# SCIENTIFIC PROGRAMS AND ACTIVITIES

April 20, 2014
 July 26-29, 2011 Conference in Harmonic Analysis and Partial Differential Equations in honour of Eric Sawyer to be held at the Fields Institute Organizer: Cristian Rios, University of Calgary

### Invited Talks:

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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.

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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).

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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.

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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.

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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.

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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.

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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.

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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.

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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.

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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".

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Carlos Perez Moreno
, University of Seville
The work of Eric Sawyer: Some high points

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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.

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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.

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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.

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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.

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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:

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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

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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)

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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.

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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.)

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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.

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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.

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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.

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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.)