Estimates for Differential Operators in Half-Space

This book serves largely as a research monograph for researchers in partial differential equations. It is based off of a series of articles written by the authors from 1972-1977. The history of this book is rather interesting. Originally written in Russian in the 1970’s, it was rejected by a Soviet publisher for political reasons, and never appeared in Russian. The book was then smuggled to East Germany, translated to German, and appeared in 1981 under the title Abschätzungen für Differentialoperatoren im Halbraum.  Almost 40 years later the book is now published in English.

 

The readers of this book should be familiar with linear algebra, functional analysis, ordinary differential equations, and partial differential equations.  The main goal of this work is to derive necessary and sufficient conditions for estimates in a half-space for a very wide class of differential operators with constant coefficients. Many examples are provided along the way, and some of the applicable differential operators include Schrödinger operators, Cauchy-Riemann operators, and more. The estimates derived are provided in complete detail and can be rather technical. To help the reader, the authors provide a great introductory section at the start of each chapter, which outlines the main results, and sketches some of the ideas for the main proofs. The reader will find these sections very helpful. Moreover, many references (mostly journal articles) are provided at the end of each chapter.

 

In addition to necessary and sufficient conditions for estimates in half-spaces, boundary estimates and estimates for maximal operators are also obtained. For the boundary estimates, a thorough description of the trace space is provided. In the chapter on estimates for maximal operators, a plethora of (still very general) examples are provided, mainly related to nonhomogeneous polynomials.

The estimates obtained by the authors in this book are then applied to well-posedness results for boundary value problems in a half-space. It is interesting to note that the corresponding estimates for \( L^{2} \) 2 functions with compact support in a domain have been studied by Hörmander in 1955.  The point is that general estimates up to the boundary had not been studied as thoroughly, although the authors provide a handful of references where such results were obtained in the 1950’s-1960’s.

 

Overall, this book is a nice reference for researchers in partial differential equations who are interested in estimates for a very large class of differential operators.

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Eric Stachura is currently an Assistant Professor of Mathematics at Kennesaw State University. He is generally interested in analysis and partial differential equations, especially treated with functional analysis techniques.