B) ≈ rn and r is small enough, then a significant portion of the support of μ| B can be covered by a uniformly n-rectifiable set. These assumptions include a flatness condition, the L2(μ) -boundedness of Tμ on a large enough dilation of B, and the smallness of the mean oscillation of Tμ at the level of B. © 2023, The Author(s).