Sharp extension theorems and Falconer distance problems for algebraic curves in two dimensional vector spaces over finite fields

In this paper we study extension theorems associated with general varieties in two dimensional vector spaces over finite fields. Applying Bezouts theorem, we obtain the sufficient and necessary conditions on general curves where sharp $L^p-L^r$ extension estimates hold. Our main result can be considered as a nice generalization of works by Mochenhaupt and Tao and Iosevich and Koh. As an application of our sharp extension estimates, we also study the Falconer distance problems in two dimensions.

Characterization of the matrix whose norm is determined by its action on decreasing sequences:The exceptional cases

Let $A=(a_{j,k})_{j,k ge 1}$ be a non-negative matrix. In this paper, we characterize those $A$ for which $|A|_{ell_p,ell_q}$ are determined by their actions on non-negative decreasing sequences, where one of $p$ and $q$ is 1 or $infty$. The conditions forcing on $A$ are sufficient and they are also necessary for non-negative finite matrices.