Abstract: The classical theorem of Privalov says that if $u$ is $Lip_\alpha$ with $0 < \alpha <1$, then its Hilbert transform $\tilde{u}$ is also $Lip_\alpha$. However, this result fails for $Lip_1$ functions. In this case, the modulus of continuity of $\tilde{u}$ behaves like $t \log 1/t$ as $t \to 0^+$. We introduce the “generalized Lipschitz class” $Lip_{\alpha(t)}$, which certainly coincides with the classical case when $\alpha(t) \equiv \alpha$, and then show that the above results, as well as some other classical results of Hardy—Littlewood, hold for $Lip_{\alpha(t)}$ functions.

Abstract: Informally speaking, the essential dimension of an algebraic object is the minimal number of independent parameters one needs to define it. In the past 15 years this notion has been investigated in several contexts by a range of techniques, and has been found to have interesting and surprising connections to many problems in algebra and algebraic geometry. I will survey some of this research in my talk.