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33
Weighted norm inequalities, offdiagonal estimates and elliptic operators, Part II: Offdiagonal estimates on spaces of homogeneous type
, 2005
"... Abstract. This is the fourth article of our series. Here, we apply the results of [AM1] to study weighted norm inequalities for the Riesz transform of the LaplaceBeltrami operator on Riemannian manifolds and of subelliptic sum of squares on Lie groups, under the doubling volume property and Poincar ..."
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Cited by 61 (15 self)
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Abstract. This is the fourth article of our series. Here, we apply the results of [AM1] to study weighted norm inequalities for the Riesz transform of the LaplaceBeltrami operator on Riemannian manifolds and of subelliptic sum of squares on Lie groups, under the doubling volume property and Poincaré inequalities. 1. Introduction and
A new approach to absolute continuity of elliptic measure, with applications to nonsymmetric equations
 Adv. in Math
"... In the late 50’s and early 60’s, the work of De Giorgi [De Gi] and Nash [N], and then Moser [Mo] initiated the study of regularity of solutions to divergence form elliptic equations with merely bounded measurable coefficients. Weak solutions in a domain Ω, a priori only in a Sobolev space W 2 1,loc ..."
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Cited by 41 (8 self)
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In the late 50’s and early 60’s, the work of De Giorgi [De Gi] and Nash [N], and then Moser [Mo] initiated the study of regularity of solutions to divergence form elliptic equations with merely bounded measurable coefficients. Weak solutions in a domain Ω, a priori only in a Sobolev space W 2 1,loc
Solvability of elliptic systems with square integrable boundary data, preprint Preprint arXiv:0809.4968v1 [math.AP
"... Abstract. We consider second order elliptic divergence form systems with complex measurable coefficients A that are independent of the transversal coordinate, and prove that the set of A for which the boundary value problem with L2 Dirichlet or Neumann data is well posed, is an open set. Furthermore ..."
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Cited by 32 (17 self)
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Abstract. We consider second order elliptic divergence form systems with complex measurable coefficients A that are independent of the transversal coordinate, and prove that the set of A for which the boundary value problem with L2 Dirichlet or Neumann data is well posed, is an open set. Furthermore we prove that these boundary value problems are well posed when A is either Hermitean, block or constant. Our methods apply to more general systems of PDEs and as an example we prove perturbation results for boundary value problems for differential forms. MSC classes: 35J25, 35J55, 47N20 1.
Weighted maximal regularity estimates and solvability of elliptic systems
 I. Invent. Math
"... Abstract. We develop new solvability methods for divergence form second order, real and complex, elliptic systems above Lipschitz graphs, with L2 boundary data. Our methods yield full characterization of weak solutions, whose gradients have L2 estimates of a nontangential maximal function or of the ..."
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Cited by 30 (11 self)
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Abstract. We develop new solvability methods for divergence form second order, real and complex, elliptic systems above Lipschitz graphs, with L2 boundary data. Our methods yield full characterization of weak solutions, whose gradients have L2 estimates of a nontangential maximal function or of the square function, via an integral representation acting on the conormal gradient, with a singular operatorvalued kernel. The coefficients A may depend on all variables, but are assumed to be close to coefficients A0 that are independent of the coordinate transversal to the boundary, in the Carleson sense ‖A − A0‖C defined by Dahlberg. We obtain a number of a priori estimates and boundary behaviour under finiteness of ‖A − A0‖C. For example, the nontangential maximal function of a weak solution is controlled in L2 by the square function of its gradient. This estimate is new for systems in such generality, even for real nonsymmetric equations in dimension 3 or higher. The existence of a proof a priori to wellposedness, is also a new fact. As corollaries, we obtain wellposedness of the Dirichlet, Neumann and Dirichlet regularity problems under smallness of ‖A−A0‖C and wellposedness for A0, improving earlier results for real symmetric equations. Our methods build on an algebraic reduction to a first order system first made for coefficients A0 by the two authors and A. McIntosh in order to use functional calculus related to the Kato conjecture solution, and the main analytic tool for coefficients A is an operational calculus to prove weighted maximal regularity estimates.
