The subject. The phrase "integral operator" (like some other mathematically informal phrases, such as "effective procedure" and "geometric construction") is sometimes defined and sometimes not. When it is defined, the definition is likely to vary from author to author. While the definition almost always involves an integral, most of its other features can vary quite considerably. Superimposed limiting operations may enter (such as L2 limits in the theory of Fourier transforms and principal values in the theory of singular integrals), IJ' spaces and abstract Banach spaces may intervene, a scalar may be added (as in the theory of the so-called integral operators of the second kind), or, more generally, a multiplication operator may be added (as in the theory of the so-called integral operators of the third kind). The definition used in this book is the most special of all. According to it an integral operator is the natural "continuous" generali zation of the operators induced by matrices, and the only integrals that appear are the familiar Lebesgue-Stieltjes integrals on classical non-pathological mea sure spaces. The category. Some of the flavor of the theory can be perceived in finite dimensional linear algebra. Matrices are sometimes considered to be an un natural and notationally inelegant way of looking at linear transformations. From the point of view of this book that judgement misses something.

The monograph presents some of the authors' recent and original results concerning boundedness and compactness problems in Banach function spaces both for classical operators and integral transforms defined, generally speaking, on nonhomogeneous spaces. Itfocuses onintegral operators naturally arising in boundary value problems for PDE, the spectral theory of differential operators, continuum and quantum mechanics, stochastic processes etc. The book may be considered as a systematic and detailed analysis of a large class of specific integral operators from the boundedness and compactness point of view. A characteristic feature of the monograph is that most of the statements proved here have the form of criteria. These criteria enable us, for example, togive var ious explicit examples of pairs of weighted Banach function spaces governing boundedness/compactness of a wide class of integral operators. The book has two main parts. The first part, consisting of Chapters 1-5, covers theinvestigation ofclassical operators: Hardy-type transforms, fractional integrals, potentials and maximal functions. Our main goal is to give a complete description of those Banach function spaces in which the above-mentioned operators act boundedly (com pactly). When a given operator is not bounded (compact), for example in some Lebesgue space, we look for weighted spaces where boundedness (compact ness) holds. We develop the ideas and the techniques for the derivation of appropriate conditions, in terms of weights, which are equivalent to bounded ness (compactness).

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.

A self-contained account of integro-differential equations of the Barbashin type and partial integral operators. It presents the basic theory of Barbashin equations in spaces of continuous or measurable functions, including existence, uniqueness, stability and perturbation results. The theory and applications of partial integral operators and linea