Let $q > p > 1$. Perhaps a very basic question one can ask in real
hypercontractivity is as follows: let $y$ be a mean zero random variable
with values in a complex plane. Given a normed space $X$, how small a real
number $r>0$ should be so that $||A+ryB||_q \leq ||A+yB||_p$ for all $A$ and $B$
in $X$? There seems to be two extreme cases studied in the literature:
Rademacher and Steinhaus random variables. In this talk I will focus on
the case when $X$ is the real line, and $y$ is distributed on the unit
circle having a certain "ultraspherical density".

Joint work with
Alexander Lindenberger, Paul F.X. Müller, and Michael Schmuckenschläger.