Kokilashvili V., Meskhi A., Rafeiro H., Samko S. Integral Operators in Non-Standard Function Spaces. Volume 1: Variable Exponent Lebesgue and Amalgam Spaces

Basel: Birkhäuser, 2016. — 567 p. — (Operator Theory: Advances and Applications 248). — ISBN: 978-3-319-21015-5; 978-3-319-21014-8.This book, the result of the authors' long and fruitful collaboration, focuses on integral operators in new, non-standard function spaces and presents a systematic study of the boundedness and compactness properties of basic, harmonic analysis integral operators in the following function spaces, among others: variable exponent Lebesgue and amalgam spaces, variable Hölder spaces, variable exponent Campanato, Morrey and Herz spaces, Iwaniec-Sbordone (grand Lebesgue) spaces, grand variable exponent Lebesgue spaces unifying the two spaces mentioned above, grand Morrey spaces, generalized grand Morrey spaces, and weighted analogues of some of them.
The results obtained are widely applied to non-linear PDEs, singular integrals and PDO theory. One of the book's most distinctive features is that the majority of the statements proved here are in the form of criteria.
The book is intended for a broad audience, ranging from researchers in the area to experts in applied mathematics and prospective students.Hardy-type Operators in Variable Exponent Lebesgue Spaces
Maximal, Singular, and Potential Operators in Variable Exponent Lebesgue Spaces with Oscillating Weights
Kernel Integral Operators
Two-weight Estimates
One-sided Operators
Two-weight Inequalities for Fractional Maximal Functions
Description of the Range of Potentials, and Hypersingular Integrals
More on Hypersingular Integrals and Embeddings into Hölder Spaces
More on Compactness
Applications to Singular Integral Equations