Nicola Arcozzi, University of Bologna
Potential Theory on Trees and Metric Spaces
Bessel (nonlinear) capacity of a subset of a Ahlfors regular
metric space can be estimated from above and below by the Bessel
capacity of a corresponding set on the boundary of a tree. This
fact is interesting for various reasons. For instance, on trees
there are recursive formulas for computing set capacities. Work
in collaboration with R. Rochberg, E. Sawyer, B. Wick.
_______________________________
Baojun Bian, Tongji University
Convexity and partial convexity for solution of partial differential
equations
In this talk, we will discuss the convexity and partial convexity
for solution of partial differential equations. We establish the
microscopic (partial) convexity principle for (partially) convex
solution of nonlinear elliptic and parabolic equations. As application,
we discuss the (partial) convexity preserving of solution for
parabolic equations. This talk is based on the joint works with
Pengfei Guan.
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Sagun Chanillo, Rutgers University
Embedding CR Three Manifolds.
We show that to globally embed three dimensional, strongly pseudo-convex
CR structures one needs the Yamabe constant to be positive and
the CR Paneitz operator to be non-negative. Conversely, the boundary
of any strictly convex domain in C^2 has positive Yamabe constant
and non-negative Paneitz operator. This is joint work with Paul
Yang and Hung-Lin Chiu.
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Seng Kee Chua, National University of Singapore
Average value problems in differential equations
Using Schauder's fixed point theorem, with the help of an integral representation
$$ y(x)-\frac{1}{v[a,b]}\int^b_a y(z)dv(z) = \frac{1}{v[a,b]}\left[\int^x_a
v[a,t]y'(t)dt -\int^b_x v[t,b]y'(t)dt\right]
$$ in `Sharp conditions for weighted 1-dimensional Poincar\'e inequalities',
Indiana Univ. Math. J., 49 (2000), 143-175, by Chua and Wheeden, we obtain
existence and uniqueness theorems and `continuous dependence of average
condition' for average value problem: $$ y'=F(x,y), \ \ \int^b_a y(x)dv
=y_0 \ \mbox{ where $v$ is any probability measure on $[a,b]$}$$ under the
usual conditions for initial value problem. We also extend our existence
and uniqueness theorems in the case where $v$ is just a signed measure with
$v[a,b]\ne 0$ and $$F: {\cal F}\subset C[a,b] \ (\mbox{ or } L_w^p[a,b])\to
L^1[a,b]\ \mbox{ is a continuous operator. } $$
We then further extend this to discuss its application to symmetric solutions
of Laplace equations $\Delta u=F(|x|,u)$ with a given average value.
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David Cruz-Uribe, Trinity College
Norm inequalities for the maximal operator\\ on variable Lebesgue
spaces
The variable Lebesgue spaces are a generalization of the classical
Lebesgue spaces, replacing the constant exponent $p$ with a variable
exponent $p(\cdot)$. More precisely, given a function $p(\cdot)
: \mathbb{R}^n \rightarrow [1,\infty]$, define $\Omega_\infty
= \{ x \in \mathbb{R}^n : p(x) =\infty\}$. A function $f\in L^{p(\cdot)}(\mathbb{R}^n)$
if for some $\lambda>0$,
% \[ \rho(f/\lambda) = \int_{\mathbb{R}^n\setminus \Omega_\infty}
\left(\frac{|f(x)|}{\lambda}\right)^{p(x)}\,dx + \lambda^{-1}\|f\|_{L^\infty(\Omega_\infty)}<
\infty. \] %
Then $L^{p(\cdot)}(\mathbb{R}^n)$ is a Banach space with the Luxemburg
norm % \[ \|f\|_{p(\cdot)} = \inf\big\{ \lambda > 0 : \rho(f/\lambda)
\leq 1 \big\}. \] % The variable Lebesgue spaces were introduced
by Orlicz in the 1930s, but became the subject of sustained interest
in the 1990s because of their applications to PDEs and variational
integrals with nonstandard growth conditions. A great deal of
work was done on extending the tools of harmonic analysis---Riesz
potentials, singular integrals, convolution operators, Rubio de
Francia extrapolation---to variable Lebesgue spaces. A fundamental
problem was to find conditions on the exponent $p(\cdot)$ for
the Hardy-Littlewood maximal operator to be bounded on $L^{p(\cdot)}(\mathbb{R}^n)$.
