The Mathematics Page
of
Thomas Bieske
University
of South Florida
I am currently an Associate Professor at the University of South Florida.
Before that, I was a NSF VIGRE Postdoctoral Fellow at the University of
Michigan working with the late Prof. Juha Heinonen and a Visiting Assistant
Professor at the University of Arkansas. I did my graduate work at the
University of Pittsburgh under Prof. Juan Manfredi. Here is a vita (current as of 4/23/21).
Personal information can be found here.
I am also moderator of the Facebook community
Avenue Carnot,
a community dedicated to disseminating news and information about analysis in metric spaces and subRiemannian geometry.
I can be reached via tbieske at mail dot usf dot edu .
The following is a list of papers with abstracts. The links are to pdf
versions of the paper.

On the Lie Algebra of polarizable Carnot groups.
Anal. and Math. Physics. (2020), 10, No. 80, 11pp.
Abstract: Balogh and Tyson (2002) established the concept of polarizable Carnot groups, which are Carnot groups for which proper polar coordinates can be constructed. They also show that the groups of Heisenbergtype are polarizable and present an open question as to which Carnot groups are polarizable. Here, we explore the hidden nuance in this question as we demonstrate that polarization is not just a property of the algebraic group law, it is also dependent upon the Lie Algebra structure. We explore generalized Heisenbergtype groups and demonstrate the impact of the Lie Algebra.
 Generalizations of the Drift Laplace Equation over the Quaternions in a Class of GrushinType Spaces
Joint with Keller Blackwell.
Submitted for publication.
Abstract: Beals, Gaveau, and Greiner in 1996 establish a formula for the fundamental solution to the Laplace equation with drift term in Grushintype planes. The first author and Childers in 2013 expanded these results by invoking a pLaplacetype generalization that encompasses these formulas while the authors explored a different natural generalization of the pLaplace equation with drift term that also encompasses these formulas. In both, the drift term lies in the complex domain. We extend these results by considering a drift term in the quaternion realm and show our solutions are stable under limits as p tends to infinity.
 Generalizations of the Drift Laplace Equation in the Heisenberg Group and a Class of GrushinType Spaces
Joint with Keller Blackwell.
Submitted for publication.
Abstract: We find fundamental solutions to pLaplace equations with drift terms in the Heisenberg group and Grushintype planes. These solutions are natural generalizations to the fundamental solutions discovered by Beals, Gaveau, and Greiner for the Laplace equation with drift term. Our results are independent of the results of Bieske and Childers, in that Bieske and Childers consider a generalization that focuses on a pLaplacetype equation while we primarily concentrate on a generalization of the drift term.
 Removability of a Level Set for Solutions of the p(x)Laplace Equation in the Heisenberg Group
Joint with Zachary Forrest and Robert Freeman.
Submitted for publication.
Abstract: We extend the proofs of Juutinen, Lukkari, and Parviainen (2010) for removability
of level sets for viscosity solutions to the p(x)Laplace equation in the Euclidean
environment to the Heisenberg Group H_n. We employ methods similar to Juutinen and Lindqvist (2004, 2005) for removability of level sets for viscosity solutions to the pLaplace equation in the Euclidean environment, combined with our arguments (2019) for the pLaplace equation in the Heisenberg Group H_n. Specifically, we show that if $1 < p(x) < \infty, \Omega \subset H_n$ is a domain, $p \in C^1_\text{sub}(\Omega)$ and $u\in C^1_\text{sub}(\Omega)$ is a viscosity solution to the p(x)Laplace equation in the set $\Omega\setminus \{\x\in\Omega: u(x)=0\}$, then u is a viscosity solution to the p(x)Laplace equation in $\Omega$.
 A Rad\'{o}type Theorem
for the pLaplace Equation in the Heisenberg Group
Joint with Zachary Forrest and Robert Freeman.
Submitted for publication.
Abstract: We extend the proofs of Juutinen and Lindqvist (2004, 2005) concerning
removability of level sets for viscosity solutions to the pLaplace
equation in the Euclidean environment to the Heisenberg Group H_n.
Specifically, we show that if $1< p < \infty, \Omega \subset H_n$ is a domain, and $u \in C_\text{sub}^1(\Omega)$ is pharmonic in the set $\Omega\setminus\{x \in \Omega:\,u(x)=0\}$, then $u$ is pharmonic in all of
$\Omega$.
 Correction to "A p(x) Poincar\'etype inequality for variable exponent Sobolev spaces with zero boundary values in Carnot groups"
Joint with Robert Freeman.
Anal. and Math. Physics. (2019), 9 (4), 16111612.
 On the pLaplace equation in a class of H\"{o}rmander Vector Fields
Joint with Robert Freeman.
