I am interested in harmonic analysis and its application to differential equations. Most of my work so far has concerned pseudo-differential operators and elliptic equations. Links to my papers are given below and some preprints have been posted on the arXiv.

In the works:

• Boundary value problems for second order elliptic equations satisfying a Carleson measure condition
  with M. Dindoš and J. Pipher (in preparation)

• On the boundedness of bilinear Fourier integral operators
  with S. Rodríguez-López and W. Stuabach (in preparation)

• Multilinear pseudodifferential operators beyond Calderón-Zygmund theory
  with N. Michalowski and W. Staubach (submitted)

• The integrability of negative powers of the solution of the Saint Venant Problem
  with A. Carbery, V. Maz'ya and M. Mitrea (submitted)

Refereed Papers:

• On the stability of a forward-backward heat equation
  with L. Boulton and M. Marletta, Integral Equations Operator Theory, vol. 73 (2012), no. 2, pp. 195-216.

  In this paper we examine spectral properties of a family of periodic singular Sturm-Liouville problems which are highly non-self-adjoint but have purely real spectrum. The problem originated from the study of the lubrication approximation of a viscous fluid film in the inner surface of a rotating cylinder and has received a substantial amount of attention in recent years. Our main focus will be the determination of Schatten class inclusions for the resolvent operator and regularity properties of the associated evolution equation.

• Weighted L^p boundedness of pseudodifferential operators and applications
  with N. Michalowski and W. Staubach, Canad. Math. Bull., published electronically, June 2011 pp.1-16.

  In this paper we prove weighted norm inequalities with weights in the A_p classes, for pseudodifferential operators with symbols in the class S^m_ρ,δ which fall outside the scope of Calderón-Zygmund theory. This paper can be viewed as a prelude to Michalowski-Rule-Staubach (below), in that the main result here is a particular case of the results in that paper. However, the techniques used here are rooted in the theory of smooth operators.

• Elliptic equations in the plane satisfying a Carleson measure condition
  with M. Dindoš, Rev. Mat. Iberoam., vol. 26 (2010), no. 3, pp. 1013-1034.

  Once again we study divergence form elliptic operators which are not assumed to be symmetric in domains in R² above the graph of a Lipschitz function. However, here we assume the coefficients satisfy a Carleson measure condition. Using a new technique of introducing an auxiliary equation, we can prove that the Neumann and regularity problems are solvable with data in L^p for some p>1 provided the Carleson measure norm is sufficiently small.

• Weighted norm inequalities for pseudo-pseudodifferential operators defined by amplitudes
  with N. Michalowski and W. Staubach, J. Funct. Anal., vol. 258, 12, pp. 4183-4209.

  We study pseudodifferential operators which are only assumed to be measurable in the spatial variable. We give conditions under which these operators are bounded on weighted L^p with weights in the Muckenhoupt classes. Some of the results are shown to be sharp with respect to these hypotheses, however, for operators of a particular form, the hypotheses can be weakened. As an application of these weighted boundedness results we show that the commutators of these operators with functions of bounded mean oscillation are bounded in L^p.

• The regularity and Neumann problem for non-symmetric elliptic operators
  with C.E. Kenig, Trans. Amer. Math. Soc., vol. 361 (2009), pp. 125-160.

  We study divergence form elliptic operators which are not assumed to be symmetric in domains in R² above the graph of a Lipschitz function. Under the assumption that the coefficients of the operator are independent of the vertical direction and measurable in the horizontal, we prove that the Neumann and regularity problems are solvable with data in L^p for some p>1. This is done via an application of David and Journé's T(b) Theorem and the extra regularity properties of solutions in R².

• Non-symmetric elliptic operators on bounded Lipschitz domains in the plane,
  Electron. J. Diff. Eqns., vol. 2007 (2007), no. 144, pp. 1-8.

  This paper extends the results of Kenig-Rule (above) to include bounded Lipschitz domains in R². We modify the arguments in Kenig-Rule to enable us to prove the boundedness of layer potentials in the more general context required for bounded domains.

Thesis:

• The regularity and Neumann problem for non-symmetric elliptic operators
  Ph.D. Thesis, University of Chicago, 2007 (supervisor: Carlos Kenig)