This was solved in a series of papers by DCU, Fiorenza, Neugebauer,
Diening, Nekvinda, Kopaliani and Lerner. A sufficient condition
is log-H\"older continuity: % \[ |p(x)-p(y)| \leq \frac{C_0}{-\log(|x-y|)},
\quad |x-y| < 1/2, \qquad |p(x)-p(\infty)| \leq \frac{C_\infty}{\log(e+|x|)}.
\] % While these conditions are optimal in terms of pointwise
regularity, they are not necessary. A necessary and sufficient
condition is known but is not well understood in terms of the
kinds of exponent functions it allows. (For example, there exist
discontinuous exponents for the which the maximal operator is
bounded.)
In this talk we will discuss these results and their deep and
surprising connections with weighted norm inequalities and the
Muckenhoupt $A_p$ condition. We will also discuss a very recent
result (joint with Fiorenza, Neugebauer, Diening and H\"ast\"o)
showing that if $p(\cdot)$ is log-H\"older continuous, then
the maximal operator satisfies the weighted inequality % \[ \|(Mf)w\|_{p(\cdot)}
\leq C\|fw\|_{p(\cdot)} \] % if and only if $w$ satisfies the
variable $A_{p(\cdot)}$ condition % \[ \sup_Q |Q|^{-1}\|w\chi_Q\|_{p(\cdot)}\|w^{-1}\chi_Q\|_{p'(\cdot)}
< \infty. \]
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Xuan Duong, Macquarie University
Boundedness of singular integrals and their commutators with
BMO functions on Hardy spaces
Let $L$ be a non-negative self-adjoint operator on $L^(X)$ where
$X$ is a doubling space. In this talk, we will establish sufficient
conditions for a singular integral $T$ to be bounded from certain
Hardy spaces $H^p_L$ (Hardy spaces associated to the operator
$L$) to Lebesgue spaces $L^p$, $0< p \le 1$, and for the commutator
of $T$ and a BMO function to be weak-type bounded on Hardy space
$H_L^1$. Our results are applicable to the following cases:
(i) $T$ is the Riesz transform or a square function associated
with the Laplace-Beltrami operator on a doubling Riemannian manifold,
(ii) $T$ is the Riesz transform associated with the magnetic Schr\"odinger
operator on an Euclidean space, and
(iii) $T = g(L) $ is the spectral multiplier of $L$.
This is a joint work with The Anh Bui (Macquarie University).
_______________________________
Pengfei Guan, McGill University
Maximum rank property and partial Legendre tranform of homegenous
Monge-Amp\`ere type equations
This is a joint work with D. Phong. The solutions to the Dirichlet
problem for two degenerate elliptic fully nonlinear equations
in $n+1$ dimensions, namely the real Monge-Amp\`ere equation and
the Donaldson equation, are shown to have maximum rank in the
space variables when $n \leq 2$. A constant rank property is also
established for the Donaldson equation when $n=3$. We also discuss
the partial Legendre transform of this type of equations are another
non-linear elliptic differential equation. In particular, the
partial Legendre transform of the Monge-Amp\`ere equation is another
equation of Monge-Amp\`ere type. In $1+1$ dimensions, this can
be applied to obtain uniform estimates to all orders for the degenerate
Monge-Amp\`ere equation with boundary data satisfying a strict
convexity condition.
_______________________________
Steve Hofmann, University of Missouri at Columbia
Harmonic Measure and Uniform Rectifiability
We present a higher dimensional, scale-invariant version of the
classical theorem of F. and M. Riesz, which established absolute
continuity of harmonic measure with respect to arc length measure,
for a simply connected domain in the complex plane with a rectifiable
boundary. More precisely, for $d\geq 3$, we obtain scale invariant
absolute continuity of harmonic measure with respect to surface
measure, along with higher integrability of the Poisson kernel,
for a domain $\Omega\subset \mathbb{R}^{d}$, with a uniformly
rectifiable boundary, which satisfies the Harnack Chain condition
plus an interior (but not exterior) corkscrew condition. We also
prove the converse, that is, we deduce uniform rectifiability
of the boundary, assuming scale invariant $L^p$ bounds, with $p>1$,
for the Poisson kernel.\end{abstract}
Joint work with J. M. Martell, and with Martell and I. Uriarte-Tuero.
