**Oleksandra Beznosova** (Missouri): *The limiting case of the Reverse
Hölder inequalities and the A*_{∞} condition.

As *p* → 1, *p*-Reverse Holder inequalities for a
weight degenerate to a reverse Jensen's inequality of the *L*
log *L* type. We show that the constant in this inequality depends
at most linearly on the A_{∞} constant of the weight .

This is a joint work with Sasha Reznikov.

**Jun Geng** (Kentucky): *W*^{1,p} estimates for elliptic
problems with Neumann boundary conditions in Lipschitz domains.

We study W^{1,p} estimates in Lipschitz domains for second order
elliptic equations of divergence form with real-valued, bounded,
measurable coefficients. For any fixed p>2, we prove that a weak
reverse Hölder inequality implies the W^{1,p} estimates for
solutions with Neumann boundary conditions. As a result, we are able to
show that if the coefficient matrix is symmetric and uniformly
continuous, the W^{1,p} estimate holds for
3/2-ε < p <
3+ε if n ≥ 3,
and for 4/3-ε < p <
4+ε if n = 2.

**Andrew Lorent** (Cincinnati): *Functions whose symmetric part of
gradient agree and a generalization of Reshetnyak's compactness theorem*.

Rigidity of functions whose gradient lies in a subject of the
conformal matrices is a classic subject, the one of the earliest theorems
being Liouville's Theorem that conformal mappings in **R**^{3}
are either affine
or Mobius. There has been extensive work on developing a quantitative
generalization of this result, one of the best known theorems has been
Reshetnyak's compactness theorem that says if u_{n} is a weakly
converging sequence in W^{1,1} and
∫ dist(Du,SO(n)) dx → 0
then u_{n} converges
strongly in W^{1,1} to an affine map whose gradient is a rotation. We
reformulate this theorem as a special case
of the question: What happens when two sequences of functions have
increasingly similar symmetric part of gradient? We will provide a sharp
answer to this question and consequently present a broad
generalization of Reshetnak's theorem.

**Cristina Pereyra** (New Mexico):
*Sharp bounds for commutators on weighted Lebesgue
spaces*.

Last year saw the solution of the A_{2}-conjecture by Tuomas
Hytönen: every Calderón-Zygmund operator obeys a linear bound on
L²(w) with respect to the A_{2}-characteristic of the weight.
Over a span of 8 years starting in 2000, one a time this was proved for a few
operators that where either dyadic to begin with, or could be realized as
suitable averages of what we now call Haar shift operators of bounded depth
(very specific ones), and each one of these linear bounds was proved using
Bellman function techniques. The last two years different techniques
were brought to the table that will now handle the whole family of Haar shift
operators, and more importantly averaging techniques which
allowed finally to cover the whole class of Calderón-Zygmund singular
integral operators. Notable are the contributions from
Lacey-Petermichl-Reguera, CruzUribe-Martell-Pérez,
and Pérez-Treil-Volberg, and Hytönen brilliantly
put together all these ingredients plus some magic touch
and was able to finish the argument.

The commutator of a Calderón-Zygmund singular integral operator with a
*BMO* function is known to be more singular than the operator
itself. This can be quantified by its lack of weak-type (1,1) estimates, and
by the fact that the commutator is dominated by M^{2}
instead of M (where M is the
Hardy-Littlewood maximal operator). It can be also quantified in terms of the
growth of the operator bound in weighted Lebesgue spaces. In joint work with
Carlos Pérez and Daewon Chung we showed that if a linear operator
obeys a linear bound in L²(w) with respect to the
A_{2}-characteristic of the weight, then the commutator
obeys a quadratic bound. In light of Hytönen's result, this behavior is
shared by all Calderón-Zygmund singular integral operators.
Our proof is based in the classical Coifman-Rochberg-Weiss argument.
However, the first results in this direction (somehow mimicking the history
of the A_{2}-conjecture), were due to Daewon Chung.
In his PhD dissertation Chung proved, using Bellman function techniques, that
the quadratic bound will hold and be sharp for the commutator of the Hilbert
transform, and for Haar shift operators, and therefore for the Riesz and
Beurling transforms. In this talk we will present these results, give some
background and give a roadmap for the proofs.

**Gideon Simpson** (Toronto): *Ill-Posedness of degenerately dispersive
equations*.

In some physical problems, such as granular media, sedimentation,
and magma dynamics, the leading order equations are degenerately dispersive.
A rigorous analysis of such equations has only recently begun and remains
incomplete. Though some cases are locally, and globally, well-posed, others
may be ill-posed.

In this talk, we consider the Rosenau-Hyman compacton equations. Inspired
by a proof of ill-posedness for a surrogate equation, we present robust
numerical evidence that the K(2,2) compacton equation is ill-posed. This is
done by constructing a family of initial conditions for a regularized
equation. As both the regularization parameter and the initial conditions
vanish, the solution, at a fixed T > 0, becomes "large." Thus, the
equations may be ill-posed in the sense that the solution operator
ceases to be continuous about the zero solution.

**Igor Verbitsky** (Missouri):
Hessian and fractional Laplacian inequalities.

Some Hessian inequalities for k-convex functions will be discussed, including
fully nonlinear analogues of the Sobolev and Poincaré inequalities,
and relations berween the k-Hessian energy
**E**_{k}[*u*] =
∫_{Ω} -*uF*_{k} [*u* dx],
Ω
**R**^{n}, and the fractional Laplacian
energy associated with (-Δ)^{k/(k+1)}.
Here *F*_{k} (k = 1,...,n)
is the k-Hessian operator, i.e. the sum of all the k × k principal
minors of the Hessian matrix of *u*.