ON A QUADRATIC ESTIMATE RELATED TO THE KATO CONJECTURE AND BOUNDARY VALUE PROBLEMS
, 810
"... Abstract. We provide a direct proof of a quadratic estimate that plays a central role in the determination of domains of square roots of elliptic operators and, as shown more recently, in some boundary value problems with L 2 boundary data. We develop the application to the Kato conjecture and to a ..."
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Cited by 23 (14 self)
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Abstract. We provide a direct proof of a quadratic estimate that plays a central role in the determination of domains of square roots of elliptic operators and, as shown more recently, in some boundary value problems with L 2 boundary data. We develop the application to the Kato conjecture and to a Neumann problem. This quadratic estimate enjoys some equivalent forms in various settings. This gives new results in the functional calculus of Dirac type operators on forms. MSC classes: 35J25, 35J55, 47N20, 47F05, 42B25 Keywords: LittlewoodPaley estimate, functional calculus, boundary value problems, second order elliptic equations and systems, square root problem 1.
Functional calculus of Dirac operators and complex perturbations of Neumann and Dirichlet problems
 Journal of Functional Analysis
"... Abstract. We prove that the Neumann, Dirichlet and regularity problems for divergence form elliptic equations in the half space are well posed in L2 for small complex L ∞ perturbations of a coefficient matrix which is either real symmetric, of block form or constant. All matrices are assumed to be i ..."
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Cited by 23 (14 self)
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Abstract. We prove that the Neumann, Dirichlet and regularity problems for divergence form elliptic equations in the half space are well posed in L2 for small complex L ∞ perturbations of a coefficient matrix which is either real symmetric, of block form or constant. All matrices are assumed to be independent of the transversal coordinate. We solve the Neumann, Dirichlet and regularity problems through a new boundary operator method which makes use of operators in the functional calculus of an underlaying first order Dirac type operator. We establish quadratic estimates for this Dirac operator, which implies that the associated Hardy projection operators are bounded and depend continuously on the coefficient matrix. We also prove that certain transmission problems for kforms are
BMO solvability and the A ∞ condition for elliptic operators
, 2009
"... We establish a connection between the absolute continuity of elliptic measure associated to a second order divergence form operator with bounded measurable coefficients with the solvability of an endpoint BMO Dirichlet problem. We show that these two notions are equivalent. As a consequence we obtai ..."
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Cited by 11 (4 self)
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We establish a connection between the absolute continuity of elliptic measure associated to a second order divergence form operator with bounded measurable coefficients with the solvability of an endpoint BMO Dirichlet problem. We show that these two notions are equivalent. As a consequence we obtain an endpoint perturbation result, i.e., the solvability of the BMO Dirichlet problem implies L p solvability for all p> p0. 1
Divergence form operators in Reifenberg flat domains
, 2009
"... We study the boundary regularity of solutions of elliptic operators in divergence form with C0,α coefficients or operators which are small perturbations of the Laplacian in nonsmooth domains. We show that, as in the case of the Laplacian, there exists a close relationship between the regularity of ..."
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Cited by 9 (5 self)
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We study the boundary regularity of solutions of elliptic operators in divergence form with C0,α coefficients or operators which are small perturbations of the Laplacian in nonsmooth domains. We show that, as in the case of the Laplacian, there exists a close relationship between the regularity of the corresponding elliptic measure and the geometry of the domain.
The Regularity problem for second order elliptic operators with complexvalued bounded measurable coefficients
"... Abstract. The present paper establishes a certain duality between the Dirichlet and Regularity problems for elliptic operators with tindependent complex bounded measurable coefficients (t being the transversal direction to the boundary). To be precise, we show that the Dirichlet boundary value pr ..."
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Cited by 7 (2 self)
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Abstract. The present paper establishes a certain duality between the Dirichlet and Regularity problems for elliptic operators with tindependent complex bounded measurable coefficients (t being the transversal direction to the boundary). To be precise, we show that the Dirichlet boundary value problem is solvable in Lp, subject to the square function and nontangential maximal function estimates, if and only if the corresponding Regularity problem is solvable in Lp. Moreover, the solutions admit layer potential representations. In particular, we prove that for any elliptic operator with tindependent real (possibly nonsymmetric) coefficients there exists a p> 1 such that the Regularity problem is wellposed in Lp. 1.