_______________________________
Alex Iosevich, University of Rochester
Three point configurations, bilinear operators and geometric
combinatorics
We are going to prove that a subset of the plane of Hausdorff
dimension greater than 7/4 determines a positive three dimensional
Lebesgue measure worth of triangles. Bilinear methods and combinatorial
reasoning play a key role.
_______________________________
Ron Kerman, Brock University
Rearrangement invariant Sobolev spaces on general domains
This talk concerns Sobolev spaces of differentiable functions
on finite measure domains in $R^n$. Such a space is determined
by a rearrangement invariant (r.i.) functional, $\rho$, like those
of Lebesgue, Lorentz or Orlicz. We have two goals. First, we seek
a functional to describe the smallest set that contains the decreasing
rearrangements of functions in an r.i. Sobolev space. Second,
we study refinements of Sobolev-Poincare imbedding inequalities
which, for spaces of functions with first order derivatives, have
the form
\[\inf_{c \in \R} \sigma(u-c) \leq A\rho(|\Delta u|).\]
Here, $\sigma$ is another r.i. functional, which we would like
to be as large as possible. It turns out that for spaces of functions
with higher order derivatives the two problems are connected.
_______________________________
Michael Lacey, Georgia Institute of Technology
Weighted Estimates for Singular Integrals
We will survey some recent results for singular integrals on
weighted spaces, including (1) refinements of the recent linear
bound in A2; (2) the state of knowledge concerning the two weight
estimate for the Hilbert transform.
_______________________________
Guozhen Lu, Wayne University
Multiparameter Hardy spaces associated to composition operators
and Littlewood-Paley theory
In this talk, we will first review some recent works on multiparameter
Hardy space theory using the discrete Littlewood-Paley theory.
We then will discuss the multiparameter structures associated
with the composition of two singular integral operators, one with
the standard homogeneity and the other with non-isotropic homogeneity
which was studied by Phong and Stein. We then discuss about the
Hardy space associated with this multiparameter structure and
prove the boundedness of the composition operators on such Hardy
spaces.
_______________________________
José María Martell, Consejo Superior de Investigaciones
Cientificas (Spain)
Higher integrability of the Harmonic Measure and Uniform Rectifiability
Consider a domain $\Omega\subset \mathbb{R}^{d}$, $d\ge 3$, with
an Ahlfors-David regular boundary, which satisfies the Harnack
Chain condition plus an interior (but not exterior) corkscrew
condition. In joint work with S. Hofmann and I. Uriarte-Tuero,
we obtain that higher integrability of the harmonic measure, via
scale invariant $L^p$ bounds, with $p>1$, for the Poisson kernel,
implies that $\partial \Omega$ is uniform rectifiable. The converse
of this result, (i.e., uniform rectifiability implies higher integrability
of the Poisson kernel), is a joint work with S. Hofmann and gives
a higher dimensional, scale-invariant version of the classical
theorem of F. and M. Riesz which established absolute continuity
of harmonic measure with respect to arc length measure, for a
simply connected domain in the complex plane with a rectifiable
boundary.
_______________________________
Camil Muscalu, Cornell University
Beyond Calderón's algebra
The goal of the talk is to describe various extensions of the
so called Calderón commutators and the Cauchy integral
on Lipschitz curves. They appear naturally when one tries to invent
a calculus which includes operators of multiplication with functions
having arbitrary "polynomial growth".
_______________________________
Carlos Perez Moreno, University of Seville
The work of Eric Sawyer: Some high points
_______________________________
Malabika Pramanik, University of British Columbia
A multi-dimensional resolution of singularities with applications
to analysis
The structure of the zero set of a multivariate polynomial is
a topic of wide interest, in view of its ubiquity in problems
of analysis, algebra, partial differential equations, probability
and geometry. The study of such sets, known in algebraic geometry
literature as resolution of singularities, originated in the pioneering
work of Jung, Abhyankar and Hironaka and has seen substantial
recent advances, albeit in an algebraic setting.
