This page is no longer being maintained. You will find my most recent papers by searching for my name at or

Papers in .pdf format.

Square functions and the A_infinity property of elliptic measure with C. Kenig, B. Kirchheim, and T. Toro,

Carleson measures and elliptic boundary value problems Proceedings of ICM 2014, CarlesonMeasure.pdf

Practical Signatures from the Partial Fourier Recovery Problem, with J. Hoffstein, j. Schanck, J. Silverman and W. Whyte,

Dyadic structure theorems for multiparameter function spaces, with J. Li and L. Ward, to appear in Revista IberoAmer. LiPipherWard_RMI_2013.pdf

The Regularity problem for second order elliptic operators with complex-valued non-smooth coefficients, with S. Hofmann, C. Kenig, S. Mayboroda, to appear Mat. Annalen, preprint available at .

Diophantine Approximation and directional discrepancy of rotated lattices, with D. Bilyk, X. Ma, and C. Spencer, bmps29.pdf to appear in TAMS,

Perturbations of elliptic operators with E. Milakis, T. Toro, to appear in Harmonic Analysis and Partial Differential Equations, Contemporary Mathematics, vol. 612, Amer. Math. Soc.,143-161

Boundary value problems for second order elliptic operators satisfying a Carleson condition, with M. Dindos, D. Rule, preprint available at

Square function/non-tangential maximal function estimates and the dirichlet problem for non-symmetric elliptic operators with Steve Hofmann, Carlos Kenig, Svitlana Mayboroda, to appear in JAMS .

Harmonic analysis on chord-arc domains with E. Milakis, T. Toro, J. Geometric Analysis, (2013), 23, 2091-2157 .

Practical Lattice-based Cryptography: NTRUEncrypt and NTRUSign with J. Hoffstein, N. Howgrave-Graham, W. Whyte, in The LLL Algorithm: Survey and Applications, Springer, 2010.

Multiparameter div-curl lemmas with M. Lacey, S. Petermichl, B. Wick, Bull. London Math. Soc. (2012) mpdivcurl.pdf

Directional discrepancy in two dimensions with D. Bilyk, X. Ma, and C. Spencer, Bulletin London Math. Soc. (2011), 43 (6), 1151-1166, bmpsLMS.pdf

Geometric-arithmetic averaging of dyadic weights with L. Ward and X. Xiao, PWX2009.pdf to appear Rev. Mat. Iberoamericana.

Weak-star convergence in multiparameter Hardy spaces, with S. Treil, Proc. Amer. Math. Soc. 139 (2011), 1445-1454, available on Arxiv here

BMO solvability and the A^infinity condition for elliptic operators, with M. Dindos, C. Kenig bmoJul29.pdf, J. Geometric Analysis (2011), 21, 78-95

Iterated Riesz Commutators: a simple proof of boundedness, with M. Lacey, S. Petermichl, B. Wick, RieszUpperBound.pdf Proceedings of Analysis at El Escorial, 2008.

Book:An Introduction to Mathematical Cryptography with J. Hoffstein, J. Silverman, Series: Undergraduate Textbooks in Mathematics, Springer, July 2008.

Multiparameter Riesz Commutators, with M. Lacey, S. Petermichl, and B. Wick, multiParameterRieszComm.pdf,, Amer. J. Math, Vol. 131, No.3, (2009) p. 731-771.

The L^p Dirichlet problem for second order elliptic operators and a p-adapted square function, with M. Dindos and S. Petermichl psqFinal.pdf , J. Funct. Anal. Vol. 249 issue 2, 2007, p. 372-392

BMO from dyadic BMO on the bidisc, with Lesley Ward BMOdyadicBMObidisc.pdf J. London Math. Soc. Vol. 77, Number 2 (2008), p. 524-544.

On Estimating the Lattice Security of NTRU, with N. Howgrave-Graham, J. Hoffstein, W. Whyte, NTRULattice-2005-1.pdf

Variations on the theme of Journe's lemma, with C. Cabrillo, M. Lacey, U. Molter, JourneLemma.pdf Houston J. Math., Vol 32, No. 3 (2006), p. 833.

Multiparameter paraproducts, with C. Muscalu, T. Tao, C. Thiele, preprint, polydisc.pdf Revista M. Ibero. Vol. 22, Number 3, 2006, p. 963-976.

Biparameter paraproducts, with C. Muscalu, T. Tao, C. Thiele, biparam.pdf , Acta Mathematica, 193, (2004), p. 269-296

A short proof of the Coifman-Meyer paraproduct theorem, with C. Muscalu, T. Tao, C. Thiele, C-M.pdf

A covering lemma for rectangles in $\R^n$, with R. Fefferman, 133 No. 11 (2005), p.3235-3241. Proc. AMS., coveringlemma.pdf

NTRUSign: Digital signatures using the NTRU lattice, with J. Hoffstein, N. Howgrave-Graham, J.Silverman, W.Whyte, Proceedings of CT-RSA 2003, NTRUSign.pdf

NTRUSign: Digital signatures using the NTRU lattice, extended version, with J. Hoffstein, N. Howgrave-Graham, J.Silverman, W.Whyte, 2002, NTRUSignextended.pdf

Lectures at the L'Institut Fourier 2002 Summer School in Cryptology on NTRU encryption and digital signatures, grenoble.pdf

NSS: An NTRU lattice-based signature scheme, with J. Hoffstein and J. Silverman, Proceedings of Eurocrypt 2001.

