Michael GoldbergProfessor Department Head | |

Office: 4114 French Hall West
Phone: 513-556-4052 Fax: 513-556-3417 E-mail: Michael . Goldberg @ uc . edu |
Mailing address:
Dept. of Mathematical Sciences University of Cincinnati Cincinnati, OH 45221-0025 |

**Research Summary**

I use techniques from Fourier analysis to study partial differential
equations.

(I also use partial differential equations as an excuse to do Fourier
analysis!)

My current projects involve linear dispersive estimates for
the Schrödinger,

wave and/or Dirac equations with a short-range potential, plus some more

classical Fourier Transform bounds.

The **13 ^{th} Ohio River Analysis Meeting** will take place March 16-17, 2024 at the University of Kentucky!

Homepages for previous meetings: 2023, 2022, 2021, 2019, 2018, 2017, 2016, 2015, 2014, 2013, 2012, 2011.

These slides from some of my past talks
discuss specific problems and results.

Since 2013 the work is supported by the Simons Foundation under
grants #281057 and #635369.

Courses taught at UC:

- Math 1062 - Calculus II [Spring 2020]
- Math 2063 - Multivariable Calculus [Fall 2016] [Spring 2019] [Spring 2021]
- Math 2073 - Ordinary Differential Equations [Fall 2017] [Fall 2018]
- Math 2074 - Dynamical Systems [Spring 2015]
- Math 6003 - Abstract Linear Algebra [Fall 2022] [Fall 2023]
- Math 6007 - PDEs and Fourier Analysis [Spring 2016]
- Math 7004 - Topology
- Math 7005 - Ordinary Differential Equations [Fall 2016] [Fall 2019]
- Math 7006 - Partial Differential Equations [Spring 2015] [Spring 2018]
- Math 8003 - Functional Analysis [Fall 2015]
- Math 251 - Calculus I
- Math 252 - Calculus II
- Math 253 - Calculus III
- Math 627-629 - Partial Differential Equations

**Publications and Preprints**

Due to copyright restrictions, some papers hosted on this site appear in
their (unrevised) **preprint version**.

For access to the peer-reviewed version of record, please click on the journal
title to visit the publisher's homepage.