Electron. J. Diff. Eqns. (2019), 2019 (35), 113.
Abstract: We find the fundamental solution to the $p$Laplace equation in a class of H\"{o}rmander vector fields that generate neither a Carnot group nor a Grushintype space. The singularity occurs at the subRiemannian points, which naturally corresponds to finding the fundamental solution of a generalized operator in Euclidean space. We then extend these solutions to a generalization of the $p$Laplace equation and use these solutions to find infinite harmonic functions and their generalizations. We also compute the capacity of annuli centered at the singularity.
 Equivalence of Weak and Viscosity Solutions to the p(x)Laplacian in Carnot Groups
Joint with Robert Freeman.
Anal. and Math. Physics. (2019), 9 (4), 15831610.
Abstract: We show the equivalence of weak and viscosity solutions to the p(x)Laplacian in Carnot groups, under certain natural restrictions on the function p(x). As a consequence, we obtain a comparison principle for viscosity solutions and thus uniqueness of viscosity solutions to the Dirichlet problem.
 A p(x)Poincar\'etype Inequality for Variable Exponent Sobolev Spaces with Zero Boundary Values in Carnot Groups
Joint with Robert Freeman.
Anal. and Math. Physics. (2018), 8 (2), 289308.
Abstract: We prove a p(x)Poincar\'etype inequality for variable exponent Sobolev spaces with zero boundary values in Carnot groups. We then establish the existence and uniqueness (up to a set of zero p(x)capacity) of a minimizer to the Dirichlet energy integral for the variable exponent case.
 The parabolic infiniteLaplace equation in Carnot groups
Joint with Erin Martin.
Mich. Math. J. (2016), 65 (3), 489509.
Abstract: By employing a Carnot parabolic maximum principle, we show existenceuniqueness of viscosity solutions to a class of equations modeled on the parabolic infiniteLaplace equation in Carnot groups. We show stability of solutions within the class and examine the limit as t goes to infinity.
 The $\infty(x)$equation in Grushintype Spaces
Electron. J. Diff. Eqns. (2016), 2016 (125), 113.
Abstract: We employ Grushin jets which are adapted to the geometry of Grushintype spaces to obtain the existenceuniqueness of viscosity solutions to the $\infty(x)$Laplace equation in Grushintype spaces. Due to the differences between Euclidean jets and Grushin jets, the Euclidean method of proof is not valid in this environment.
 The $\infty(x)$equation in Riemannian Vector Fields
Electron. J. Diff. Eqns. (2015), 2015 (164), 19.
Abstract: We employ Riemannian jets which are adapted to the Riemannian geometry to obtain the existenceuniqueness of viscosity solutions to the $\infty(x)$Laplace equation in Riemannian vector fields. Due to the differences between Euclidean jets and Riemannian jets, the Euclidean method of proof is not valid in this environment.
 The parabolic pLaplace equation in Carnot groups
Joint with Erin Martin.
Ann. Acad. Sci. Fenn. (2014), 39, 605623.
Abstract: By establishing a parabolic maximum principle, we show uniqueness of viscosity solutions to the parabolic pLaplace equation and then examine the limit as t goes to infinity. Additionally, we explore the limit as p goes to infinity.
 Generalizations of a LaplacianType Equation in the Heisenberg Group and a Class of GrushinType Spaces
Joint with Kristen Childers.
Proc. Amer. Math. Soc. (2014), 142 (3), 9891003
Abstract: Beals, Gaveau and Greiner (1996) find the fundamental solution to a 2Laplacetype equation in a class of subRiemannian spaces. This solution is related to the wellknown fundamental solution to the pLaplace equation in Grushintype spaces (BieskeGong, 2006) and the Heisenberg group (Capogna, Danielli, Garofalo, 1997). We extend the 2Laplacetype equation to a pLaplacetype equation. We show that the obvious generalization does not have desired properties, but rather, our generalization preserves some natural properties.
 A SubRiemannian Maximum Principle and
its application to the pLaplacian in Carnot Groups
Ann. Acad. Sci. Fenn. (2012), 37, 119134.
Abstract: We prove a subRiemannian maximum principle for semicontinuous functions. We apply this principle to Carnot groups to provide a ``subRiemannian" proof of the uniqueness of viscosity infinite harmonic functions and to establish the equivalence of weak solutions and viscosity solutions to the pLaplace equation. This result extends the author's previous work in the Heisenberg group.
 Fundamental solutions to PLaplace
equations in Grushin vector fields
Electron. J. Diff. Eqns. (2011), 2011 (84), 110.
Abstract: We find the fundamental solution to the PLaplace equation in
Grushintype spaces. The singularity occurs at the subRiemannian points,
which naturally corresponds to finding the fundamental solution of a
generalized Grushin operator in Euclidean space. We then use this solution
to find an infinite harmonic function with specific boundary data and to
compute the capacity of annuli centered at the singularity.
