(𝑃,𝑄) complex hypercontractivity

Let ξ \xi be a standard normal random vector in R k \mathbb {R}^{k} . Under some mild growth and smoothness assumptions on any increasing P , Q : [ 0 , ∞ ) → [ 0 , ∞ ) P, Q : [0, \infty ) \to [0, \infty ) we show ( P , Q ) (P,Q) complex hypercontractivity Q − 1 ( E Q ( | T z f ( ξ ) | ) ) ≤ P − 1 ( E P ( | f ( ξ ) | ) ) \begin{equation*} Q^{-1}(\mathbb {E} Q(|T_{z}f(\xi )|))\leq P^{-1}(\mathbb {E}P(|f(\xi )|)) \end{equation*} holds for all polynomials f : R k → C f:\mathbb {R}^{k} \to \mathbb {C} , where T z T_{z} is the hermite semigroup at complex parameter z , | z | ≤ 1 z, |z|\leq 1 , if and only if | t P ( t ) P ′ ( t ) − z 2 t Q ( t ) Q ′ ( t ) + z 2 − 1 | ≤ t P ( t ) P ′ ( t ) − | z | 2 t Q ( t ) Q ′ ( t ) + 1 − | z | 2 \begin{align*} \left |\frac {tP(t)}{P’(t)}-z^{2}\frac {tQ(t)}{Q’(t)}+z^{2}-1\right |\leq \frac {tP(t)}{P’(t)}-|z|^{2}\frac {tQ(t)}{Q’(t)}+1-|z|^{2} \end{align*} holds for all t > 0 t>0 provided that F > 0 F>0 , and F ′ / F F’/F is concave, where F = Q ∘ P − 1 F = Q\circ P^{-1} . This extends Hariya’s result from real to complex parameter z z . Several old and new applications are presented for different choices of P P and Q Q . The proof uses heat semigroup arguments, where we find a certain map C ( s ) C(s) , which interpolates the inequality at the endpoints. The map C ( s ) C(s) itself is composed of four heat flows running together at different times.