Weighted Hardy-Littlewood averages on central total Morrey-Guliyev spaces

In this paper, we study the weighted Hardy-Littlewood average Hψ and its commutator Hψbf=bHψf−Hψ(bf) on central total Morrey-Guliyev spaces. Our main results are as follows. First, we obtain a necessary and sufficient condition for the boundedness of Hψ on these spaces and identify the sharp operator norm, which is expressed in terms of a weighted integral of ψ. Second, we establish a necessary and sufficient condition for the boundedness of the commutator Hψb on central total Morrey-Guliyev spaces, showing that the symbol function b must belong to a suitable central BMO space. Moreover, we obtain analogous sharp constants for the Cesàro operator Gψ (the dual of Hψ) and establish necessary and sufficient conditions for the boundedness of its commutator.