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

Abstract

AbstractWe consider a uniformly elliptic operator $$L_A$$ L A in divergence form associated with an $$(n+1)\times (n+1)$$ ( n + 1 ) × ( n + 1 ) -matrix A with real, merely bounded, and possibly non-symmetric coefficients. If "Equation missing"then, under suitable Dini-type assumptions on $$\omega _A$$ ω A , we prove the following: if $$\mu $$ μ is a compactly supported Radon measure in $$\mathbb {R}^{n+1}$$ R n + 1 , $$n \ge 2$$ n ≥ 2 , and $$ T_\mu f(x)=\int \nabla _x\Gamma _A (x,y)f(y)\, \textrm{d}\mu (y) $$ T μ f ( x ) = ∫ ∇ x Γ A ( x , y ) f ( y ) d μ ( y ) denotes the gradient of the single layer potential associated with $$L_A$$ L A , then $$\begin{aligned} 1+ \Vert T_\mu \Vert _{L^2(\mu )\rightarrow L^2(\mu )}\approx 1+ \Vert {\mathcal {R}}_\mu \Vert _{L^2(\mu )\rightarrow L^2(\mu )}, \end{aligned}$$ 1 + ‖ T μ ‖ L 2 ( μ ) → L 2 ( μ ) ≈ 1 + ‖ R μ ‖ L 2 ( μ ) → L 2 ( μ ) , where $${\mathcal {R}}_\mu $$ R μ indicates the n-dimensional Riesz transform. This allows us to provide a direct generalization of some deep geometric results, initially obtained for $${\mathcal {R}}_\mu $$ R μ , which were recently extended to $$T_\mu $$ T μ associated with $$L_A$$ L A with Hölder continuous coefficients. In particular, we show the following: If $$\mu $$ μ is an n-Ahlfors-David-regular measure on $$\mathbb {R}^{n+1}$$ R n + 1 with compact support, then $$T_\mu $$ T μ is bounded on $$L^2(\mu )$$ L 2 ( μ ) if and only if $$\mu $$ μ is uniformly n-rectifiable. Let $$E\subset \mathbb {R}^{n+1}$$ E ⊂ R n + 1 be compact and $${\mathcal {H}}^n(E)<\infty $$ H n ( E ) < ∞ . If $$T_{{\mathcal {H}}^n|_E}$$ T H n | E is bounded on $$L^2({\mathcal {H}}^n|_E)$$ L 2 ( H n | E ) , then E is n-rectifiable. If $$\mu $$ μ is a non-zero measure on $$\mathbb {R}^{n+1}$$ R n + 1 such that $$\limsup _{r\rightarrow 0}\tfrac{\mu (B(x,r))}{(2r)^n}$$ lim sup r → 0 μ ( B ( x , r ) ) ( 2 r ) n is positive and finite for $$\mu $$ μ -a.e. $$x\in \mathbb {R}^{n+1}$$ x ∈ R n + 1 and $$\liminf _{r\rightarrow 0}\tfrac{\mu (B(x,r))}{(2r)^n}$$ lim inf r → 0 μ ( B ( x , r ) ) ( 2 r ) n vanishes for $$\mu $$ μ -a.e. $$x\in \mathbb {R}^{n+1}$$ x ∈ R n + 1 , then the operator $$T_\mu $$ T μ is not bounded on $$L^2(\mu )$$ L 2 ( μ ) . Finally, we prove that if $$\mu $$ μ is a Radon measure on $${\mathbb {R}}^{n+1}$$ R n + 1 with compact support which satisfies a proper set of local conditions at the level of a ball $$B=B(x,r)\subset {\mathbb {R}}^{n+1}$$ B = B ( x , r ) ⊂ R n + 1 such that $$\mu (B)\approx r^n$$ μ ( B ) ≈ r n and r is small enough, then a significant portion of the support of $$\mu |_B$$ μ | B can be covered by a uniformly n-rectifiable set. These assumptions include a flatness condition, the $$L^2(\mu )$$ L 2 ( μ ) -boundedness of $$T_\mu $$ T μ on a large enough dilation of B, and the smallness of the mean oscillation of $$T_\mu $$ T μ at the level of B.