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. 9511000
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 A_{2} 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. RodríguezLópez and W. Staubach, Trans. Amer. Math. Soc., vol. 367 (2015), pp. 69716995
We prove the global L^{2} ✕ L^{2} → L^{1} boundedness of bilinear oscillatory integral operators with amplitudes satisfying a Hörmandertype condition and phases satisfying appropriate growth as well as the strong nondegeneracy 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 SeegerSoggeStein theorem for bilinear Fourier integral operators
with S. RodríguezLópez and W. Staubach, Adv. Math., vol. 264 (2014) pp. 154.
We establish the regularity of bilinear Fourier integral operators with bilinear amplitudes in S^{m}_{1,0}(n,2) and nondegenerate phase functions, from L^{p} ✕ L^{q} → L^{r} under the assumptions that m ≤ (n1)(1/p1/2+1/q1/2) and 1/p+1/q=1/r. This is a bilinear version of the classical theorem of SeegerSoggeStein concerning the L^{p} → L^{p} boundedness of linear Fourier integral operators. Moreover, our result goes beyond the aforementioned theorem in that it also includes the case of quasiBanach 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. 465531.
We initiate the study of the finiteness condition
∫_{Ω} u(x)^{β} dx ≤ C(Ω,β) < +∞ where
Ω ⊆ R^{n} 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 C^{1} domains
in R^{n}, n ≥ 2, with conical singularities (in particular
polygonal domains in the plane), polyhedra in R^{3}, and
bounded domains which are locally of class C^{2} and which
have (finitely many) outwardly pointing cusps. For example, we show that
if u_{N} is the solution of the Saint Venant problem in the regular polygon
Ω_{N} with N sides circumscribed by the unit disc in the plane,
then for each β ∈ (0,1) the following asymptotic formula holds:
∫_{ΩN} u_{N}(x)^{β} dx ≤ C(Ω,β) = 4βπ/(1  β) + O(N^{β  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 CalderónZygmund theory
with N. Michalowski and W. Staubach, J. Math. Anal. Appl., vol. 414 (2014), no. 1, pp. 149165.
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 pseudopseudodifferential operators. Secondly, we investigate the boundedness of bilinear pseudodifferential operators with symbols in the Hörmander S^{m}_{ρ,δ} classes. These results are new in the case ρ < 1, that is, outwith the scope of multilinear CalderónZygmund theory.
On the stability of a forwardbackward heat equation
with L. Boulton and M. Marletta, Integral Equations Operator Theory, vol. 73 (2012), no. 2, pp. 195216.
In this paper we examine spectral properties of a family of periodic singular SturmLiouville problems which are highly nonselfadjoint 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., vol. 55 (2012), no. 3, pp. 555570.
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 CalderónZygmund theory. This paper can
be viewed as a prelude to MichalowskiRuleStaubach (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. 10131034.
Once again we study divergence form elliptic operators which are not assumed to be symmetric in domains in R^{2} 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 pseudopseudodifferential operators defined by amplitudes
with N. Michalowski and W. Staubach, J. Funct. Anal., vol. 258, 12, pp. 41834209.
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 nonsymmetric elliptic operators with C.E. Kenig, Trans. Amer. Math. Soc., vol. 361 (2009), pp. 125160.
We study divergence form elliptic operators which are not assumed to be symmetric in domains in R^{2} 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 R^{2}.
Nonsymmetric elliptic operators on bounded Lipschitz domains in the plane,
Electron. J. Diff. Eqns., vol. 2007 (2007), no. 144, pp. 18.
This paper extends the results of KenigRule (above) to include bounded Lipschitz domains in R^{2}.
We modify the arguments in KenigRule to enable us to prove the
boundedness of layer potentials in the more general context required for
bounded domains.
Thesis:
