L2 -Boundedness of Gradients of Single Layer Potentials for Elliptic Operators with Coefficients of Dini Mean Oscillation-Type
Molero, Alejandro (Universitat Autònoma de Barcelona. Departament de Matemàtiques)
Mourgoglou, Mihalis (Universidad del País Vasco. Departamento de Matemáticas)
Puliatti, Carmelo (Universidad del País Vasco. Departamento de Matemáticas)
Tolsa Domènech, Xavier (Centre de Recerca Matemática)

Data:	2023
Resum:	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.
Ajuts:	Agencia Estatal de Investigación BES-2017-081272 Ministerio de Economía y Competitividad MTM-2016-77635-P Agencia Estatal de Investigación PID2020-118986GB-I00 Agencia Estatal de Investigación PGC2018-094522-B-I00 European Commission 101018680 Agencia Estatal de Investigación CEX2020-001084-M Agencia Estatal de Investigación PID2020-114167GB-I00
Drets:	Aquest document està subjecte a una llicència d'ús Creative Commons. Es permet la reproducció total o parcial, la distribució, la comunicació pública de l'obra i la creació d'obres derivades, fins i tot amb finalitats comercials, sempre i quan es reconegui l'autoria de l'obra original.
Llengua:	Anglès
Document:	Article ; recerca ; Versió publicada
Matèria:	Riesz transform ; Layer potentials ; Second order elliptic equations ; Dini mean oscillation ; David-Semmes problem ; Uniform rectifiability ; Rectifiability
Publicat a:	Archive for Rational Mechanics and Analysis, Vol. 247 (April 2023) , art. 38, ISSN 1432-0673

DOI: 10.1007/s00205-023-01852-1

59 p, 781.7 KB

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