Rational multipliers and analytical properties of their transforms
Banach Center Publications
The results presented in this paper are concentrated around the analysis of translation invariant singular operators acting in the scale of L p spaces (1 ≤ p ≤ ∞). We study the operators acting on function spaces on abelian groups: d-dimensional euclidean spaces R d and d-dimensional tori T d . A linear operator T acting on the space L p (G) (where G is one of the groups mentioned above) is called invariant iff S a T = T S a for every S a where S a denotes the operator of shift by a ∈ G. The
... ft by a ∈ G. The invariant operator T : L p (G) → L p (G) can be expressed in terms of the Fourier transform " " and its inverse " ∨ ": The function m appearing in this formula is called a multiplier. Conversely, the operator T m corresponding to the multiplier m is called a multiplier transform of m. The fundamental example of invariant operators are differential operators with constant coefficients. Multipliers which they generate are rational functions of several variables. These operators are our main object of study. They arise in the natural way in the theory of Sobolev spaces and they are the main analytical tools to study their properties. The behavior of such operators as p → 1 (we are going to study the dependence of their L p norm on p ∈ [0, ∞]) is of particular interest. We stress that very often we deal with the singularities stronger that those described by the Calderón-Zygmund conditions (cf. [CZ], [St]). The main class of operators which we study, as well as questions about their boundedness, arise from the theory of anisotropic Sobolev spaces. We shall present the multiplier theorems in this context, showing their natural motivations. To describe more precisely this class we present briefly basic definitions concerning the Sobolev spaces (cf. e.g. [BBPW], [P]).