Inequalities and Integral Operators in Function Spaces

The modern theory of functional spaces and operators, built on powerful analytical methods, continues to evolve in the search for more precise, universal, and effective tools. Classical inequalities such as Hardy’s inequality, Remez’s inequality, the Bernstein-Nikolsky inequality, the Hardy-Littlewood-Sobolev inequality for the Riesz transform, the Hardy-Littlewood inequality for Fourier transforms, O’Neil’s inequality for the convolution operator, and others play a fundamental role in analysis, and their influence is hard to overestimate. With the development of new interpolation methods, new functional spaces, and novel problem formulations for functions of many variables, these inequalities have undergone significant advancements. Inequalities and Integral Operators in Function Spaces focuses primarily on new approaches to the interpolation of spaces, which significantly extend the classical framework of the methods developed by Lions and Peetre. The book demonstrates how the use of net spaces and modern interpolation techniques not only provides a deeper understanding of the structure of functional spaces but also leads to stronger results that cannot be achieved within the traditional framework. Features: - Can be used for specialized courses in harmonic analysis focusing on interpolation - Suitable for both researchers in the field of real analysis and mathematicians interested in applying these methods to related areas - Contains new and interesting results, previously unpublished