Moduli of Continuity and Average Decay of Fourier Transforms: Two-sided Estimates and a Cantor Type Example

Following D. B. H. Cline [Cl], we study the connections between the behavior of the spherical mean of an L modulus of continuity corresponding to the finite difference of an arbitrary order m > 0 of a function f ∈ L(R), 1 ≤ p ≤ ∞, d ∈ N, and the average decay of the Fourier transform f̂ . We obtain an estimate which refines one result from [Cl] for 1 ≤ p ≤ 2. Our main interest is in two-sided estimates. Using the mentioned refinement, and also a result from [Cl], in the case p = 2, we prove an equivalence between the two-sided estimates on the modulus of continuity on one hand, and on the tail of the Fourier transform, on the other. We also construct for any d ∈ N and 0 < β ≤ 1, an example of a spherically symmetric compact set Ω ⊂ R such that the d-dimensional Lebesgue measure of Ω \ (Ω− h) is bounded from above and from below by the multiples of |h| for |h| ≤ 1. The results of this paper are applied in [Gi2] in the proof of a sharp remainder estimate in a certain Szegö type asymptotic formula for integral operators with discontinuous symbols acting in R, d ∈ N.