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.