My research is in classical analysis. I am interested in understanding how to resolve singularities of real-analytic functions, for applications in analysis. In addition, I am generally interested in oscillatory integrals, exponential sums, and maximal operators.
Fourier transforms of indicator functions, lattice point discrepancy, and the stability of integrals, submitted.
L^p Sobolev regularity for a class of Radon and Radon-like transforms of various codimension, J. Four. Anal. Appl. (2018) doi: 10.1007/s00041-018-09647-9
L^p Sobolev regularity of averaging operators over hypersurfaces and the Newton polyhedron, J. Funct. Anal. (2018) doi: 10.1016/j.jfa.2018.05.014.
Smooth and singular maximal averages over 2D hypersurfaces and associated Radon transforms, submitted.
Fourier transforms of irregular mixed homogeneous hypersurface measures , Math. Nachr. 291 (2018), no. 7, 1075-1087.
Fourier transforms of powers of well-behaved 2D real analytic functions, Forum Math. 30 (2018), no. 3, 723–732.
Convolution kernels of 2D Fourier multipliers based on real analytic functions, J. Geom. Anal. 28 (2018), no. 2, 787–816.
Singular integral operators with kernels associated to negative powers of real-analytic functions, J. Funct. Anal. 269 (2015), no. 11, 3663-3687.
Uniform bounds for Fourier transforms of surface measures in R^3 with nonsmooth density, Trans. Amer. Math. Soc. 368 (2016), no. 9, 6601-6625.
Estimates for Fourier transforms of surface measures in R^3 and PDE applications, Rev. Mat. Iberoam vol 32 (2016), no. 2, 419–446.
Applications of an elementary resolution of singularities algorithm to exponential sums and congruences modulo p^n, Israel J. Math. 212 (2016), no. 1, 315-335.
Stability of oscillatory integral asymptotics in two dimensions. J. Geom. Anal. vol 24, #1 (2014), 417-444
A constructive elementary method for local resolution of singularities. This is an extension of my earlier resolution of singularities algorithms to general local fields of characteristic zero, with a substantially improved exposition. Most referees felt this was too similar to my earlier papers to accept, so for now this should be considered as an expository paper, not for publication.
L^p boundedness of maximal averages over hypersurfaces in R^3. Trans. Amer. Math. Soc. vol 365 (2013), no. 4, 1875-1900.
Maximal averages over hypersurfaces and the Newton polyhedron. J. Funct. Anal. vol 262 #5 (2012) 2314--2348.
Resolution of singularities in two dimensions and the stability of integrals. Adv. Math., 226 #2 (2011) 1772--1802.
Resolution of singularities, asymptotic expansions of integrals, and applications. J. Analyse Math. vol 111 #1 (2010) 221-245.
Oscillatory integral decay, sublevel set growth, and the Newton polyhedron. Math. Annalen vol. 346 #4 (2010), 857-890.
The asymptotic behavior of degenerate oscillatory integrals in two dimensions. J. Funct. Anal. vol 257 #6 (2009), 1759--1798.
A coordinate-dependent local resolution of singularities with applications. J. Funct. Anal. vol. 255, #8 (2008) 1957--1994.
Simply nondegenerate multilinear oscillatory integral operators with smooth phase. Math. Res. Lett. vol. 15 #4 (2008) 653-660.
A T(1) theorem for singular Radon transforms. Math Annalen vol 339 no. 3, (2007) 599-626.
An analog to a theorem of Fefferman and Phong for averaging operators along curves with fractional integral kernel. GAFA, vol 17, no. 4 (2007), 1106--1138.
Newton polygons and local integrability of negative powers of smooth functions in the plane., Trans. Amer. Math. Soc. vol 358 (2006), #2, 657-670.
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.
Sharp estimates for oscillatory integral operators with C-infinity phase. American J of Math. vol 127 (2005) #3 659-695.
A direct resolution of singularities for functions of two variables with applications to analysis, J. Anal. Math. 92 (2004), 233--257.
Scalings, metrics, and smoothing of translation-invariant Radon transforms along curves, J. Funct. Anal. 206 (2004), no. 2, 307-321.
Boundedness of singular Radon transforms on L^p spaces under a finite-type condition. Amer. J. of Math. (2001).
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.