# FPAS-2001-10-19

# First Prairie Analysis Seminar

October 19-20, 2001

### Kansas State University - Manhattan, Kansas

## Featured Speaker:## Prof. Fang-Hua Lin, Courant Institute, NYU."These two lectures will address some analytical and topological issues concerning Sobolev mappings betweenmanifolds.We shall discuss local and global topological obstructions for smooth maps to be strongly dense or to be weakly sequentially dense in Sobolev mapping spaces,and the topology of Sobolev mappings.We shall also outline a generalized varifold theory for Sobolev mappings and their applications in variational problems." Invited Speakers: ## Prof. Robert Hardt, Rice UniversitySize Minimization and Approximating Problems(Thierry De Pauw (Orsay) and Robert Hardt (Rice)) "A k dimensional rectifiable current is given by an oriented k ## Prof. Yisong Yang , Poltytechnic Unversity and IAS (Princeton):Nonlinear Problems in Born-Infeld Theory "In 1933, M. Born and L. Infeld developed a geometric theory of electromagnetism to accommodate a finite-energy |

# Schedule

### Friday afternoon:

One 1 hour lecture by Fang-Hua Lin, one 1 hour invited lecture, and several 25 minute contributed talks.

### Saturday:

Morning: One 1 hour invited lecture and several 25 minute contributed talks.

Afternoon: One 1 hour lecture by Fang-Hua Lin and several 25 minute contributed talks.

All talks will be in Cardwell Hall, room to be announced.## Contributed Speakers and Titles of their talks:

David Auckly (Kansas State University)Title: A framework for analysis on noncompact quasiconformal 4-manifolds.

Sun-Sig Byun (University of Iowa)

Title: Geometric approach to W^1,p estimates.

Luca Capogna (University of Arkansas)

Title: Wave maps with target in the Heisenberg group.

Thierry de Pauw (Universite de Paris-Sud, and Rice University)

Title: Nearly flat almost monotone measures are big pieces of Lipschitz graphs.

Brian Hollenbeck (Emporia State University)

Title: Best Constants for Operators Involving the Hilbert Transform.

Yaozhong Hu (University of Kansas)

Title: On Logaritmic Sobolev and some other inequalities.

Frank Jochmann (Universität Leipzig)

Title: Asymptotic behavior of solutions to nonlinear polarization models.

Lev Kapitanski (Kansas State University)

Title: $S^3$ and $S^2$ nonlinear $\sigma$-models.

Gocha Lepsveridze (Telavi State University, Georgia)

Title: The Rate of Growth of Integral Means from Orlicz Clases.

Mircea Martin (Baker University)

Title: Multidimensional Generalizations of Alexander's Inequality.

Gabriel Nagy (Kasas State University)

Title: Idemptotents in finite AW* factors.

Richard Rochberg (Washington University, St. Louis)

Title: Hankel and Schrodinger Forms on Dyadic Trees.

Sharon Schaffer Vestal (Missouri Western State College)

Title: Using Functional Analysis to relate a wavelet GMRA to a

multiwavelet MRA.

Eric Weber (Texas A & M University)

Title: Frame Representations of Groups and Sampling Theory.

Shihshu Walter Wei (Oklahoma University)

Title: On the structure of minimal submanifolds in nonpositively curved manifolds.

Karen Yagdjian (Kansas State University)

Title: Parametric resonance and the global solutions to nonlinear hyperbolic equations.

# Abstracts:

David Auckly

Title: A framework for analysis on noncompact quasiconformal 4-manifolds.

Abstract: This talk will report on analytical parts of a project joint

with V. Kapovitch. An Alexandrov metric is a special structure that is

studied by differential geometers. The existance question for Alexandrov

metrics leads to the need for a gauge theory that would be applicable to

non-compact quasiconformal 4-manifolds. In order to define a gauge

theoretic moduli space, one needs to have a function space that

compactly embeds into the continous functions, and is well defined when

one considers the overlap maps. In 4 dimensions, the Sobolev space W^4_1,

does not embed into the continous functions. On the other hand, the

spaces, W^p_1 are not preserved by quasiconformal overlap maps for p>4.

We define a weighted modified Sobolev space sitting between W^4_1 and each

W^p_1. As an application of the weighted, modified Sobolev space, we will

prove that any solution to a certain elliptic partial differential

equation that grows slower than a given exponential function, must in

fact decay exponentially.

Sun-Sig Byun

Title: Geometric approach to W^1,p estimates.

Abstract: We consider a Neumann problem for divergence form elliptic

equations with discontinuous coefficients. We will prove a W^1,p

esimate by a geometric approach which employs varified Vitali's

covering lemma, Hardy-Littlewood maximal function and compactness

method. Our methods can be easily extended to parabolic

equations.

Luca Capogna

Title: Wave maps with target in the Heisenberg group

Abstract: This is a joint project with Jalal Shatah (Courant/NYU)

concerning well posedness of the Cauchy problem for wave maps with

target in the Heisenberg group. Such maps are solution of systems of

quasilinear wave equations and satisfy differential constraints in the

form of transport equations with rough coefficients. We can show local

well-posedness for a large class of initial data.