The Dirichlet problem for elliptic equations with drift terms, with C. Kenig, Publicaciones Matematiques 45 (2001), 199-217 Dir.dvi

A new approach to absolute continuity of elliptic measure, with applications to non-symmetric equations. Adv. Math. 153 (2000) , no. 2, 231--298. w/Kenig, C.; Koch, H.; Toro, T. AbsoluteContinuity.pdf

NTRU: a ring-based public key cryptosystem. Algorithmic number theory (Portland, OR, 1998), 267--288, Lecture Notes in Comput. Sci., 1423 , Springer, Berlin, 1998. w/Hoffstein, Jeffrey; Silverman, Joseph H.

The inhomogeneous Dirichlet problem for $\Delta\sp 2$ in Lipschitz domains. J. Funct. Anal. 159 (1998), no. 1, 137--190. w/Adolfsson, Vilhelm

The absolute continuity of elliptic measure revisited. J. Fourier Anal. Appl. 4 (1998) , no. 4-5, 463--468. w/Kenig, Carlos E.

Vector potential theory on nonsmooth domains in ${R}\sp 3$ and applications to electromagnetic scattering. J. Fourier Anal. Appl. 3 (1997) , no. 2, 131--192. w/Mitrea, Dorina; Mitrea, Marius;

Area integral estimates for higher order elliptic equations and systems. Ann. Inst. Fourier (Grenoble) 47 (1997) , no. 5, 1425--1461. w/Dahlberg, B. E. J.; Kenig, C. E.; Verchota, G. C.

Littlewood-Paley estimates: some applications to elliptic boundary value problems. Partial differential equations and their applications (Toronto, ON, 1995), 221--238, CRM Proc. Lecture Notes, 12 , Amer. Math. Soc., Providence, RI, 1997. fieldsinst.dvi

Multiparameter operators and sharp weighted inequalities. Amer. J. Math. 119 (1997) , no. 2, 337--369. w/Fefferman, R.

Review of: Harmonic Analysis Techniques in Second Order Elliptic PDE, by Carlos Kenig, Bulletin of the AMS 33 (2) (1996), p. 229-236, review.pdf

The Neumann problem for elliptic equations with nonsmooth coefficients. II. A celebration of John F. Nash, Jr. Duke Math. J. 81 (1995) , no. 1, 227--250 (1996). w/Kenig, Carlos E.

Boundary value problems for higher order operators in Lipschitz and $C\sp 1$ domains. Fourier analysis and partial differential equations (Miraflores de la Sierra, 1992) , 297--309, Stud. Adv. Math., CRC, Boca Raton, FL, 1995.

Maximum principles for the polyharmonic equation on Lipschitz domains. Potential Anal. 4 (1995) , no. 6, 615--636. w/Verchota, G. C.

Dilation invariant estimates and the boundary Gårding inequality for higher order elliptic operators. Ann. of Math. (2) 142 (1995) , no. 1, 1--38. w/Verchota, Gregory C.

The Neumann problem for elliptic equations with nonsmooth coefficients. Invent. Math. 113 (1993) , no. 3, 447--509. w/Kenig, Carlos E.

A maximum principle for biharmonic functions in Lipschitz and $C\sp 1$ domains. Comment. Math. Helv. 68 (1993) , no. 3, 385--414. w/Verchota, G.

The Dirichlet problem in $L\sp p$ for the biharmonic equation on Lipschitz domains. Amer. J. Math. 114 (1992) , no. 5, 923--972. w/Verchota, Gregory

A martingale inequality related to exponential square integrability. Proc. Amer. Math. Soc. 118 (1993) , no. 2, 541--546.

The theory of weights and the Dirichlet problem for elliptic equations. Ann. of Math. (2) 134 (1991) , no. 1, 65--124. w/Fefferman, R. A.; Kenig, C. E.

Partial differential equations with minimal smoothness and applications. Proceedings of the IMA Participating Institutions Conference held at the University of Chicago , Chicago, Illinois, March 21--25, 1990. Edited by B. Dahlberg, E. Fabes, R. Fefferman, D. Jerison, C. Kenig and J. Pipher. The IMA Volumes in Mathematics and its Applications, 42. Springer-Verlag, New York, 1992. xii+220 pp. ISBN: 0-387-97774-0

Area integral estimates for the biharmonic operator in Lipschitz domains. Trans. Amer. Math. Soc. 327 (1991) , no. 2, 903--917. w/Verchota, Gregory

The $h$-path distribution of the lifetime of conditioned Brownian motion for nonsmooth domains. Probab. Theory Related Fields 82 (1989) , no. 4, 615--623. w/Kenig, Carlos E.

Hardy spaces and the Dirichlet problem on Lipschitz domains. Rev. Mat. Iberoamericana 3 (1987) , no. 2, 191--247. w/Kenig, Carlos E.

Oblique derivative problems for the Laplacian in Lipschitz domains. Rev. Mat. Iberoamericana 3 (1987) , no. 3-4, 455--472.

The oblique derivative problem on Lipschitz domains with $L\sp p$ data. Amer. J. Math. 110 (1988) , no. 4, 715--737. w/Kenig, Carlos E.

Bounded double square functions. Ann. Inst. Fourier (Grenoble) 36 (1986) , no. 2, 69--82.

Journé's covering lemma and its extension to higher dimensions. Duke Math. J. 53 (1986) , no. 3, 683--690.