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10
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 23 (6 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 13 (3 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
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 7 (6 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
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 5 (5 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.
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 3 (3 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.
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 1 (0 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
In memory of E.Fabes
, 2002
"... Abstract. We consider the Dirichlet problem Lu = 0 in D u = g on ∂D for two second order elliptic operators Lku = ∑n i,j=1 ai,j k (x) ∂iju(x), k = 0, 1, in a bounded Lipschitz domain D ⊂ IR n. The coefficients a i,j k belong to the space of bounded mean oscillation BMO with a suitable small BMO modu ..."
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Abstract. We consider the Dirichlet problem Lu = 0 in D u = g on ∂D for two second order elliptic operators Lku = ∑n i,j=1 ai,j k (x) ∂iju(x), k = 0, 1, in a bounded Lipschitz domain D ⊂ IR n. The coefficients a i,j k belong to the space of bounded mean oscillation BMO with a suitable small BMO modulus. We assume that L0 is regular in Lp (∂D, dσ) for some p, 1 < p < ∞, that is, ‖Nu‖Lp ≤ C ‖g‖Lp for all continuous boundary data g. Here σ is the surface measure on ∂D and Nu is the nontangential maximal operator. The aim of this paper is to establish sufficient conditions on the difference of the coefficients εi,j (x) = a i,j 1 (x) − ai,j 0 (x) that will assure the perturbed operator L1 to be regular in Lq (∂D, dσ) for some q, 1 < q < ∞. 1.
THE DIRICHLET PROBLEM FOR ELLIPTIC EQUATIONS IN DIVERGENCE AND NONDIVERGENCE FORM WITH SINGULAR DRIFT TERM
, 2005
"... Abstract. Given two elliptic operators L0 and L1 in nondivergence form, with coefficients Aℓ and drift terms bℓ, ℓ = 0, 1 satisfying sup Y −X ≤ δ(X) ..."
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Abstract. Given two elliptic operators L0 and L1 in nondivergence form, with coefficients Aℓ and drift terms bℓ, ℓ = 0, 1 satisfying sup Y −X ≤ δ(X)
A NEW APPROACH TO SOLVABILITY OF SOME ELLIPTIC PDE’S WITH SQUARE INTEGRABLE BOUNDARY DATA
, 802
"... Abstract. We consider second order elliptic divergence form equations with complex measurable coefficients A that are independent of the transversal coordinate, and prove that the set of A ∈ L∞(R n;C n+1) for which boundary value problems with L2 Dirichlet or Neumann data are well posed, is an open ..."
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Abstract. We consider second order elliptic divergence form equations with complex measurable coefficients A that are independent of the transversal coordinate, and prove that the set of A ∈ L∞(R n;C n+1) for which boundary value problems with L2 Dirichlet or Neumann data are 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. In particular, our methods give a new proof of a theorem of Dahlberg, Jerison and Kenig on the comparability of square functions and non tangential maximal functions of solutions to real symmetric equations. MSC classes: 35J25, 35J55, 47N20 1.
ON HOFMANN’S BILINEAR ESTIMATE
, 812
"... Abstract. Using the framework of a previous article joint with Axelsson and McIntosh, we extend to systems two results of S. Hofmann for real symmetric equations and their perturbations going back to a work of B. Dahlberg for Laplace’s equation on Lipschitz domains, The first one is a certain biline ..."
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Abstract. Using the framework of a previous article joint with Axelsson and McIntosh, we extend to systems two results of S. Hofmann for real symmetric equations and their perturbations going back to a work of B. Dahlberg for Laplace’s equation on Lipschitz domains, The first one is a certain bilinear estimate for a class of weak solutions and the second is a criterion which allows to identify the domain of the generator of the semigroup yielding such solutions.