Research

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 (accepted by Comm. Pure Appl. Math.)

Refereed Papers:

  • Boundary value problems for degenerate elliptic equations and systems
    with P. Auscher and A. Rosén, Ann. Sci. Éc. Norm. Supér. (4) vol. 48 (2015) pp. 951-1000

    We study boundary value problems for degenerate elliptic equations and systems with square integrable boundary data. We can allow for degeneracies in the form of an A2 weight. We obtain representations and boundary traces for solutions in appropriate classes, perturbation results for solvability and solvability in some situations. The technology of earlier works of the first two authors can be adapted to the weighted setting once the needed quadratic estimate is established and we even improve some results in the unweighted setting. The proof of this quadratic estimate does not follow from earlier results on the topic and is the core of the article.

  • On the boundedness of certain bilinear oscillatory integral operators
    with S. Rodríguez-López and W. Staubach, Trans. Amer. Math. Soc., vol. 367 (2015), pp. 6971-6995

    We prove the global L2 ✕ L2 → L1 boundedness of bilinear oscillatory integral operators with amplitudes satisfying a Hörmander-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 pseudodifferential operators, to the case of oscillatory integral operators.

  • A Seeger-Sogge-Stein theorem for bilinear Fourier integral operators
    with S. Rodríguez-López and W. Staubach, Adv. Math., vol. 264 (2014) pp. 1-54.

    We establish the regularity of bilinear Fourier integral operators with bilinear amplitudes in Sm1,0(n,2) and non-degenerate phase functions, from Lp ✕ Lq → Lr under the assumptions that m ≤ -(n-1)(|1/p-1/2|+|1/q-1/2|) and 1/p+1/q=1/r. This is a bilinear version of the classical theorem of Seeger-Sogge-Stein concerning the Lp → Lp boundedness of linear Fourier integral operators. Moreover, our result goes beyond the aforementioned theorem in that it also includes the case of quasi-Banach target spaces.

  • The integrability of negative powers of the solution of the Saint Venant Problem
    with A. Carbery, V. Maz'ya and M. Mitrea, Ann. Sc. Norm. Super. Pisa Cl. Sci. (5), vol. 13 (2014) no. 2, pp. 465-531.

    We initiate the study of the finiteness condition Ω u(x) dx ≤ C(Ω,β) < +∞ where Ω ⊆ Rn is an open set and u is the solution of the Saint Venant problem Δ u = -1 in Ω, u=0 on ∂Ω. The central issue which we address is that of determining the range of values of the parameter β > 0 for which the aforementioned condition holds under various hypotheses on the smoothness of Ω and demands on the nature of the constant C(Ω,β). Classes of domains for which our analysis applies include bounded piecewise C1 domains in Rn, n ≥ 2, with conical singularities (in particular polygonal domains in the plane), polyhedra in R3, and bounded domains which are locally of class C2 and which have (finitely many) outwardly pointing cusps. For example, we show that if uN is the solution of the Saint Venant problem in the regular polygon ΩN with N sides circumscribed by the unit disc in the plane, then for each β ∈ (0,1) the following asymptotic formula holds: ΩN uN(x) dx ≤ C(Ω,β) = 4βπ/(1 - β) + O(Nβ - 1) as N → ∞. One of the original motivations for addressing the aforementioned issues was the study of sublevel set estimates for functions v satisfying v(0) = 0, ∇ v(0) = 0 and Δ v = c ≥ = 0.

  • Multilinear pseudodifferential operators beyond Calderón-Zygmund theory
    with N. Michalowski and W. Staubach, J. Math. Anal. Appl., vol. 414 (2014), no. 1, pp. 149-165.

    We consider two types of multilinear pseudodifferential operators. First, we prove the boundedness of multilinear pseudodifferential operators with symbols which are only measurable in the spatial variables in Lebesgue spaces. These results generalise earlier work of the present authors concerning linear pseudo-pseudodifferential operators. Secondly, we investigate the boundedness of bilinear pseudodifferential operators with symbols in the Hörmander Smρ,δ classes. These results are new in the case ρ < 1, that is, outwith the scope of multilinear Calderón-Zygmund theory.

  • 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 Lp boundedness of pseudodifferential operators and applications
    with N. Michalowski and W. Staubach, Canad. Math. Bull., vol. 55 (2012), no. 3, pp. 555-570.

    In this paper we prove weighted norm inequalities with weights in the Ap classes, for pseudodifferential operators with symbols in the class Smρ,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 Lp 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 Lp 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 Lp.

  • 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 Lp 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)


David Rule
david.rule@liu.se
Last updated 11/8/2016