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

[摘要] 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 Frechet spaces L-0(p) (R-d), the intersection (over all epsilon > 0) of the spaces L-epsilon(p) (R-d) of functions such that integral(Rd) e(-epsilon vertical bar x vertical bar 2)vertical bar f(x)vertical bar(p) dx < infinity. These spaces consist of functions that are 'large at infinity', and L-0(1)(R-d) 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 L-0(p) (R-d) into L-0(q)(R-d), 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 L--epsilon(p) (R-d) of very-rapidly decreasing functions such that integral(Rd) e(epsilon vertical bar x vertical bar 2)vertical bar f(x)vertical bar(p) dx < infinity. We show that (L-p epsilon(p) (R-d))' = L--q epsilon(q) (R-d) (with 1 < p < infinity and (p, q) conjugate), and that the heat flow on L-epsilon(p) (R-d) is the adjoint of the flow on L--delta(q) (R-d) for an appropriate (time-dependent) choice of delta. (C) 2020 Elsevier Inc. All rights reserved.

[发布日期] 2020-11-15