Home page of Dr. Steve Hofmann

Info for Students FS 2017

Math 2300

*Calendar

*Homework

*Syllabus 2300

___________________________________*Syllabus 8630

Math 8630

*vita

**Research interests:**
Harmonic analysis with applications to partial differential
equations and geometric measure theory.

__Note: this site is
under construction. Only items marked with * are linked to a file!__

NOTE: The
author acknowledges support of the National Science
Foundation (currently grant number DMS-1664047)

Selected Publications and Preprints

Recent Preprints and Publications

(with P. Le, J. M. Martell, and K. Nystr\"om), Anal. PDE 10 (2017), no. 3, 513–558.

__*Absolute
Contin uity between the surface measure and harmonic measure implies rectifiability__(with J. M. Martell,
S. Mayboroda, X. Tolsa and A. Volberg), preprint

__*Rectifiability of harmonic
measure
__(with J. Azzam, J.
M. Martell, S. Mayboroda, M. Mourgoglou, X. Tolsa and A.
Volberg), Geom. Funct. Anal. 26 (2016), no.
3, 703–728.

Duke Math. J. 165 (2016), no. 12, 2331–2389

(with C. Kenig, S. Mayboroda, and J. Pipher), Math. Annalen. 361 (2015) 863-907.

*Uniform Rectiﬁability and Harmonic Measure I, (with J. M. Martell), Ann. Sci. Éc. Norm. Supér. 47 (2014) 577-654.

*Uniform Rectiﬁability and Harmonic Measure II: Poisson kernels in Lp imply uniform rectﬁability,

(with J. M. Martell and I. Uriarte-Tuero), Duke Math. J. 163 (2014) 1601-1654.

*Uniform Rectiﬁability and Harmonic Measure III: Riesz transform bounds imply uniform rectﬁability of boundaries of 1-sided NTA domains,

(with J. M. Martell and S. Mayboroda) International Mathematics Research Notices 2014 (2014) 2702-2729.

*(with C. Kenig, S. Mayboroda, and J. Pipher) Square function/Non-

tangential maximal function estimates and the Dirichlet problem for non-

symmetric elliptic operators, Jour. Amer. Math. Soc. 28 (2015), 483-529.

with M. Alfonseca, P. Auscher, A. Axelsson and S. Kim, Adv. Math. 226 (2011), no. 5, 4533–4606.

*Singular integrals and elliptic boundary problems on regular Semmes-Kenig-Toro domains, with M. Mitrea and M. Taylor,

with J. M. Martell, Trans. Amer. Math. Soc. 364 (2012), no. 1, 65–101.

*Hardy spaces and regularity for the inhomogeneous Dirichlet and Neumann problems, with X. T. Duong, D. Mitrea, M. Mitrea and L. Yan,

Rev. Mat. Iberoam. 29 (2013), no. 1, 183–236

*Hardy spaces, singular integrals and the geometry of Euclidean domains of locally finite perimeter, with E. Marmolejo-Olea, M. Mitrea,

S. Perez-Esteva and M. Taylor,

*Hardy spaces associated to non-negative self-adjoint operators satisfying Davies-Gaffney estimates, with G. Lu, D. Mitrea, M. Mitrea and L. Yan,

Mem. Amer. Math. Soc. 214 (2011), no. 1007, vi+78 pp. ISBN: 978-0-8218-5238-5

*$L^p-$bounds for the Riesz transforms associated with the Hodge Laplacian in Lipschitz subdomains of Riemannian manifolds, with M. Mitrea and S. Monniaux,

Ann. Inst. Fourier (Grenoble) 61 (2011), no. 4, 1323–1349 (2012)

*A local

*Second order elliptic operators with complex bounded measurable coefficients in $L^p$, Sobolev and Hardy spaces, with S. Mayboroda and A. McIntosh,

Ann. Sci. Éc. Norm. Supér. (4) 44 (2011), no. 5, 723–800.

*The solution of the Kato problem in two dimensions, with A. McIntosh, Proceedings of the conference on harmonic analysis and PDE held at El Escorial, June 2000, Publ. Mat. Vol. extra, 2002 pp. 143-160.

Papers on the Kato problem

For survey articles on this topic see below.

