Let 0<γ<1. A function f∈Lp⁢(ℝ,A⁢(x)⁢d⁢x) is said to be in the Lipschitz class, denoted by Lip⁡(γ,p), if it satisfies

Let f be in the class Lip⁡(γ,p), where 0<γ<1 and 1<p≤2, and let p′ be the conjugate component of p. Then F⁢f belongs to Lδ([0,+∞[,dμ(λ)) for all δ satisfying

Let α>-12, α≥β≥-12 and x0>0. Then, for all λ∈R, there exist c>0 and η>0 such that the function φλ satisfies

Proof.

By [1, Lemma 9], there exists a constant c2>0 such that for all 0≤x≤x0, the function φλ satisfies

where jα, α>-12, is the normalized Bessel function of the first kind given by

(i) The asymptotic formulas for the Bessel function imply that jα⁢(x)→0 as x→∞. Consequently, there exists a number x0>0 such that, with x≥x0, the inequality |jα⁢(x)|<12 is true. Let m=minx∈[1,x0]|1-jα⁢(x)|. Then for all x≥1, we have |1-jα⁢(x)|≥c1, where c1=min⁡(m,12). We obtain relation (2) with c=c1⁢c2.

(ii) From (5), we see that limx→0⁡jα⁢(x)-1x2≠0. Then there exists η>0 such that |1-jα⁢(x)|≥c3⁢x2 for all |x|<η. From relation (4), we conclude that for all |λ⁢x|<η, we have inequality (3), where c=c2⁢c3. ∎

Proof of Theorem 1.

Let f be in Lp⁢(ℝ,A⁢(x)⁢d⁢x) satisfying relation (1). Using the Hausdorff–Young inequality, we obtain

On the other hand, we have

Using relation (3), we deduce that

For X>1, put ψ⁢(X)=∫1X|λ2⁢ℱ⁢f⁢(λ)|δ⁢𝑑μ⁢(λ). Then for δ≤p′ and applying Hölder’s inequality, we get

Hence, from the properties of the function |c⁢(λ)|-2 and (6) for 0<h<η, we obtain

Now, using [3, Theorem 8.17] and integrating by parts, we obtain

and this is bounded as X→∞ if -2⁢δ+(2-γ)⁢δ+(2⁢α+2)⁢(1-δp′)≤0, as δ≤p′, then (2⁢α+2)⁢p′γ⁢p′+2⁢α+2<δ≤p′. ∎

Let f∈Lp⁢(ℝ,A⁢(x)⁢d⁢x), 1≤p<∞. We say that f belongs to the Dini–Lipschitz class if it satisfies

We show that contrary to the Lipschitz conditions, the imposition of Dini–Lipschitz conditions on our functions does not improve upon the conclusion of the Hausdorff–Young inequality. Indeed, with the same tools used in the proof of the previous theorem, we prove that δ=p′.