A Laplace transform approach to linear equations with infinitely many derivatives and zeta-nonlocal field equations

We study existence, uniqueness and regularity of solutions for linear equations in infinitely many derivatives. We develop a natural framework based on Laplace transform as a correspondence between appropriate $L^p$ and Hardy spaces: this point of view allows us to interpret rigorously operators of the form $f(partial_t)$ where $f$ is an analytic function such as (the analytic continuation of) the Riemann zeta function. We find the most general solution to the equation begin{equation*} f(partial_t) phi = J(t) ; , ; ; ; t geq 0 ; , end{equation*} in a convenient class of functions, we define and solve its corresponding initial value problem, and we state conditions under which the solution is of class $C^k,, k geq 0$. More specifically, we prove that if some a priori information is specified, then the initial value problem is well-posed and it can be solved using only a {em finite number} of local initial data. Also, motivated by some intriguing work by Dragovich and Arefu0027eva-Volovich on cosmology, we solve explicitly field equations of the form begin{equation*} zeta(partial_t + h) phi = J(t) ; , ; ; ; t geq 0 ; , end{equation*} in which $zeta$ is the Riemann zeta function and $h u003e 1$. Finally, we remark that the $L^2$ case of our general theory allows us to give a precise meaning to the often-used interpretation of $f(partial_t)$ as an operator defined by a power series in the differential operator $partial_t$.