*The Solution of the Kato Problem for Divergence Form Elliptic Operators with Gaussian Heat Kernel Bounds, with M. Lacey and A. McIntosh, Annals of Math. 156 (2002), pp 623-631.

*The solution of the Kato square root problem for elliptic operators on $\mathbb{R}^n$, with P. Auscher, M. Lacey, A. McIntosh and P. Tchamitchian, Annals of Math. 156 (2002), pp 633-654.

*Extrapolation of Carleson measures and the analyticity of Kato's square root operators, with P. Auscher, J. L. Lewis, and P. Tchamitchian, Acta Math. 187 (2001), pp 161-190.

*The Kato square root problem
for higher order elliptic operators and systems on ${\Bbb
R}^n$, with P. Auscher, A. McIntosh, and P.
Tchamitichian, dedicated to the memory of Tosio Kato. J. Evol.
Equ. 1 (2001) no. 4 pp. 361-385.

**$L^p$ estimates for Riesz
transforms**

*
For
survey
articles on this topic see below
*

*Riesz transforms on
manifolds and heat kernel regularity, with P. Auscher, T. Coulhon and
X. Duong, Annales Scientifiques de L'ENS (4) 37 (2004),
911-957.

*$L^p$ bounds for Riesz transforms and square roots associated to second order elliptic operators, with J. Martell, Pub. Mat. 47 (2003), 497-515.

**
Decay of Fourier
transforms and applications
**

__*Circular averages and
Falconer/Erdos distance conjecture in the plane for random
metrics,__ with A. Iosevich, PAMS
** 133 ** (2005), 133--143.

*Lattice points
inside random ellipsoids, with A. Iosevich and D.
Weidinger, Mich. Math. J. 52 (2004), pp. 13-21.

*Sharp rate of decay
of the Fourier transform of a bounded set, with L.
Brandolini, and A. Iosevich, GAFA 13 (2003), 671-680.

**Parabolic and elliptic equations and SIOs**

*For survey articles on
this topic see below.*

*Hardy and BMO spaces associated to divergence form elliptic operators, with S. Mayboroda, to appear, Math. Ann. 344 (2009) 37-116.

*Functional calculus of
Dirac operators and complex perturbations of Neumann and Dirichlet problems,

with P. Auscher and A. Axelsson, JFA 255 (2008), 374-448.

*Dahlberg's bilinear estimate for solutions of divergence form complex elliptic equations, Proc. Amer. Math. Soc. 136 (2008), 4223-4233.

*Geometric
and
Transformational
Properties
of
Lipschitz Domains, Semmes-Kenig-Toro
Domains, and Other Classes of Finite Perimeter Domains,

with M. Mitrea and M. Taylor, J. Geom. Analysis 17
(2007), 593-647.

*The Green function estimates for strongly elliptic systems of second order, with S.Kim, Manuscripta Math. 124 (2007), pp 139-172.

*The $L^p$ Neumann
problem for the heat equation in noncylindrical domains,
with J.L. Lewis, J. Functional Analysis, Vol 220 (2005),
pp. 1-54.

*Gaussian
estimates for fundamental solutions
to certain parabolic systems, with Seick Kim, Pub. Mat. 48
(2004), 481-496.

*Caloric Measure
in Parabolic Flat Domains, with J. L. Lewis and K.
Nystrom, Duke Math. J. 122 (2004), no. 2, 281--346.

__*Existence of big pieces of
graphs for parabolic problems, __ with
J. L. Lewis and K. Nystrom, Ann. Acad. Sci. Fenn. Math.
28 (2003), 355--384.

*Spectral properties of parabolic layer potentials and transmission boundary problems in nonsmooth domains, with J. L. Lewis and M. Mitrea, Ill. J. Math. 47 (2003), no. 4, 1345--1361.

*The Dirichlet problem for parabolic operators with singular drift terms, with J. Lewis, Mem. Amer. Math. Soc. 151 (2001) no. 719.

*Square functions of Calderon type and applications, with J. Lewis, Rev. Mat. Iberoamericana 17 (2001) no.1 pp.1-20.

A generalized characterization of commutators of parabolic singular integrals, with X. Li and D. Yang, Canad. Math. Bull. 42 (1999) no.4 pp. 463-477.