 The Infinite
Dirac Operator
Joint with John Ryan.
2010 J. Phys.: Conf. Ser. 254 012003.
Abstract: In this article, we define the infinite Dirac operator and explore some key properties, particularly its conformal invariance. En route, we also establish the conformal invariance of the pDirac equation. We also introduce the infinite Dirac operator on the sphere $S^{n}$ and establish the relationship between the two infinite Dirac operators via the Cayley transformation. Also we introduce an infinite Laplace operator on $S^{n}$.
 The CarnotCarath\'eodory distance visavis the
eikonal equation and the infinite Laplacian
Bull. London Math. Soc. (2010), 42 (3), 395404.
Abstract: In R^n equipped with the Euclidean metric, the distance from the
origin (smoothly) satisfies the eikonal equation and is (smoothly)
infinite harmonic everywhere except the origin. Dragoni (2007) has shown
that the CarnotCarath\'eodory distance satisfies the eikonal equation in
the viscosity sense outside of the origin but Bieske, Dragoni, Manfredi
(2008) have shown that the distance is not viscosity infinite harmonic at
all points outside the origin. We examine the behavior of the negative
distance function and show that it is a viscosity solution to the eikonal
equation exactly where it is viscosity infinite harmonic.

The CarnotCarath\'eodory
distance and the infinite Laplacian
Joint with Federica Dragoni and Juan Manfredi.
J. of Geo. Anal. (2009), 19 (4), 737754.
Abstract: In R^n equipped with the Euclidean metric, the distance from the
origin is smooth and infinite harmonic everywhere except the origin. Using
geodesics, we find a geometric characterization for when the distance from
the origin in an arbitrary CarnotCarath\'eodory space is a viscosity
infinite harmonic function at a point outside the origin. We show that at
points in the Heisenberg group and Grushin plane where this condition
fails, the distance from the origin is not a viscosity infinite harmonic
subsolution. In addition, the distance function is not a viscosity
infinite harmonic supersolution at the origin.
 Properties of Infinite Harmonic Functions on
Grushintype Spaces
Rocky Mtn J. of Math. (2009), 39 (3), 729756.
Abstract: In this paper, we examine potentialtheoretic and geometric
properties of viscosity infinite harmonic functions in Grushintype
spaces, which are subRiemannian spaces lacking a group structure. In
particular, we prove such functions enjoy comparison with Grushin cones.
As a consequence, the distance function is viscosity infinite
superharmonic, but we show that it is not necessarily viscosity infinite
subharmonic.
 Parabolic equations relative to vector
fields
Electron. J. Diff. Eqns. (2008), 2008 (124), 17.
Abstract: We define two notions of viscosity solutions to parabolic
equations defined using vector fields, depending on whether the test
functions concern only the past or both the past and the future. Using the
parabolic maximum principle, we then prove a comparison principle for a
class of parabolic equations and show the sufficiency of considering the
test functions that concern only the past.
 Properties of Infinite Harmonic Functions
relative to Riemannian Vector Fields
Le Matematiche (2008), LXIII (2), 1937.
Abstract: We employ Riemannian jets which are adapted to the Riemannian
geometry to obtain the existenceuniqueness of infinite harmonic functions
in Riemannian spaces. We then show such functions are equivalent to those
that enjoy comparison with Riemannian cones. Using comparison with cones,
we show that the Riemannian distance is a supersolution to the infinite
Laplace equation, but is not necessarily a solution. We find some
geometric conditions under which the Riemannian distance is infinite
harmonic and under which it fails to be infinite harmonic.
 A Comparison principle for a class of
subparabolic equations in Grushintype spaces
Electron. J. Diff. Eqns. (2007), 2007 (30), 19.
Abstract: We define two notions of viscosity solutions to subparabolic
equations in Grushintype spaces, depending on whether the test functions
concern only the past or both the past and the future. We then prove a
comparison principle for a class of subparabolic equations and show the
sufficiency of considering the test functions that concern only the past.
 The
PLaplace Equation on a class of Grushintype Spaces
Joint with Jasun Gong.
Proceedings Amer. Math. Soc. (2006), 134 (12), 35853594.
Abstract: We find the fundamental solution to the PLaplace equation in
Grushintype spaces. The singularity occurs at the subRiemannian points,
which naturally corresponds to finding the fundamental solution of a
generalized Grushin operator in Euclidean space. We then use this solution
to find an infinite harmonic function with specific boundary data and to
compute the capacity of annuli centered at the singularity. A solution to
the 2Laplace equation in a wider class of spaces is presented.
