David J. Rule

Elliptic equations in the plane satisfying a Carleson measure condition

In this paper we settle (in dimension n = 2) the open question whether for a divergence form equation div(A∇u) = 0 with coefficients satisfying certain minimal smoothness assumption (a Carleson measure condition), the Lp Neumann and Dirichlet regularity problems are solvable for some values of p ∈ (1,∞). The related question for the Lp Dirichlet problem was settled (in any dimension) in 2001 by Kenig and Pipher [11].

On The Boundedness Of Certain Bilinear Oscillatory Integral Operators

We prove the global L2 × L2 → L1 boundedness of bilinear oscillatory integral operators with amplitudes satisfying a Hormander type condition and phases satisfying appropriate growth as well as the strong non-degeneracy conditions. This is an extension of the corresponding result of R. Coifman and Y. Meyer for bilinear pseudo-differential operators, to the case of oscillatory integral operators.

Weighted

In this paper we prove weighted norm inequalities with weights in the Ap classes, for pseudodifferential operators with symbols in the class S n(ρ−1) ρ,δ which fall outside the scope of Calderón-Zygmund theory. Our weighted norm inequalities also yield Lp boundedness of commutators of functions of bounded mean oscillation with a wide class of operators in OPSm ρ,δ .