Some special spaces of distributions

Abstract

In an existence theory for partial differential equations it is important to give precise statements concerning the regularity of the solutions obtained. Now a condition on the regularity of a distribution or function u (with compact support) can also be regarded as a condition on the behavior at infinity of the Fourier transform û. To classify this behavior one may for example ask for which weight functions k it is true that kû ∈ L _p . The set of all such temperate distributions u is denoted by ℬ _p,k ,here. Only the cases þ = 2, þ = ∞ and þ = 1 are really interesting. Concerning k we shall make some assumptions (Definition 2.1.1) which ensure that ℬ _p,k , is a module over C ^∞₀ and which are suggested by the theory developed in the following chapters.