Thierry de Pauw

Title: Nearly flat almost monotone measures are big pieces of Lipschitz graphs.

Abstract: Mass minimizing integral currents and stationary varifolds have strong

regularity properties. For instance their support contains an open dense

set which is an embedded smooth submanifold of the ambient space. The

question asked in this talk is to what extent these regularity properties

depend only upon some monotonicity property. We show that monotonicity

implies partial regularity of the supports.

Brian Hollenbeck

Title: Best Constants for Operators Involving the Hilbert Transform.

Abstract: We calculate sharp constants in inequalities of the form, $\|Sf\|_{L^p} \le

C_p\|f\|_{L^p}$, $1< p < \infty$, where $f$ is a complex-valued function and

$S$ is an operator involving the Hilbert transform, $H$, and the

identity operator, $I$. This is equivalent to finding the $L^p$-norm of

$S$.

In particular, we calculate the norm of $aI + bH$, where $a, b \in \R$.

We also prove $\|(I + iH)f\|_{L^p} \le 2 \csc \pip \|f\|_{L^p}$ for

$1 < p < \infty$. This immediately

gives the norm of the Riesz projection to be $\csc \pip$, solving a conjecture

made by Gohberg and Krupnik in 1968 and a problem posed by Pe\l czy\'nski

in 1985.

Yaozhong Hu

Title: On Logaritmic Sobolev and some other inequalities.

Abstract: In this talk, I will present the Nelson's

hypercontractivity inequality, Logarithmic

Sobolev inequality and the motivation

for this inequalities. I will also present

a more general inequality which includes

logarithmic Sobolev inequality, Poincare

inequality as its particular cases.

A simple proof of this general inequality

will also be given.

Some related inequalities such as Meyer's inequality,

Interpolation inequality, correlation inequality

will also presented. The correlation conjecture

will also be presented.

Frank Jochmann

Title: Asymptotic behavior of solutions to nonlinear polarization models.

Abstract: This talk is concerned with the Maxwell-Bloch system and the anharmonic

oscillator model describing the electromagnetic field

in polarizable media. The main subject is the asymptotic behavior of the

solutions to these models, in particular decay properties and

covergence to stationary states.

Lev Kapitanski

Title: $S^3$ and $S^2$ nonlinear $\sigma$-models.

Abstract: The Skyrme model (1961) was one of the first attempts to describe

elementary particles as localized in space solutions of nonlinear PDEs.

The fields take their values in SU(2)=S^3 and stabilize at spatial infinity.

Thus, the configuration space splits into different sectors (homotopy classes)

with a constant integer topological charge (the degree) in each sector.

Faddeev's model (1975) was designed to provide additional internal structure

(knottedness) to the localized solutions. The fields take their values

in the two-dimensional sphere and the topological charge is the Hopf invariant.

I will discuss some old and new results for these models.

Gocha Lepsveridze

Title: The Rate of Growth of Integral Means from Orlicz Clases.

Abstract: Let $f\in L({\bf R}^n )$ be any function.

For every $x\in {\bf R}^n$

we consider integral means $1/|I|\int_{I} f$, where

$I$ is an $n$ dimensional interval

in ${\bf R}^n$. We obtain the certain weak type

maximal inequalities from which

are derived some exact estimates on growth order of

these means for functions from

Orlicz classes $L\Phi(L)({\bf R}^2)$ . In

particular,

for any function $f\in L\log_+^{\beta}({\bf

R}^2),0<\beta<1$ the expression

$\frac{1}{|I|}\int_I

f/\log^{1-\beta}\Big(\frac{1}{M(I)}\Big)$ tends to $0$

as

$\diam(I)\to 0$, where $M(I)$ denotes the length of

the biggest side of the interval $I$.

Estimates of growth order of Multiple Fourier series

are implied.

Mircea Martin

Title: Multidimensional Generalizations of Alexander's Inequality.

Abstract: Alexander's inequality says that if

$\Omega$ is a compact set in the complex plane and

$C(\Omega)$ is the Banach algebra of complex-valued

continuous functions on $\Omega$, then

$$ {\rm dist}_{C(\Omega)}[\bar{z},R(\Omega)]\leq\left[\frac{1}{\pi}{\rm

area}(\Omega)\right]^{1/2}, $$

where $\bar{z}$ is the complex conjugate coordinate

function, and $R(\Omega)$ stands for the uniform closure in

$C(\Omega)$ of rational functions that are analytic on open

neighborhoods of $\Omega$.

This inequality with many interesting applications is

just a quantitative form of the classical Hartogs-Rosenthal

theorem: Whenever ${\rm area}(\Omega)=0$, it follows that

$R(\Omega)$ is a subalgebra of $C(\Omega)$ that contains $z$

and $\bar{z}$, and by the Stone-Weierstrass theorem one gets

$R(\Omega)=C(\Omega)$.

