Characterising Extended Lipschitz Type Conditions with Moduli of Continuity

Asymptotic Estimates for the Growth of Deformed Hankel Transform by Modulus of Continuity

“…The classical results used the Fourier transform as a basic model to give an asymptotic estimate of the growth of the norm of functions belonging to the Lipschitz class. Recently, it was proved that the aforementioned classical results can be extended to a more general family of growths called generalised modulus (moduli) of continuity [14], which, in particular, recover the Titchmarsh and Younis results. As a one of the simplest example of this class we can take the function t γ ln ln 1 t λ , where γ ∈ (0, 1) and λ ∈ R, which gives rise to new results [14].…”

“…Recently, it was proved that the aforementioned classical results can be extended to a more general family of growths called generalised modulus (moduli) of continuity [14], which, in particular, recover the Titchmarsh and Younis results. As a one of the simplest example of this class we can take the function t γ ln ln 1 t λ , where γ ∈ (0, 1) and λ ∈ R, which gives rise to new results [14]. For less trivial examples we can take t γ+ C ln λ 1 t .…”

Titchmarsh theorems on Damek-Ricci spaces via moduli of continuity of higher order

“…The Titchmarsh theorem has been extensively studied in many different contexts on various groups, for instances, the higher dimensional Euclidean spaces [31,5], the Vilenkin groups [32], the special linear group of real matrices of order two SL 2 (R) [33], the rank one symmetric spaces of non-compact type [25,15], the p-adic groups [26] and the compact homogeneous manifolds [12]. In terms of the moduli of continuity, the theorem has been explored on R [30,13] and the rank one symmetric spaces [16]. See [6,17,18] for some growth properties of the Fourier transform on certain spaces via moduli of continuity.…”

Generalized Lipschitz and Besov spaces in terms of decay of Dunkl transforms in the space L2(Rd, wl(x)dx)

“…Other studies of Lipschitz conditions have been done in [22,23,24,25] in term of the modulus of continuity. However, our aim in this paper is to extend the classical Titchmarsh theorems in case of functions of the wider Lipschitz class in the space L 2 (R d , w l (x)dx) in terms of the moduli of continuity of higher orders.…”