- Dispersive estimates for higher order Schrödinger operators with scaling-critical potentials (with M. B. Erdogan and W. Green), preprint 2023. [pdf]
- Spectral multipliers and wave propagation for Hamiltonians with a scalar potential (With M. Beceanu),
J. Funct. Anal.
**286**(2024), no. 6, Article 110300. [pdf] - Counterexamples to L
^{p}boundedness of wave operators for classical and higher order Schröodinger operators (with M. B. Erdogan and W. Green) J. Funct. Anal.**285**(2023), no. 5, Article 110008. [pdf] - On the L
^{p}Boundedness of the Wave Operators for Fourth Order Schrödinger Operators (with W. Green), Trans. Amer. Math. Soc.**374**(2021), 4075-4092. [pdf] - Time Integrable Weighted Dispersive Estimates for the Fourth
Order Schrödinger Equation in Three Dimensions
(with W. Green),
Bull. London Math. Soc.
**54**(2022), no. 2, 428-448. [pdf] - Strichartz Estimates for the Schrödinger Equation
with a Measure-Valued Potential
(with M. B. Erdogan and
W. Green),
Proc. Amer. Math. Soc. Ser. B,
**8**(2021), 336-348. [pdf] - Restrictions of Higher Derivatives of the Fourier Transform
(with D. Stolyarov),
Trans. Amer. Math. Soc. Ser. B
**7**(2020), 46-96. [pdf] - The Massless Dirac Equation in Two Dimensions:
Zero-Energy Obstructions and Dispersive Estimates
(with M. B. Erdogan and
W. Green),
J. Spectr. Theory
**11**(2021), no. 3, 935-979. [pdf] - Limiting Absorption Principle and Strichartz Estimates for
Dirac Operators in Two and Higher Dimensions
(with M. B. Erdogan and
W. Green),
Comm. Math. Phys.
**367**(2019), no. 1, 241-263. [pdf] - On the L
^{p}Boundedness of Wave Operators for Two-Dimensional Schrödinger Operators with Threshold Obstructions (with M. B. Erdogan and W. Green), J. Funct. Anal.**274**(2018), no. 7, 2139-2161. [pdf] - On the L
^{p}Boundedness of Wave Operators for Four-Dimensional Schrödinger Operators with a Threshold Eigenvalue (with W. Green), Ann. Henri Poincaré**18**(2017), no. 4, 1269-1288. [pdf] - The L
^{p}Boundedness of Wave Operators for Schrödinger Operators with Threshold Singularities (with W. Green), Adv. Math.**303**(2016), 360-389. [pdf] - The Helmholtz Equation with L
^{p}Data and Bochner-Riesz Multipliers. Math. Res. Lett.**23**(2016), no. 6, 1665-1679. [pdf] - Dispersive Estimates for Higher Dimensional Schrödinger
Operators with Threshold Eigenvalues I: The Odd Dimensional Case (with
W. Green).
J. Funct. Anal.,
**269**(2015), no. 3, 633-682. [pdf] - Dispersive Estimates for Higher Dimensional Schrödinger
Operators with Threshold Eigenvalues II: The Even Dimensional Case (with
W. Green),
J. Spectr. Theory
**7**(2017), no. 1, 33-86. [pdf] - Dispersive Estimates for Four Dimensional Schrödinger
and Wave Equations with Obstructions at Zero Energy (with
M. B. Erdogan and
W. Green).
Comm. PDE,
**39**(2014), no. 10, 1936-1964. [pdf] - The Klein-Gordon Equation on
**Z**^{2}and the Quantum Harmonic Lattice (with V. Borovyk). J. Math. Pures Appl. (9)**107**(2017), no. 6, 667-696. [pdf] - Strichartz Estimates and Maximal Operators for the Wave
Equation in
**R**^{3}(with M. Beceanu). J. Funct. Anal.**266**(2014), no. 3, 1476-1510. [pdf] - Dispersive Estimates for Schrödinger Operators with
Measure-Valued Potentials in
**R**^{3}. Indiana Univ. Math. J.**61**(2012), no. 6, 2123-2141. [pdf] - Schrödinger Dispersive Estimates for a Scaling-Critical
Class of Potentials (with
M. Beceanu),
Comm. Math. Phys.
**314**(2012), no. 2, 471-481. [pdf] - A Dispersive Bound for Three-Dimensional Schrödinger
Operators with Zero Energy Eigenvalues,
Comm. PDE
**35**(2010), 1610-1634. [pdf] - Strichartz Estimates for Schrödinger Operators with
a Non-Smooth Magnetic Potential,
Discrete Contin. Dyn. Syst.
**31**(2011), no. 1, 109-118. [pdf] - Strichartz Estimates for the Schrödinger Equation with
Time-Periodic L
^{n/2}Potentials, J. Funct. Anal.**256**(2009), 718-746. [dvi] [ps] [pdf] - Strichartz and Smoothing Estimates for Schrödinger
Operators with Almost Critical Magnetic Potentials in Three and Higher Dimensions
(with M. B. Erdogan and
W. Schlag),
Forum Math.
**21**(2009), no. 4, 687-722. [dvi] [ps] [pdf] - Strichartz and Smoothing Estimates for Schrödinger
Operators with Large Magnetic Potentials in
**R**^{3}(with M. B. Erdogan and W. Schlag), J. Eur. Math. Soc.**10**(2008), no. 2, 507-531. [dvi] [pdf] - Transport in the One-Dimensional Schrödinger Equation,
Proc. Amer. Math. Soc.
**135**(2007), 3171-3179. [pdf] - Counterexamples of Strichartz Inequalities for Schrödinger
Equations with Repulsive Potentials (with
L. Vega and
N. Visciglia),
Intl. Math. Res. Not.
**2006**(2006), Article ID 13927, 16pp. [dvi] [pdf] - A Counterexample to Dispersive Estimates for Schrödinger
Operators in Higher Dimensions (with
M. Visan),
Comm. Math. Phys.
**266**(2006), no. 1, 211-238. [dvi] [pdf] - Dispersive Bounds for the Three-Dimensional Schrödinger
Equation with Almost Critical Potentials,
Geom. and Funct. Anal.
**16**(2006), no. 3, 517-536. [dvi] [ps] [pdf] - Dispersive Estimates for the Three-Dimensional Schrödinger
Equation with Rough Potentials,
Amer. J. Math.
**128**(2006) 731-750. [dvi] [ps] [pdf] - A Limiting Absorption Principle for the Three-Dimensional
Schrödinger Equation with
*L*Potentials (with W. Schlag), Intl. Math. Res. Not.^{p}**2004:75**(2004), 4049-4071. [dvi] [ps] [pdf] - Dispersive Estimates for Schrödinger Operators in
Dimensions One and Three (with
W. Schlag),
Comm. Math. Phys.
**251**(2004), no. 1, 157-178. [dvi] [ps] [pdf] - Matrix
*A*Weights via Maximal Functions, Pac. J. Math._{p}**211**(2003), 201-220. [pdf] - Asymptotic Properties of the Vector Carleson Embedding Theorem,
Proc. Amer. Math. Soc.
**130**(2002), 529-531. [pdf] - Vector
*A*_{2}Weights and a Hardy-Littlewood Maximal Function (with M. Christ), Trans. Amer. Math. Soc.**353**(2001), 1995-2002. [pdf]

**Education**

AB., Mathematics,
Princeton University, 1997

Ph.D., Mathematics, University of
California, Berkeley, 2002

Here is my full
**Curriculum Vitae
**.

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