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).
