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In this paper, we prove the boundedness of Hardy operator for monotone functions in grand Lebesgue spaces  Lp ) (0, 1) (0 &lt;  p  ≤ 1). In particular, we prove similar results for the Hardy operator in weighted Lebesgue spaces. Also, it is proved that the grand Lebesgue space  Lp ) (0, 1) is a quasi-Banach function space. Finally, we establish necessary and sufficient conditions for the boundedness of some integral operator in weighted quasi-Banach Lebesgue spaces.

On Hardy type inequalities in grand Lebesgue spaces L p) for 0 < p ≤ 1