## Chapter 10 exercises

Let $\boldsymbol{X}$ be a random variable and let $1 \leq r \leq \infty$. Recall that the triangle (Minkowski) inequality implies that for real-valued functions $f_1, f_2$, $\|f_1(\boldsymbol{X}) + f_2(\boldsymbol{X})\|_r \leq \|f_1(\boldsymbol{X})\|_r + \|f_2(\boldsymbol{X})\|_r.$ More generally, if $w_1, \dots, w_m$ are nonnegative reals summing to $1$ and $f_1, \dots, f_m$ are real functions [...]

In this chapter we complete the proof of the Hypercontractivity Theorem for uniform $\pm 1$ bits. We then generalize the $(p,2)$ and $(2,q)$ statements to the setting of arbitrary product probability spaces, proving the following:
The General Hypercontractivity Theorem Let $(\Omega_1, \pi_1), \dots, (\Omega_n, \pi_n)$ be finite probability spaces, in each of which [...]