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 L_A in divergence form associated with an (n+1)times (n+1)-matrix A with real, merely bounded, and possibly non-symmetric coefficients. If then, under suitable Dini-type assumptions on omega _A, we prove the following: if mu is a compactly supported Radon measure in mathbb {R}^{n+1}, n ge 2, and T_mu f(x)=int nabla _xGamma _A (x,y)f(y), textrm{d}mu (y) denotes the gradient of the single layer potential associated with L_A, then 1+‖Tμ‖L2(μ)→L2(μ)≈1+‖Rμ‖L2(μ)→L2(μ),\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\begin{aligned} 1+ \\Vert T_\\mu \\Vert _{L^2(\\mu )\\rightarrow L^2(\\mu )}\\approx 1+ \\Vert {\\mathcal {R}}_\\mu \\Vert _{L^2(\\mu )\\rightarrow L^2(\\mu )}, \\end{aligned}$$\\end{document}where {mathcal {R}}_mu indicates the n-dimensional Riesz transform. This allows us to provide a direct generalization of some deep geometric results, initially obtained for {mathcal {R}}_mu , which were recently extended to T_mu associated with L_A with Hölder continuous coefficients. In particular, we show the following: If mu is an n-Ahlfors-David-regular measure on mathbb {R}^{n+1} with compact support, then T_mu is bounded on L^2(mu ) if and only if mu is uniformly n-rectifiable.Let Esubset mathbb {R}^{n+1} be compact and {mathcal {H}}^n(E)<infty . If T_{{mathcal {H}}^n|_E} is bounded on L^2({mathcal {H}}^n|_E), then E is n-rectifiable.If mu is a non-zero measure on mathbb {R}^{n+1} such that limsup _{rrightarrow 0}tfrac{mu (B(x,r))}{(2r)^n} is positive and finite for mu -a.e. xin mathbb {R}^{n+1} and liminf _{rrightarrow 0}tfrac{mu (B(x,r))}{(2r)^n} vanishes for mu -a.e. xin mathbb {R}^{n+1}, then the operator T_mu is not bounded on L^2(mu ).Finally, we prove that if mu is a Radon measure on {mathbb {R}}^{n+1} with compact support which satisfies a proper set of local conditions at the level of a ball B=B(x,r)subset {mathbb {R}}^{n+1} such that mu (B)approx r^n and r is small enough, then a significant portion of the support of mu |_B can be covered by a uniformly n-rectifiable set. These assumptions include a flatness condition, the L^2(mu )-boundedness of T_mu on a large enough dilation of B, and the smallness of the mean oscillation of T_mu at the level of B.