Last update: 29-06-2017

Michael T. Lacey (Georgia Institute of Technology)

Title: Sparse Bounds: A Survey.

Abstract: The notion of a sparse bound for a bilinear form provides an upper bound in terms of a sum of local averages of the input functions. As a corollary, one deduces a range of weighted and vector valued inequalities. The weighted inequalities are completely quantitative, and frequently sharp in the \(A_p\) characteristic. The arguments that lead to these bounds are natural extensions of the Calderon-Zygmund theory, but their application in this way has only recently been established.

Alexander Nagel (University of Wisconsin-Madison)

Title: Estimates for Bergman kernels in Reinhardt and tube domains.

Abstract: We obtain geometric estimates of the Bergman kernel and its derivatives in tube domains over bounded convex domains. Then using the Poisson summation formula we obtain estimates in Reinhardt domains which are bounded away from the coordinate axes.

Fulvio Ricci (Scuola Normale Superiore, Pisa)

Title: A maximal restriction theorem and Lebesgue points of Fourier transforms.

Abstract: This is joint work with D. Müller and J. Wright.
Let \(S\) be the graph in \(\mathbb R^2\) of a convex \(C^2\) function, with affine arclength measure \(d\mu=\kappa^\frac13ds\), \(\kappa\) denoting the curvature of \(S\). We prove that, for \( 1 < p < \frac87 \) and \(q=p'/3\), the following maximal restriction inequality holds: \[ \big\|(M\hat f)_{|_S}\big\|_{L^q(S,\mu)}\le C\big\|f\|_{L^p(\mathbb R^n)}\ , \] where \(M\) is the strong maximal function. This implies that, given \(f\in L^p(\mathbb R^2)\) with \(p<\frac87\) and \(S\) as above, \(\mu\)-almost every point of \(S\) is a Lebesgue point of \(\hat f\). Moreover, the regularized values of \(\hat f\) at these points coincide \(\mu\)-a.e. with the values of the image \(\mathcal R f\) of \(f\) under the restriction operator \( \mathcal R\).

Terence Tao (University of California, Los Angeles)

Title: Failure of the pointwise ergodic theorem on the free group at the \(L^1\) endpoint.

Abstract: A well known theorem of Nevo and Stein shows that the pointwise ergodic theorem on the free group holds in \(L^p\) for every \(p>1\). We give a construction that shows that the \(L^1\) endpoint of this theorem is false.