In this paper, we consider the deformed Hankel transform \({\mathscr {F}}_{\kappa } \), which is a deformation of the Hankel transform by a parameter \(\kappa >\frac{1}{4}\). We introduce, via modulus of continuity, a function subspace of \(L^p(d\mu _{\kappa })\) that we call deformed Hankel Dini–Lipschitz spaces. In the case \(p = 2\), we provide equivalence theorem: we get a characterization of those spaces by means of asymptotic estimate growth of the norm of their \({\mathscr {F}}_{\kappa }\) transform for \(0< \gamma < 1\) and \(\alpha \ge 0\). As a consequence we have the analogous of generalized Titchmarsh theorem in \( L^{2}(d\mu _{\kappa }) \). Moreover, we introduce the modulus of smoothness related to \({\mathscr {F}}_\kappa \) for which we study some properties on the Sobolev type space.

在本文中，我们考虑变形的 Hankel 变换\({\mathscr {F}}_{\kappa } \)，它是 Hankel 变换通过参数\(\kappa >\frac{1}{4 }\)。我们通过连续性模引入了一个函数子空间\(L^p(d\mu_{\kappa })\)，我们称之为变形 Hankel Dini-Lipschitz 空间。在\(p = 2\)的情况下，我们提供了等价定理：我们通过其\({\mathscr {F}}}_{\kappa }\)的范数的渐近估计增长来表征这些空间变换\(0< \gamma < 1\)和\(\alpha \ge 0\)。因此，我们有类似于广义 Titchmarsh 定理的\( L^{2}(d\mu _{\kappa }) \)。此外，我们引入了与\({\mathscr {F}}_\kappa \)相关的平滑模数，我们研究了 Sobolev 类型空间的一些属性。