Abstract
In recent years, grand Lebesgue spaces, grand Lorentz spaces, and their generalizations have been extensively studied in functional analysis. This is because it has become evident that most known functional spaces are insufficient for modeling applied problems such as electrorheological fluids, thermorheological fluids, image processing, differential equations with non-standard growth, and other fields. Therefore, new precise scales of functional spaces have been introduced, namely variable exponent spaces and grand spaces. In this article, using the definition of anisotropic grand Lorentz spaces and their previously proven properties, we derive a previously unproven Hölder's inequality in this space. To prove these inequalities, we utilize the properties of decreasing rearrangements of functions. The study employs methodologies developed for multidimensional cases, including the analysis of relationships between ordered and rearranged versions of functions. The duality of these spaces is established by means of Hölder’s inequality. The results obtained are not only of theoretical importance but also find applications in practical problems, such as solving differential equations and studying integral operators. The findings presented in this article contribute to deepening the theory of functional spaces and expanding their areas of application.