# Reception

Thursday, May 31, 2012 - 3:30pm - 5:00pm

Lind 400

**Poster - Uniqueness of extremizers for an endpoint inequality of the k-plane transform**

Taryn Flock (University of California, Berkeley)

The $k$-plane transform is a bounded operator from $L^p(\mathbb{R}^d)$ to $L^q$ of the Grassmann manifold of all affine hyperplanes in $\R^n$ for certain exponents depending on $k$ and $n$. When $k=1$ this is the X-ray transform, and when $k=n-1$ it is the Radon transform. In the endpoint case, $q=n+1$. We show that all extremizers have the form $c(1+\phi(x)^2)^{-(k+1)/2}$ where $\phi$ an affine transformation of $\R^n$.**Poster - Locally linear analysis with applications**

We study the related problems of denoising images corrupted by impulsive noise and blind inpainting

(i.e., inpainting when the deteriorated region is unknown). Our basic approach is to

model the set of patches of pixels in an image as a union of low dimensional subspaces, corrupted

by sparse but perhaps large magnitude noise. For this purpose, we develop a robust and iterative

method for single subspace modeling and extend it to an iterative algorithm for

modeling multiple subspaces. We prove convergence for both algorithms and carefully compare

our methods with other recent ideas for such robust modeling. We demonstrate state of the art

performance of our method for both imaging problems.**Poster - Weighted multiparameter Hardy spaces associated with the Zygmund dilation**

Yayuan Xiao (Wayne State University)

We apply the discrete Littlewood-Paley-Stein analysis to establish the duality theorem of weighted multi-parameter Hardy spaces H^p_Z(w) associated with Zygmund dilations, i.e., (H^p_Z(w))^*= CMO^p_Z(w) for 0**Poster: Asymptotic Estimates For Unimodular Fourier Multipliers On**

Modulation Spaces

Lijing Sun (University of Wisconsin)

Recently, it has been shown that the unimodular Fourier

multipliers \ $e^{it\Delta ^{\frac{\alpha }{2}}}$ are bounded on all

modulation spaces. In this paper, using the almost orthogonality of

projections and some techniques on oscillating integrals, we obtain

asymptotic estimates for the unimodular Fourier multipliers \ $e^{it\Delta

^{\frac{\alpha }{2}}}$ \ on the modulation spaces. As applications, we give

the grow-up rates of the solutions for the Cauchy problems for the free Schr%

\{o}dinger equation, the wave equation and the Airy equation with the

initial data in a modulation space. We also obtain a quantitative form about the solution to the Cauchy problem of the nonlinear dispersive

equations.**Poster - Logarithmic bounds for maximal directional singular integrals in the plane**

Francesco Di Plinio (Indiana University)

We discuss the boundedness of the maximal directional singular integrals in the plane along a finite set V of N directions. This maximal directional operator can be thought of as a discrete model of the Hilbert transform along a smooth vector field in the plane, whose boundedness has been conjectured by Stein. Logarithmic dependence on N of the L^p bounds (for p greater than 2) are established for a set of arbitrary structure.

Sharp bounds are proved for lacunary and Vargas sets of directions. The latter include the case of uniformly distributed directions and the finite truncations of the Cantor set.

We make use of both classical harmonic analysis methods and product-BMO based time-frequency analysis techniques.**Poster - Regularity of solutions of degenerate quasilinear equations**

Lyudmila Korobenko (University of Calgary)

In their paper of 1983 C. Fefferman and D. H. Phong have

characterized subellipticity of linear second order PDEs. 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. We extend the result to the case of

non-doubling metric spaces. Our main result is continuity of weak

solutions of infinitely degenerate quasilinear equations. This

together with a recent result by Rios, Sawyer and Wheeden completes

the proof of hypoellipticity of a certain class of infinitely

degenerate quasilinear operators.**Poster: Laminates Meet Burkholder Functions**

Nicholas Boros (Michigan State University)

We will explain how to compute the exact Lp operator norm of a quadratic perturbation of the real part of the Ahlfors--Beurling operator. For the lower bound estimate we use a new approach of constructing a sequence of laminates (probability measures for which Jensen's inequality holds, but for rank one concave functions) to give an almost extremal sequence to approximate the operator.