The Laplace Transform And Nonlocal Field Equations

We use the Laplace transform as a correspondence between appropriate L ^p and Hardy spaces in order to interpret operators of the form f (∂t) in which the “symbol” f is an analytic function. This framework allows us to find the most general solution to the equation f(∂t)φ=J(t) t≥0, in a convenient class of functions, and to define and solve initial value problems. We state conditions under which the solution φ is of class C ^k , k ≥ 0, and we observe that if some a priori information is specified, then the initial value problem is well-posed and it can be solved using a finite number of local initial data. The present approach is motivated by recent work on field theory in which the (analytic continuation of the) Riemann zeta function is used as a symbol.