In this talk, I will discuss a few situations in analysis where
the study of polynomial zero sets play a critical role, and discuss
prior work in this analytical framework in two dimensions. Our
main result (joint with Tristan Collins and Allan Greenleaf) is
a formulation of an algorithm for resolving singularities of a
multivariate real-analytic function with a view to applying it
to a class of problems in harmonic analysis.
_______________________________
Richard Rochberg, Washington University at Saint Louis
Toeplitz Operators and Hankel Forms on Model Spaces
A Model Space is the orthocomplement of a shift invariant subspace
of the Hardy space. A Truncated Toeplitz Operator (TTO) is the
compression to a Model Space of a Toeplitz operator on the Hardy
space. A basic question is if/when a bounded TTO has a bounded
symbol. In 2009 it was shown that this is not always the case.
More recent work has shown that the question is closely related
to a weak factorization construction which also shows up in the
study of boundedness of Hankel forms.
I will introduce a notion of a Truncated Hankel Form (THF) on
the Model Space. The natural conjugation operator on the Model
Space gives a conjugate linear isometric isomorphism between the
space of TTO’s and the space of THF’s. This helps in
understanding why weak factorization plays a role in both theories.
It also suggests new questions and new approaches to existing
questions.
_______________________________
Rodolfo Torres, University of Kansas
A new geometric regularity condition for the end-point estimates
of bilinear Calder\'on-Zygmund operators.
A new minimal regularity condition involving certain integrals
of the kernels of bilinear Calder\'on-Zygmund operators over appropriate
families of dyadic cubes is presented. This regularity condition
ensures the existence of end-point estimates for such operators
and is weaker than other typical regularity assumptions considered
in the literature. This is joint work with Carlos Pérez.
_______________________________
Ignacio Uriarte-Tuero, Michigan State University
Two conjectures of Astala on distortion of sets under quasiconformal
maps and related removability problem
Quasiconformal maps are a certain generalization of analytic
maps that have nice distortion properties. They appear in elasticity,
inverse problems, geometry (e.g. Mostow's rigidity theorem)...
among other places. In his celebrated paper on area distortion
under planar quasiconformal mappings (Acta 1994), Astala proved
that if $E$ is a compact set of Hausdorff dimension $d$ and $f$
is $K$-quasiconformal, then $fE$ has Hausdorff dimension at most
$d' = \frac{2Kd}{2+(K-1)d}$, and that this result is sharp. He
conjectured (Question 4.4) that if the Hausdorff measure $\mathcal{H}^d
(E)=0$, then $\mathcal{H}^{d'} (fE)=0$. UT showed that Astala's
conjecture is sharp in the class of all Hausdorff gauge functions
(IMRN, 2008). Lacey, Sawyer and UT jointly proved completely Astala's
conjecture in all dimensions (Acta, 2010). The proof uses Astala's
1994 approach, geometric measure theory, and new weighted norm
inequalities for Calder\'{o}n-Zygmund singular integral operators
which cannot be deduced from the classical Muckenhoupt $A_p$ theory.
These results are related to removability problems for various
classes of quasiregular maps. I will mention sharp removability
results for bounded $K$-quasiregular maps (i.e. the quasiconformal
analogue of the classical Painleve problem) recently obtained
jointly by Tolsa and UT. I will further mention recent results
related to another conjecture of Astala on Hausdorff dimension
of quasicircles obtained jointly by Prause, Tolsa and UT.
_______________________________
Richard Wheeden, Rutgers University
Norm inequalities for rough Calderon-Zygmund operators; Regularity
of weak solutions of degenerate quasilinear equations with rough
coefficients
Part 1 concerns results obtained with D. Watson about one and
two weight norm estimates for homogeneous singular integral operators
whose kernels belong to L log L on the unit sphere. Part 2 involves
generalizations to degenerate quasilinear equations of work initiated
by J. Serrin in the nondegenerate case. The equations typically
have nonsmooth coefficients. The results, obtained with S. Rodney
and D. Monticelli, show that weak solutions have local regularity
such as boundedness and Holder continuity.
