Weighted Inequalities for a Class of Multilinear Integral Operators via Sharp Maximal Functions
In the fields of harmonic analysis, functional analysis and operator theory, studies of inequalities of classical operators (= singular, maximal, Riesz potentials, etc.) in various function spaces have a very important place. The maturation of many subjects in the field of analysis by introducing them as a result of various needs and developing them to respond to the problems of the time has led to the emergence of many studies and works on these subjects. Since Morrey spaces can be considered as an extension of Lebesgue spaces, it is natural and important to examine the boundednesses of multilinear integral operators on Morrey spaces. In this context, many studies focus on function spaces based on Morrey spaces to fill some gaps in the theory of Morrey-type spaces. Many researchers have begun to delve more and more into Morrey-type inequality problems due to the relationship of Morrey-type spaces with differential equations and the fact that differential equations are widely used in inequalities in Morrey-type spaces. For this reason, the importance of this issue is increasing. However, since these topics are beyond the scope of here, we will not go into details here. In this work, weighted boundednesses of some multilinear operators related to the Littlewood-Paley operator and the Marcinkiewicz operator are obtained by using sharp estimates of multilinear operators on generalized weighted Morrey spaces. This study has two purposes. The first is to create some sharp inequalities for the multilinear Littlewood-Paley and Marcinkiewicz operators. The second is to use these inequalities to prove the weighted boundedness of multilinear operators on generalized weighted Morrey spaces. This work is also suitable for graduate students who have some knowledge of real analysis and measure theory.
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