The heat flow in an optimal Fréchet space of unbounded initial data in Rd

Abstract

In this paper we show that solutions of the heat equation that are given in terms of the heat kernel define semigroups on the family of Fréchet spaces Lp0 (Rd ), the intersection (over all ε > 0) of the spaces Lpε (Rd ) of functions such that ∫ Rd e−ε|x|2 |f (x)|p dx < ∞. These spaces consist of functions that are ‘large at infinity’, and L10 (Rd ) is the maximal space in which one can use the heat kernel to obtain globally-defined solutions of the heat equation. We prove suitable estimates from Lp0 (Rd ) into Lq0 (Rd ), q ≥ p, for these semigroups. We then consider the heat semigroup posed in spaces that are dual to these spaces of functions, namely the spaces Lp−ε (Rd ) of very-rapidly decreasing functions such that ∫ Rd eε|x|2 |f (x)|p dx < ∞. We show that (Lppε (Rd ))' = Lq−qε (Rd ) (with 1 <p< ∞ and (p, q) conjugate), and that the heat flow on Lpε (Rd ) is the adjoint of the flow on Lq−δ (Rd ) for an appropriate (time-dependent) choice of δ.