An improvement of Bernstein’s inequality for functions in Orlicz spaces with smooth Fourier image

Abstract

Let Φ:[0,+∞)→[0,+∞] be an arbitrary Young function and K be a compact set in ℝn having (O)-property. Then there exists a constant CK,Φ<∞ independent of f such that ∥Dαf∥(Φ)≤CK,Φsupz∈K|zα|∥f∥ℒK,3 for all α∈ℤ+n and f∈ℒK,3, where ℒK,3={f∈𝒮′(ℝn):suppf^⊂K,D(3,3,…,3)f^∈C(ℝn)}, ∥f∥ℒK,3=∥D(3,3,…,3)f^∥∞, f^ is the Fourier transform of f and ∥⋅∥(Φ) is the Luxemburg norm. As an application, a new Paley–Wiener theorem is given.