Investigation of common properties of $Lip$ and $\mathcal{H}^\infty$ functions (preliminary report)

This is an expository talk based on two continuing projects with José Bonet, Verónica Dimant, Luis Carlos García Lirola, and Manuel Maestre. Both studies center on spaces of Lipschitz functions and bounded holomorphic functions.
For a metric space $(M,d)$ and a fixed point $x_0 \in M,$ let $Lip_{x_0}(M) :\equiv \{f:M \to \mathbb{K} \ | \ f(x_0) = 0 {\rm \ and\ for \ some \ C } \geq 0,$
$$ |f(x) - f(y)| \leq C d(x,y), \forall x,y \in M\}.$$
$Lip_{x_0}(M)$ is a Banach space with norm $L(f) :\equiv \inf \{ C \ | \ $ the above inequality holds $\}.$
For a complex Banach space $X$ with open unit ball $B_X,$ let $\mathcal H^\infty(B_X) :\equiv \{ f:B_X \to \C \ | \ f $ is holomorphic and bounded$\},$ which is a Banach space with the $sup-$norm.
Although these spaces are quite different, they have a number of similarities which we investigate. For one thing, both $Lip_{x_0}(M)$ and $\mathcal H^\infty(B_X)$ are dual spaces, and we look at characterizing norm attaining elements of the two spaces. We also describe our study of a “combination” of these two types of spaces.