Essential properties of L_p,q spaces (the amalgams) and the implicit function theorem for equilibrium analysis in continuous time

To extend the analysis of continuous-time general-equilibrium macro models we study 2 parameter variants L_p,q of the Lebesgue spaces, thus gaining separate control on the asymptotic behaviour (p) and the local behaviour (q): they behave w.r.t. p like the spaces ℓ_p and w.r.t. q like the spaces L_q on a probability space. Such spaces might naturally contain equilibrium variables (paths) as well as time-dependent policies of a macro model. Convolution behaves very well on those spaces, which can be used as a basis for the classical "comparative statics" (see e.g. Mertens and Rubinchik (2011)). Finally, we generalise the classical implicit function theorem (ift) for a family of Banach spaces, with the resulting implicit function having derivatives that are locally Lipschitz to very strong operator norms.