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We introduce the minimal operator on weighted grand Lebesgue spaces, discuss some weighted norm inequalities and characterize the conditions under which the inequalities hold. We also prove that the John-Nirenberg inequalities in the framework of weighted grand Lebesgue spaces are valid provided that the weight function belongs to the Muckenhoupt class.
This paper is devoted to the study of noncommutative weak Orlicz spaces and martingale inequalities. The Marcinkiewicz interpolation theorem is extended to include noncommutative weak Orlicz spaces as interpolation classes. As an application, we prove the weak type Φ-moment Burkholder-Gundy inequality for noncommutative martingales through establishing a weak type Φ-moment noncommutative Khinchin inequality for Rademacher random variables.
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