Weighted integrability of Laguerre–Bessel transforms

The problem of weighted integrability of the Laguerre–Bessel transform in terms of the moduli of smoothness related to generalized translations is considered. Sufficient conditions are given to solve this problem. These results generalize a famous Titchmarsh’s theorem, due to [L. Rakhimi and R. Daher, An analog of Titchmarsh’s theorem for the Laguerre–Bessel transform, Arab. J. Math. Sci. 2023, 10.1108/AJMS-04-2022-0101] in the Laguerre–Bessel hypergroup. Also some results connected with the integrability of Laguerre–Bessel transforms are given.