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

• Published in: Archive for rational mechanics and analysis Vol. 247; no. 3; p. 38
• Main Authors: Molero, Alejandro, Mourgoglou, Mihalis, Puliatti, Carmelo, Tolsa, Xavier
• Format: Journal Article
• Language: English
• Published: Berlin/Heidelberg Springer Berlin Heidelberg 01.06.2023; Springer Nature B.V
• Subjects: Classical Mechanics; Coefficients; Complex Systems; Digital Object Identifier; Fluid- and Aerodynamics; Integrals; Mathematical analysis; Mathematical and Computational Physics; Operators (mathematics); Physics; Physics and Astronomy; Radon; Theoretical; Second order elliptic equations; Rectifiability; 28A75; Uniform rectifiability; Dini mean oscillation; 33C55; 42B20; David–Semmes problem; Riesz transform; Layer potentials; 35J15; 42B37
• ISSN: 0003-9527; 1432-0673; 1432-0673
• DOI: 10.1007/s00205-023-01852-1

We consider a uniformly elliptic operator L A in divergence form associated with an ( n + 1 ) × ( n + 1 ) -matrix A with real, merely bounded, and possibly non-symmetric coefficients. If then, under suitable Dini-type assumptions on ω A , we prove the following: if μ is a compactly supported Radon measure in R n + 1 , n ≥ 2 , and 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 , then 1 + ‖ T μ ‖ L 2 ( μ ) → L 2 ( μ ) ≈ 1 + ‖ R μ ‖ L 2 ( μ ) → L 2 ( μ ) , where R μ indicates the n -dimensional Riesz transform. This allows us to provide a direct generalization of some deep geometric results, initially obtained for R μ , which were recently extended to T μ associated with L A with Hölder continuous coefficients. In particular, we show the following: If μ is an n -Ahlfors-David-regular measure on R n + 1 with compact support, then T μ is bounded on L 2 ( μ ) if and only if μ is uniformly n -rectifiable. Let E ⊂ R n + 1 be compact and H n ( E ) < ∞ . If T H n | E is bounded on L 2 ( H n | E ) , then E is n -rectifiable. If μ is a non-zero measure on R n + 1 such that lim sup r → 0 μ ( B ( x , r ) ) ( 2 r ) n is positive and finite for μ -a.e. x ∈ R n + 1 and lim inf r → 0 μ ( B ( x , r ) ) ( 2 r ) n vanishes for μ -a.e. x ∈ R n + 1 , then the operator T μ is not bounded on L 2 ( μ ) . Finally, we prove that if μ is a Radon measure on R n + 1 with compact support which satisfies a proper set of local conditions at the level of a ball B = B ( x , r ) ⊂ R n + 1 such that μ ( B ) ≈ r n 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 L 2 ( μ ) -boundedness of T μ on a large enough dilation of B , and the smallness of the mean oscillation of T μ at the level of B .