 Equivalence
of Weak and Viscosity Solutions to the PLaplace Equation in the
Heisenberg Group
Ann. Acad. Sci. Fenn. Math. (2006), 31, 363379.
Abstract: We prove weak and viscosity solutions to the PLaplace equation
in the Heisenberg group coincide. In particular, the viscosity
sub(super)solutions coincide with the potential theoretic
Psub(super)harmonic functions. We are then able to obtain a comparison
principle for the PLaplacian.
 Comparison principle for parabolic
equations in the Heisenberg group
Electron. J. Diff. Eqns. (2005), 2005 (95), 111.
Abstract: We prove a comparison principle for a class of parabolic
equations in the Heisenberg group and show the sufficiency of considering
test functions that concern only the past.
 The Maximum Principle for Vector Fields
Joint with Frank Beatrous and Juan Manfredi.
Contemp. Math. 370, Amer. Math. Soc., Providence, RI, 2005. 19.
Abstract: We present an extension of Jensen's uniqueness theorem for
viscosity solutions of second order partial differential equations to the
case of equations generated by vector fields.
 The
AronssonEuler equation for Absolutely minimizing Lipschitz extensions
with respect to CarnotCarath\'eodory metrics
Joint with Luca Capogna.
Trans AMS (2005), 357 (2), 795823.
Abstract: We derive the EulerLagrange equation (also known in this
setting as the AronssonEuler equation) for absolute minimizers of the
L^{\infty} variational problem ``inf \nabla_0 u_{L^{\infty}(Omega)''
subject to the condition that u=g is Lipschitz on the boundary of Omega,
where Omega is an open subset of a Carnot group, \nabla_0 u denotes the
horizontal gradient of a realvalued function u on Omega, and the
Lipschitz class is defined in relation to the CarnotCarath\'eodory
metric. In particular we show that absolute minimizers are infinite
harmonic in the viscosity sense. As a corollary we obtain the uniqueness
of absolute minimizers in a large class of groups. This result extends previous
work of Jensen (1993) and Crandall, Evans and Gariepy (2001). We also
derive the AronssonEuler equation for more ``regular" absolutely
minimizing Lipschitz extensions corresponding to those
CarnotCarath\'eodory metrics which are associated to ``free" systems
of vector fields.
 Lipschitz Extensions on generalized Grushin spaces
Mich. Math J. (2005), 53 (1), 331.
Abstract: In Viscosity
solutions on Grushintype planes , viscosity solutions to a class of
nonlinear differential equations are defined and Euclidean results are
extended to Grushintype planes, a subRiemannian environment without a
group structure. In this paper, we examine the same class of equations but
now consider generalized Grushintype spaces of higher dimension. In
addition, we show that C^1_{sub} absolute minimizers are viscosity
infinite harmonic.
 Absolute Minimizers on Carnot Groups.
Future Trends in Geometric Function Theory. RNC Workshop.
Jyv\"askyl\"a 2003.
University of Jyvaskyla, Dept. of Mathematics and
Statistics, Report 92, 1521.
Abstract: In this note, we extend the concepts of viscosity solutions and
absolute minimizers to the setting of Carnot groups. In particular, the
existenceuniqueness of infinite harmonic functions in the viscosity sense
and the relationship between absolute minimizers and infinite harmonic
functions are discussed. As a consequence, the uniqueness of absolute
minimizers follows.

Viscosity
solutions on Grushintype planes
Illinois J. Math. (2002), 46, 893911.
Abstract: This paper examines viscosity solutions to a class of fully
nonlinear equations on Grushintype planes. First, viscosity solutions are
defined, using subelliptic second order superjets and subjets. Then, a
Grushin maximum principle is proved, and as an application, comparison
principles for certain types of nonlinear functions follow. This is
accomplished by establishing a natural relationship between Euclidean and
subelliptic jets, in order to use the viscosity solution technology of
Crandall, Ishii, and Lions (1992). The particular example of infinite
harmonic functions on certain Grushintype planes is examined in further
detail.
 On
Infinite Harmonic Functions on the Heisenberg Group
Comm. PDE. (2002), 27 (3&4), 727761.
Abstract: This paper examines infinite harmonic functions in the viscosity
sense on the Heisenberg group by extending Aronsson's concept of Absolute
Minimizing Lipschitz Extensions (1967) to the Heisenberg group. Existence
of infinite harmonic functions in the viscosity sense is proved following
the scheme of Bhatthacharya, DiBenedetto, and Manfredi (1989). Uniqueness
of infinite harmonic functions is proved using an extension of Jensen's
proof (1993). Both the existence and uniqueness proofs utilize the concept
of subelliptic jets. By establishing a natural relationship between
Euclidean and subelliptic jets, the technology of viscosity solutions
found in Crandall, Ishii, and Lions (1992) can be used.