We will present several proper multidimensional

generalizations of Alexander's inequality in the framework

of Clifford analysis. The compact space $\Omega$ is now a

subset of $\mathbb{R}^{m+1}$, $m\geq1$, and instead of

$C(\Omega)$ we take the Banach algebra

$C(\Omega,\frak{A}_m)$ of $\frak{A}_m$-valued continuous

functions on $\Omega$, where $\frak{A}_m$ is the Clifford algebra

with $m$ generators. The analog of $R(\Omega)$, denoted by

$R(\Omega,\frak{A}_m)$, is defined as the uniform closure in

$C(\Omega,\frak{A}_m)$ of functions Clifford-analytic on

open neighborhoods of $\Omega$.

As a direct generalization of Alexander's inequality, we

will show that $$ {\rm dist}_{C(\Omega,\frak{A}_m)}[\bar{x},R(\Omega,\frak{A}_m)]\leq

A_m[{\rm vol}(\Omega)]^{1/(m+1)}, $$

where $\bar{x}$ is the Clifford conjugate of the identity

function on $\mathbb{R}^{m+1}$, $A_m$ is a universal

constant that only depends on $m$, and ${\rm vol}(\Omega)$

is the Lebersgue measure of $\Omega$ in $\mathbb{R}^{m+1}$.

Actually, this result will be derived from a more general

inequality that estimates the distance in

$C(\Omega,\frak{A}_m)$ from an

arbitrary smooth $\frak{A}_m$-valued function to

$R(\Omega,\frak{A}_m)$. In particular, that general

inequality will imply

$R(\Omega,\frak{A}_m)=C(\Omega,\frak{A}_m)$, whenever

${\rm vol}(\Omega)=0$, so, once more, we end up with a

quantitative form of the Hartogs-Rosenthal theorem, but now

this theorem is in the setting of Clifford analysis.

Gabriel Nagy

Title: Idemptotents in finite AW* factors

Abstract: The talk addresses a problem posed by Kaplansky in the 1950's,

which conjectures that an AW* factor is a von Neumann algebra.

In connection with this question, we prove that the quasitrace of an

idempotent in an AW* factor of type II_1 is equal to the dimension

function of its left (or right) support. Based on this result, we discuss

some linear algebraic reformulations of Kaplansky's conjecture.

Richard Rochberg

Title: Hankel and Schrodinger Forms on Dyadic Trees

Abstract: Hankel forms on dyadic trees can be viewed as discrete models

for Hankel forms on the Dirichlet space or as discrete models for

Schrodinger forms. I will describe how these discrete models are related

to the classical questions and will describe boundedness criteria for the

discrete forms. The criteria for those forms to be in Schatten-Von

Neumann classes is not known.

Sharon Schaffer Vestal

Title: Using Functional Analysis to relate a wavelet GMRA to a multiwavelet MRA

Abstract: It is well-known that wavelets have an associated subspace structure

calleda multiresolution analysis (MRA). There are other wavelets, minimally

supported frequency (MSF) wavelets, which are associated with a generalized

multiresolution analysis (GMRA). We will present a theorem that links the

two structures and give examples illustrating this relationship.

Eric Weber

Title: Frame Representations of Groups and Sampling Theory.

Abstract: We consider unitary representations of Abelian groups that give

rise to a frame sequence. By analyzing the group we can get information

regarding the corresponding analysis operator; in particular, we have a

way of "parametrizing" the range. Such information is significant for

multiplexing schemes. We then demonstrate how this can be applied to

sampling theory.

Shihshu Walter Wei

Title: On the structure of minimal submanifolds in nonpositively curved manifolds.

Abstract: We provide a topological obstruction for a complete submanifold

with a specific uniform bound involving Ricci curvature to be minimally immersed

in any complete simply-connected manifold of nonpositive sectional curvature.

We prove that such minimal submanifolds of dimension greater

than two have only one topological end. The proof uses the Liouville

theorem for bounded harmonic functions on minimal submanifolds of this sort

due to Yau, and also adapts a technique of Cao-Shen-Zhu to show the

existence of nonconstant bounded harmonic functions based on the Sobolev inequality of Hoffman-Spruck. This extends the work

of Yau. The same phenomena occur in a wider class of $n$-submanifolds with bounded mean

curvature in an $L^n$ sense. By improving the techniques in Cao-Shen-Zhu, one can obtain the

topological conclusion in the intrinsic settings. These generalize and

unify the structure theorems in the extrinsic settings.

Karen Yagdjian

Title: Parametric resonance and global solutions to nonlinear hyperbolic equations.

Abstract: We show how parametric resonance can affect global existence of

solutions to the Cauchy problem for nonlinear hyperbolic equations. Namely

we give some examples of nonlinear hyperbolic equations and systems such that

for arbitrary small smooth initial data, and for arbitrary large space dimension

there are blowing up solutions.

The Prairie Analysis Seminar is a joint project of the Department of Mathematics of Kansas State University and the

Department of Mathematics of The University of Kansas.