Boundary Value Problems for Higher Order Operators in Lipschitz and Domains
Bounded double square functions
We extend some recent work of S. Y. Chang, J. M. Wilson and T. Wolff to the bidisc. For , we determine the sharp order of local integrability obtained when the square function of is in . The Calderón-Torchinsky decomposition reduces the problem to the case of double dyadic martingales. Here we prove a vector-valued form of an inequality for dyadic martingales that yields the sharp dependence on p of in .
Oblique derivative problems for the laplacian in Lipschitz domains.
The aim of this paper is to extend the results of Calderón [1] and Kenig-Pipher [12] on solutions to the oblique derivative problem to the case where the data is assumed to be BMO or Hölder continuous.
The Absolute Continuity of Elliptic Measure Revisited.
The Neumann problem for elliptic equations with non-smooth coefficients.
Hardy spaces and the Dirichlet problem on Lipschitz domains.
Our concern in this paper is to describe a class of Hardy spaces H(D) for 1 ≤ p < 2 on a Lipschitz domain D ⊂ R when n ≥ 3, and a certain smooth counterpart of H(D) on R, by providing an atomic decomposition and a description of their duals.
The Dirichlet problem for elliptic equations with drift terms.
We establish absolute continuity of the elliptic measure associated to certain second order elliptic equations in either divergence or nondivergence form, with drift terms, under minimal smoothness assumptions on the coefficients.
Multi-parameter paraproducts.
We prove that classical Coifman-Meyer theorem holds on any polidisc T or arbitrary dimension d ≥ 1.
Area integral estimates for higher order elliptic equations and systems
Let be an elliptic system of higher order homogeneous partial differential operators. We establish in this article the equivalence in norm between the maximal function and the square function of solutions to in Lipschitz domains. Several applications of this result are discussed.
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