Integrability of $q$-Bessel Fourier transforms with Gogoladze–Meskhia type weights

In the paper, we consider the $q$-integrability of functions $\lambda(t)|\mathcal F_{q, \nu}(f)(t)|^r$, where $\lambda(t)$ is a Gogoladze-Meskhia-Moricz type weight and $\mathcal F_{q, \nu}(f)(t)$ is the $q$-Bessel Fourier transforms of a function $f$ from generalized integral Lipschitz classes. There are some corollaries for power type and constant weights, which are analogues of classical results of Titchmarsh et al. Also, a $q$-analogue of the famous Herz theorem is proved.