My research is in classical analysis. I am interested in understanding how to resolve singularities of polynomials or more generally real-analytic functions, for applications in analysis. In addition, I am generally interested in oscillatory integrals and Radon transforms.
"Oscillatory integral decay, sublevel set growth, and the Newton polyhedron," submitted. PDF PS
The methods of the n-dimensional resolution of singularities theorem below are used to generalize Varchenko's well-known theorem relating the decay of oscillatory integrals to the Newton polyhedron of the phase. In particular, it is shown that the decay rate given through the Newton polyhedron still holds for a large class of functions that do not satisfy Varchenko's nondegeneracy condition on the faces of the Newton polyhedron. Analogues are given for sublevel set integrals. Some stronger results are given in lower dimension. Also, some estimates are proven for more degenerate phases, including some whose oscillatory integral decay rate is not given through the Newton polyhedron.
"Resolution of singularities, asymptotic expansions of sublevel sets, and applications," preprint. PDF PS
The elementary resolution of singularities algorithm of my paper below is developed further, replacing the quasibump functions in the blown-up coordinates with the characteristic function of a rectangle times a smooth function. Such functions are easier to deal with, and as application the existence of asymptotic expansions for oscillatory integrals and related objects is given an elementary proof. In addition, some more detailed information about these expansions is given.
"Simply nondegenerate multilinear oscillatory integral operators with smooth phase," to appear, Math. Res. Lett. PDF PS
This paper extends the results of Christ-Li-Tao-Thiele on multilinear oscillatory integral operators in the simply nondegenerate situation. Instead of assuming polynomial phase, one only assumes the phase is smooth and decay is shown whenever the phase doesn't vanish to infinite order. Some consequences are given.
"A coordinate-dependent local resolution of singularities with applications," submitted. PDF PS
This gives a local resolution of singularities algorithm for real analytic functions in n dimensions. Other than using the implicit function theorem and some basic linear algebra, the paper is entirely self-contained. The procedure is done explicitly in coordinates, making extensive use of Newton polyhedra. Two applications are given. The first and primary application is a general theorem regarding the existence of critical integrability exponents. The second application is a new proof of a well-known inequality of Lojasiewicz. The main theorem is stated in terms of a sort of partition of unity with respect to the zeroes of the function being resolved.
"A T(1) theorem for singular Radon transforms", to appear, Math Annalen. PS PDF
Over the last few years I've found ways to significantly simplify and extend the arguments of my first two papers, and relate them to Carnot-Caratheodory geometry. These methods are used here to prove a version of the T(1) theorem for singular Radon transforms.
"Stability of sublevel set estimates and sharp L^2 regularity of Radon transforms in the Plane", Math Res Letters, v.12 (2005) #1, 1-17. PS PDF
The stability of certain sublevel set estimates in two dimensions are shown to imply $L^2$ regularity of associated Radon transforms. The estimates are sharp under certain conditions that can be expressed in terms of an associated Newton polygon.
"An analog to a theorem of Fefferman and Phong for averaging operators along curves with fractional integral kernel", to appear, GAFA. PS PDF
This paper shows that for Radon transforms with fractional singular kernels, if the singularity is strong enough, one can prove sharp L^2 regularity of the operator in terms of the metric of my second singular Radon transform paper. The estimates are in many ways analogous to the Fefferman-Phong subelliptic estimates which are expressible in terms of their metric.
"Sharp estimates for oscillatory integral operators with C-infinity phase", 33 pages, American J of Math. vol 127 (2005) #3 659-695. PS PDF
A generalization of paper #6 for smooth phases. It uses a stopping time argument in conjunction with an appropriate set of Bernstein inequalities, and uses the various Phong-Stein operator Van der Corput-type lemmas, as well the type of resolution of singularities that one uses in paper #5 below. One of Stein's grad students, Rytchkov, proved a theorem of this type (not including certain exceptional cases) using Weierstrass factorizations and the like, more in the spirit of Phong and Stein's original paper.
"A direct resolution of singularities for functions of two variables with applications to analysis", J. Anal. Math. 92 (2004), 233--257. PS PDF
In their major 1997 Acta paper, Phong and Stein showed how the boundedness properties of one-dimensional oscillatory integral operators with real-analytic phase can be expressed in terms of the Newton polygon of the Hessian of the phase function. The proof uses facts from algebraic geometry such as the Weierstrass Preparation Theorem which are not easily generalizable to higher-dimensional situations. In this paper, I directly resolve the singularities of an arbitrary two-dimensional real-analytic function and use it in conjunction with various analytic lemmas like the ones Phong and Stein use, giving a proof of their main result. This proof therefore isn't totally independent of theirs but hopefully it will make higher-dimensional generalizations more accessible.
"Newton polygons and local integrability of negative powers of smooth functions in the plane", Trans. Amer. Math. Soc. vol 358 (2006), #2, 657-670. PS PDF
Basically, what the title says. It shows when the Newton polygon gives the most negative power of a smooth function that is integrable in a neighborhood of the origin. Endpoint cases are also analyzed. It uses resolution of singularities ideas.
"Scalings, metrics, and smoothing of translation-invariant Radon transforms along curves", J. Funct. Anal. 206 (2004), no. 2, 307--321. PS
This gives L^p_a to L^q_b estimates for translation-invariant Radon transforms along curves, using a method inspired by my thesis. For fixed a and b, the estimates are sharp up to endpoints for p near 2. This has been completely rewritten and now is fully coherent."Boundedness of singular Radon transforms on L^p spaces under a finite-type condition", 43 pages, American Journal of Mathematics, (2001). PS
This paper started out as a generalization of my thesis to singular Radon transforms over surfaces of higher codimension but grew into something quite different. A generalization of the notion of finite-type to the nontranslation-invariant case is defined (equivalent to the curvature condition of Christ-Nagel-Stein-Wainger), and geometric and analytic methods are used to prove L^p boundedness of singular Radon transforms under this finite-type condition, over surfaces of any codimension. There is a significant relation to the Nagel-Stein-Wainger constructions, but to have a general theorem it was necessary to develop things from scratch. A natural metric is associated to a given singular Radon transform, and the methods used in this paper are related to singular integral theory with respect to this metric.
"A method for proving L^p boundedness of singular Radon transforms in codimension one for 1 < p < infinity", Duke Math Journal (108) 2001, 363-393. PS
This is a slightly touched-up version of my thesis under E. M. Stein. I develop a method of proving L^p boundedness of singular Radon transforms over hypersurfaces under a finite-type condition. This paper has antecedents in Phong/Stein's 1986 Acta paper on singular Radon transforms. Other than using almost-orthogonality and the Calderon-Zygmund theorem for non-Euclidean metrics, the arguments contained therein are elementary.