In this article, we introduce a variable-order Bessel–Riesz integral operator and investigate its boundedness properties on variable Lebesgue spaces. We first establish the boundedness in the diagonal case under suitable regularity assumptions on the exponent function. For the nondiagonal setting, we derive a sharp pointwise estimate, which enables us to obtain boundedness results from $L^{p(\cdot )}({\mathbb{R}}_{+})$ to $L^{q(\cdot )}({\mathbb{R}}_{+})$ under appropriate relations between the exponent functions. Moreover, we establish the boundedness through an analog of Young's inequality. Because the classical Young's inequality generally fails in variable Lebesgue spaces due to the lack of translation invariance, we develop an alternative framework adapted to this setting. In particular, we identify sufficient conditions under which the associated kernel belongs to an appropriate variable Lebesgue space and use these conditions to derive the corresponding boundedness results. To the best of our knowledge, this class of operators has not been studied previously in the literature, and the study provides a systematic way to develop the boundedness of variable-order fractional integral operators in variable Lebesgue spaces.

In the present paper, we defined and then studied Hardy spaces related to spherical mean operators. We proved Hardy-type inequalities, then we showed refined Sobolev-type inequalities between homogeneous Riesz-type spaces, homogeneous Besov-type spaces, and Lorentz-type spaces. Next, we introduced and studied Hausdorff operators on generalized Hardy spaces. Finally, we investigated maximal Bochner-Riesz operators on generalized Hardy spaces.

In this paper, we introduce the notion of a wavelet multiplier in the setting of the linear canonical deformed Hankel transform (LCDHT), which depends on a symbol and two bounded functions. Then, we study the boundedness and compactness of these operators according to the symbol and the bounded functions. We will then show that, under certain assumptions, the wavelet multiplier is equal to the well-known time-frequency restriction operator. Then, we show that a function that is almost time- and band-limited can be approximated by its projection on the subspace spanned by the first eigenfunctions of such an operator, corresponding to the greatest eigenvalues, which are near one. This study for the LCDHT includes, in particular, some known transforms, such as the deformed Hankel, the Fresnel, and the fractional deformed Hankel transforms.