Contributed Short Talks:
_______________________________
Julio Delgado-Valencia, Universidad del Valle
Degenerate Elliptic Operators on the Torus
In this talk we begin by some basics on the theory of pseudo-differential
operators on the euclidean space. We define a non-homogeneous
class of symbols corresponding to a family of degenerate elliptic
operators and study the periodic version of those classes
_______________________________
Xiaolong Han, Wayne State University
Hardy-Littlewood-Sobolev inequalities on $\R^N$ and the Heisenberg
group
The talk will surround non-weighted and weighted Hardy-Littlewood-Sobolev
inequalities on both Euclidean spaces and the Heisenberg group.
Most attention will be on the sharp versions of the inequalities
including the best constants and existence, uniqueness and formulae
of the maximizers, recent development on singularity analysis
and asymptotic behavior of the maximizers will also be mentioned.
(Joint work with Guozhen Lu and Jiuyi Zhu)
_______________________________
Lyudmila Korobenko, University of Calgary
Regularity of solutions of degenerate quasilinear equations
In their paper of 1983 C. Fefferman and D. H. Phong have characterized
subellipticity of linear second order differential operators.
The characterization is given in terms of subunit metric balls
associated to the differential operator. An extension of this
result to the case of the operator with non-smooth coefficients
has been given by E. Sawyer and R. Wheeden in 2006. In collaboration
with C. Rios we are working on the extention of this result to
the case of infinitely degenerate operators. In this talk I am
going to discuss the key points of the subunit metrics approach
and its possible application to the question of hypoellipticity
of the operators with infinite vanishing.
_______________________________
Henri Martikainen, University of Helsinki
Tb Theorems on Upper Doubling Spaces
We will discuss Tb theorems in the framework of non-homogeneous
analysis in metric spaces. One main point is the randomization
of the metric dyadic cubes of M. Christ -- such constructions
are also useful in generalizing other results of harmonic analysis
to metric spaces (like the $A_2$ theorem). Some direct applications
and examples of our Tb theory will also be presented. (Joint work
with Tuomas Hytönen.)
_______________________________
Nguyen Cong Phuc, Louisiana State University
A nonlinear Calderón-Zygmund theory for quasilinear
operators and its applications
We discuss the boundedness of nonlinear singular operators arising
from a class of quasilinear elliptic PDEs in divergence form.
Applications are given to quasilinear Riccati type equations with
supernatural growth in the gradients and measure data.
_______________________________
Treven Wall, Johns Hopkins University
The $L^p$ Dirichlet problem for second-order, non-divergence
form operators
In describing my recent joint work with Martin Dindo\v{s}, I
will touch on the history of our problem and will highlight the
significant role of perturbation theorems. In addition, I will
give an outline of the proof of our perturbation theorem, which
leads to new results with less restrictive hypotheses for solvability
in the non-divergence form case.
_______________________________
Xiao Yuayuan, Wayne State University
Wolff potentials and integral systems on homogeneous spaces
This is a joint work with Guozhen Lu and Xiaolong Han. We first
establish the comparison between Wolf and Riesz potentials on
homogeneous spaces, followed by a Hardy-Littlewood-Sobolev type
inequality for Wolf potentials. Then we consider a Lane-Emden
type integral system and derive integrability estimates of positive
solutions to the system. Furthermore, we prove that the positive
solutions are also Lipschitz continuous.
_______________________________
Xiangwen Zhang, McGill University
Schauder estimate for the complex Monge-Amp\'ere equation
In the talk, a regularity result for the complex Monge-Amp\`ere
equation will be presented. We will prove that any $C^{1,1}$ plurisubharmonic
solution u to the complex Monge-Amp\`ere equation $\det(u_{i\bj})
= f$ with $f$ strictly positive and H\"older continuous has
in fact H\"older continuous second derivatives. For smoother
f this follows from the classical Evans-Krylov theory, yet in
our case it cannot be applied directly. (This is a joint work
with S. Dinew and Xi Zhang.)
