Weighted Paley–Wiener theorem on the Hilbert transform

Two power-weight inequalities for the Hilbert transform on the cones of monotone functions

“…The case, when Àq À 155À1 and Àp À 155À1, is reduced to the inequality Z The proof of (20) is similar to the proof of (4) provided (6) and (17) and (18) hold. g Remark 2 Note that sufficiency in Theorems 1 and 2 for the case p ¼ q ¼ 1 and for general monotone functions [18,19] also follows from [5]. …”

“…hold for w 2 A 1 (in particular, w(x) ¼ jxj , À15 0), even under the assumption of oddness or evenness of f [5]. In order to extend Hardy-Littlewood's and Flett's results for the case p ¼ 1, one needs to consider even or odd function f, respectively, such that f is monotone on R þ :¼ [0, 1).…”

“…The characterizations of (4) and (5) are given in Sections 2 and 3, respectively. In this article we assume that the right-hand side integrals in inequalities (4) and (5) are finite. We use signs :¼ and ¼: for determining new quantities and Z for the set of all integers.…”

On Fourier Re-Expansions

“…Though an odd function always satisfies (7), not every odd integrable function belongs to H 1 (R), for a counterexample see, e.g., [10]. Paley-Wiener's theorem (see [12]; for alternative proof and discussion, see Zygmund's paper [16]) asserts that if g ∈ L 1 (R) is an odd and monotone decreasing on R + function, then Hg ∈ L 1 , i.e., g is in H 1 (R).…”

Interaction between the Fourier transform and the Hilbert transform

“…An odd function always has mean zero. However, not every odd integrable function belongs to H 1 (R): in [17] an example of an odd function with non-integrable Hilbert transform is given. To this end, take g(t)…”