Bernstein inequality for multivariate functions with smooth Fourier images

Abstract

UDC 517.5 Let K be a compact set in \BbbR n with ( O ) -property and let 1 ≤ p ≤ ∞ . Then there exists a constant C K < ∞ independent of f and α such that ‖ D α f ‖ p ≤ C K sup ξ ∈ K | ξ α | ‖ f ‖ ℋ p , K ,3  for all α ∈ ℤ + n and f ∈ ℋ p , K ,3 , where  ℋ p , K ,3 = { f ∈ L p ( \BbbR n ) : s u p p f ^ ⊂ K , D ( 3,3 , … ,3 ) f ^ ∈ C ( \BbbR n ) } , , and f ^ is the Fourier transform of f . Note that K is said to have the ( O ) -property if there exists a constant C > 0 such that  for all α ∈ ℤ + n and j = 1,2 , … , n .