*The $L^p$ regularity problem for the heat equation in non-cylindrical domains, with J. Lewis, Illinois Math. J. 43 (1999) no. 4 pp. 752-769.

The Calderon commutator along a parabola, with A. Carbery and J. Wright, Math. Proc. Camb. Phil. Soc. 126 (1999) no.3 pp. 543-553.

An off-diagonal $T(1)$ theorem and applications. With an appendix "The Mary Weiss lemma" by Loukas Grafakos and the author. J. Functional Analysis 160 (1998) no.2 pp. 581-622.

*Parabolic singular integrals of Calderon-type, rough operators, and caloric layer potentials, Duke Math. J. 90 (1997) no.2 pp. 209-259.

*$L^2$ solvability and representation by caloric layer potentials in time varying domains, with J. L. Lewis, Annals of Math. 144 (1996), 349-420.

Boundedness criteria for rough singular integrals, Proc. London Math. Soc. (3) 70 (1995) no.2 pp.386-410.

On singular integrals of Calderon-type in ${\Bbb R}^n$, Rev. Mat. Iberoamer. 10 (1994) no.3 pp. 467-505.

A characterization of commutators of parabolic singular integrals, Fourier analysis and partial differential equations (Miraflores de la Siera, 1992) Stud. Adv. Math. CRC, Boca Raton, FL (1995) pp. 195-210.

On certain non-standard Calderon-Zygmund operators, Studia Math. 109 (1994), no.2 pp. 105-131.

Weighted norm inequalities and vector valued inequalities for certain rough operators, Indiana Univ. Math. J. 42 (1993) no.1 pp.1-14.

A note on weighted Sobolev spaces and regularity of commutators and layer potentials associated to the heat equation, Proc. Amer. Math. Sci. 118 (1993) no.4 pp.1087-1096.

Weighted inequalities for commutators of rough singular integrals, Indiana Univ. Math. J. 39 (1990) no.4 pp. 1275-1304.

Singular integrals with power weights, Proc. Amer. Math. Soc. 110 (1990) no.2 pp. 343-353.

Weighted weak-type (1,1) inequalities for rough operators, Proc. Amer. Math. Soc. 107 (1989) no.2 pp. 423-435.

Weak (1,1)
boundedness of singular integrals with non-smooth kernel,
Proc. Amer. Math. Soc. 103 (1988) no.1 pp. 260-264.

**Expository and Survey
articles**

*(with
A. McIntosh) Boundedness and applications of singular integrals and square functions: a survey,*
Bull. Math. Sci.*

*A proof of the local
$Tb$ Theorem for standard Calderon-Zygmund operators,

(unpublished manuscript)

*Local Tb Theorems and applications in PDE, Proceedings of the ICM Madrid 2006.

*Local *
Harmonic analysis and partial differential equations, *
29–52, Contemp.
Math.,
505, *Amer. Math. Soc., Providence, RI,* 2010

*Carleson Measures,
Trees, Extrapolation, and Tb Theorems, with P. Auscher, C.
Muscalu, T. Tao, C. Thiele,

Pub. Math. 46 (2002), 257-325

The solution of Kato's conjectures, with P. Auscher, M. Lacey, J. Lewis, A. McIntosh and P. Tchamitchian, C. R. Acad. Sci. Paris Ser. I Math. 332 (2001) no.7, pp. 601-606.

*A short course on the Kato problem, Contemp. Math. 289 (2001) pp. 61-77.

The solution of the Kato problem, Contemp. Math. 277 Amer. Math. Soc. Providence, RI (2001).

On parabolic and elliptic measure, with J. Lewis, Acta Univ. Upsaliensis Skr. Uppsala Univ. C Organ. Hist. 64 (1999) , 179-186.

__The $L^p$ Neumann
and regularity problems for the heat equation in
non-cylindrical domains, __ with J. L.
Lewis,

Journées "Équations aux Dérivées Partielles"
(Saint-Jean-de-Monts, 1998), Exp. No. VI, 7 pp., Univ.
Nantes, Nantes, 1998.

*Heat kernels and
Riesz Transforms, Cont. Math. 398
(2006).