Read more
The purpose of this book is to give a comprehensive exposition of the theory of pointwise multipliers acting in pairs of spaces of differentiable functions. The theory was essentially developed by the authors during the last thirty years and the present volume is mainly based on their results.
Part I is devoted to the theory of multipliers and encloses the following topics: trace inequalities, analytic characterization of multipliers, relations between spaces of Sobolev multipliers and other function spaces, maximal subalgebras of multiplier spaces, traces and extensions of multipliers, essential norm and compactness of multipliers, and miscellaneous properties of multipliers.
Part II concerns several applications of this theory: continuity and compactness of differential operators in pairs of Sobolev spaces, multipliers as solutions to linear and quasilinear elliptic equations, higher regularity in the single and double layer potential theory for Lipschitz domains, regularity of the boundary in $L_p$-theory of elliptic boundary value problems, and singular integral operators in Sobolev spaces.
List of contents
Description and Properties of Multipliers.- Trace Inequalities for Functions in Sobolev Spaces.- Multipliers in Pairs of Sobolev Spaces.- Multipliers in Pairs of Potential Spaces.- The Space M(B m p ? B l p ) with p > 1.- The Space M(B m 1 ? B l 1).- Maximal Algebras in Spaces of Multipliers.- Essential Norm and Compactness of Multipliers.- Traces and Extensions of Multipliers.- Sobolev Multipliers in a Domain, Multiplier Mappings and Manifolds.- Applications of Multipliers to Differential and Integral Operators.- Differential Operators in Pairs of Sobolev Spaces.- Schrödinger Operator and M(w 1 2 ? w ?1 2).- Relativistic Schrödinger Operator and M(W ½ 2 ? W ?½ 2).- Multipliers as Solutions to Elliptic Equations.- Regularity of the Boundary in L p -Theory of Elliptic Boundary Value Problems.- Multipliers in the Classical Layer Potential Theory for Lipschitz Domains.- Applications of Multipliers to the Theory of Integral Operators.
About the author
In July 2009 the Senior Whitehead Prize of the London Mathematical Society was awarded to Professor Maz'ya. He has also received the Celcius Gold Medal from the Swedish Royal Society in Uppsala (2004), the Verdaguer Prize from the Academie de France (2003), and the Humboldt Research Prize (1999) in Germany. In 2002 he was elected to the Royal Swedish Academy of Sciences and in 2001 he became Corresponding Fellow of the Scottish National Academy. He was an invited speaker at the International Congress of Mathematicians (2002) and on the occasion of his 70th birthday (2008) two international conferences in Rome and Stockholm were organized. In 2009 five volumes dedicated to him were published in USA, Italy and Germany.
Summary
The first part of this book offers a comprehensive overview of the theory of pointwise multipliers acting in pairs of spaces of differentiable functions. The second part of the book explores several applications of this theory.
Additional text
From the reviews:
“This very interesting book … collects the multitude of new results in this area, essentially obtained by the authors or inspired by their work during the past thirty years. … This comprehensive monograph is very well written and structured. It will certainly become an extremely useful reference for mathematicians working in functional analysis and in the theories of partial differential, integral, and pseudodifferential operators. … it recommendable both for experts and mathematicians with a wider field of interest.” (Dorothee D. Haroske, Mathematical Reviews, Issue 2010 d)
“The main goal of Sobolev Multipliers is to present complete characterizations and applications of multipliers Mf acting from a Sobolev space … . it will be of interest to a great variety of serious readers, from graduate students to experts in analysis … as well as mathematical physicists, differential geometers, and applied mathematicians. … a valuable reference and guide to the literature, as well as a unique collection of methods and results–indispensable for anyone who is using Sobolev spaces in their work.” (I. E. Verbitsky, Bulletin of the American Mathematical Society, Vol. 48 (1), January, 2011)
Report
From the reviews:
"This very interesting book ... collects the multitude of new results in this area, essentially obtained by the authors or inspired by their work during the past thirty years. ... This comprehensive monograph is very well written and structured. It will certainly become an extremely useful reference for mathematicians working in functional analysis and in the theories of partial differential, integral, and pseudodifferential operators. ... it recommendable both for experts and mathematicians with a wider field of interest." (Dorothee D. Haroske, Mathematical Reviews, Issue 2010 d)
"The main goal of Sobolev Multipliers is to present complete characterizations and applications of multipliers Mf acting from a Sobolev space ... . it will be of interest to a great variety of serious readers, from graduate students to experts in analysis ... as well as mathematical physicists, differential geometers, and applied mathematicians. ... a valuable reference and guide to the literature, as well as a unique collection of methods and results-indispensable for anyone who is using Sobolev spaces in their work." (I. E. Verbitsky, Bulletin of the American Mathematical Society, Vol. 48 (1), January, 2011)
