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In this paper, we introduce the generalized grand Morrey spaces in the framework of probability space setting in the spirit of the martingale theory and grand Morrey spaces. The Doob maximal inequalities on the generalized grand Morrey spaces are provided. Moreover, we present the boundedness of fractional integral operators for regular martingales in this new framework.

The Boundedness of Doob’s Maximal and Fractional Integral Operators for Generalized Grand Morrey-Martingale Spaces