ON THE STRONG TYPE MULTIPLIER NORMS OF RATIONAL FUNCTIONS IN SEVERAL VARIABLES

摘要

Let G be a locally compact abelian group, Γ its dual. For φ ∈ L~∞ (Γ) denote by T_φ the L~2(G) multiplier transform defined by φ. If T_φ extends to a bounded operator on L~p(G) we put N_p(φ) = ‖T_φ: L~p(G) → L~p(G)‖. Otherwise we put N_p(φ) = ∞. Denote by M(G) the space of regular complex-valued Borel measures on G with the total variation (denoted ‖ · ‖_(M(G))) as norm. We deal with the models G = R~d (d-dimensional Euclidean space) and G = T~d (the d-dimensional torus). In the present paper we study the dependence on p of the function p |→ N_p(φ).