# When is the bound in Riesz-Thorin Interpolation Theorem attained?

Theorem (Riesz-Thorin): Let $(X,\mu)$ and $(Y,\nu)$ be $\sigma$-finite measure spaces, let $1\leq p_0\leq p_1\leq \infty$ and let $1\leq q_0\leq q_1\leq \infty$. Suppose that $T\colon L^{p_0}(X,\mu)+L^{p_1}(X,\mu)\to L^{q_0}(X,\mu)+L^{q_1}(X,\mu)$ is a linear map such that:

(1) $T$ restricts to a bounded linear map $T\colon L^{p_0}(X,\mu) \to L^{q_0}(X,\mu)$. Denote its norm by $M_0$.

(2) $T$ restricts to a bounded linear map $T\colon L^{p_1}(X,\mu) \to L^{q_1}(X,\mu)$. Denote its norm by $M_1$.

Given $\theta$ in $(0,1)$, set $\frac{1}{p_\theta}=\frac{1-\theta}{p_0}+\frac{\theta}{p_1}$ and $\frac{1}{q_\theta}=\frac{1-\theta}{q_0}+\frac{\theta}{q_1}$. Then $T$ restricts to a bounded linear map $T\colon L^{p_\theta}(X,\mu) \to L^{q_\theta}(X,\mu)$, and moreover

$$\|T\|_{\mathcal{B}(L^{p_\theta}(X,\mu),L^{q_\theta}(X,\mu)}\leq M_0^{1-\theta}M_1^\theta.$$

My question is under what circumstances the norm bound is actually attained. Marcin Bownik has pointed out to me that in the classical applications of this theorem (Hausdorff-Young and Young inequalities), the bounds are not optimal. However, it seems possible to me that by imposing some (hopefully not too drastic) conditions on the exponents, the spaces, and the operator $T$, one may get equality.

In my situation, $X=Y=\mathbb{Z}$ with counting measure, and $p_0$ and $q_0$ are conjugate exponents (same with $p_1,q_1$).

If the measure space is a locally compact group and the operator $T$ is given by convolution (and thus one is considering Young's inequality), then one has that the interpolating constant is strictly less than that given by Riesz-Thorin if the space/group is unimodular with no compact open subgroups.
Moreover, if $G$ is unimodular with open compact subgroups then a pair of functions $(f,g)$ that satisfy equality in Young's inequality (for an `internal' pair of exponents) is called a maximal pair. It is an old theorem of Hardy and Littlewood that the only maximal pairs are those that arise from characters in various obvious ways.