CHAPTER 6 - Theory of multipliers in spaces of differentiable functions and applications

Abstract

This chapter discusses theory of multipliers in spaces of differentiable functions and applications. It presents some results on multipliers on the Sobolev–Slobodeckii spaces Wlp(Rn), the Besov space Blp(Rn), etc. Some of the theorems stated are related to multipliers on each of the aforementioned spaces—although their proofs are often specific). In such cases the chapter gives unique formulations, using one of the symbols S or Slp for any space. A multiplier on the space S means a function γ such that the operator u → γu maps S into S. The space of multipliers on S is denoted by MS. Because the operator of multiplication of a function γ ∈ MS by elements of the space S is closed, it is bounded. The norm of this operator serves as a norm of γ in MS. Some of the results presented in the chapter concern the spaces M(S1 → S2) of multipliers mapping S1 into S2. The chapter also discusses properties of multiplier spaces.