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Bulletin (new Series) of the American Mathematical Society

This monumental book, to which for brevity we will be referring to as Sobolev Multipliers, was published by Springer in the prestigious series Grundlehren der mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences]. It is devoted to a broadly developed theory of pointwise multipliers (or more precisely, multiplication operators) acting in pairs of Sobolev spaces, as well as other spaces of differentiable functions, along with their applications in analysis, partial differential equations, and mathematical physics. To distinguish pointwise multipliers from well-studied Fourier multipliers and to emphasize the underlying class of function spaces, they are conveniently called Sobolev multipliers. The book under review consists of two parts. Part I presents a general theory of multipliers in pairs of Sobolev spaces and their generalizations, while Part II is concerned with applications to the Schrödinger operator and its relativistic counerpart, as well as to studies of solutions of certain elliptic partial differential equations, both linear and quasi-linear, in divergence and nondivergence form, along with related pseudo-differential and integral equations. Questions of optimal regularity of the boundary, which are characterized completely in many cases in multiplier terms, as well as the singleand double-layer potential theory, including higher regularity, in Lipschitz domains, and calculus of singular integral operators with symbols in multiplier spaces compose a substantial part of this study. It is worth mentioning that Sobolev Multipliers includes most of the material of the previous book by the same authors entitled Theory of Multipliers in Spaces of Differential Functions (Pitman, 1985). It was reviewed by David R. Adams in the Bulletin of the American Mathematical Society, 15, No. 2 (1986), 254–259. That material has been vastly expanded and modernized, both in the scope and methods of the theory, and many new applications have been considered. (The only subject omitted in the present book is multipliers in spaces of analytic functions, which would require a substantial expansion in volume.) Multiplicative properties of functions in Sobolev spaces are notoriously hard to investigate. Although concrete examples of multipliers acting in a Sobolev space W k,p had been known for a long time, a study of pointwise multipliers in spaces of differentiable functions was started in earnest by Devinatz and Hirshman [DH], [Hir] (for p = 2, disguised as Fourier multipliers of l spaces with a power weight) in the late 1950s and early 1960s, and it was continued by Peetre [P], Polking [Pol], and Strichartz [Str] in the 1960–70s. In particular, in the latter paper ([Str]) a complete characterization was found for multipliers acting in the space of Bessel potentials H(R) in the simpler case kp > n.