A generalized three lines lemma in Hardy-like spaces

In this paper we address the following question: given a holomorphic function with prescribed $L^p(\mathbb{R})$ and $L^q(\mathbb{R})$ norm (with $1\le p,q\le \mathrm{\infty }$) along two parallel lines in the complex plane, then what is the maximum value that this function can achieve at a given point between these lines. Here we show that this problem is well-posed in suitable Hardy-like spaces on the strip. Moreover, in this setting we completely solve this problem by providing not only an explicit formula for the optimizers but also for the optimal values. In addition, we briefly discuss some applications of these results to interpolation theory and to Lieb-Thirring inequalities.