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In this set of lecture notes, the author includes some of the latest research on the theory of Morrey Spaces associated with Harmonic Analysis. There are three main claims concerning these spaces that are covered: determining the integrability classes of the trace of Riesz potentials of an arbitrary Morrey function; determining the dimensions of singular sets of weak solutions of PDE (e.g. The Meyers-Elcart System); and determining whether there are any "full" interpolation results for linear operators between Morrey spaces.
This book will serve as a useful reference to graduate students and researchers interested in Potential Theory, Harmonic Analysis, PDE, and/or Morrey Space Theory.
List of contents
Introduction.- Function Spaces.- Hausforff Capacity.- Choquet Integrals.- Duality for Morrey Spaces.- Maximal Operators and Morrey Spaces.- Potential Operators on Morrey Spaces.- Singular Integrals on Morrey Spaces.- Morrey-Sobolev Capacities.- Traces of Morrey Potentials.- Interpolation of Morrey Spaces.- Commutators of Morrey Potentials.- Mock Morrey Spaces.- Morrey-Besov Spaces and Besov Capacities.- Morrey Potentials and PDE I.- Morrey Potentials and PDE II.- Morrey Spaces on Complete Riemannian Manifolds.
About the author
David R. Adams is a Professor in the Department of Mathematics at the University of Kentucky. He received his Ph.D. from the University of Minnesota in 1969. His research areas include analysis and partial differential equations.
Summary
In this set of lecture notes, the author includes some of the latest research on the theory of Morrey Spaces associated with Harmonic Analysis. There are three main claims concerning these spaces that are covered: determining the integrability classes of the trace of Riesz potentials of an arbitrary Morrey function; determining the dimensions of singular sets of weak solutions of PDE (e.g. The Meyers-Elcart System); and determining whether there are any “full” interpolation results for linear operators between Morrey spaces.
This book will serve as a useful reference to graduate students and researchers interested in Potential Theory, Harmonic Analysis, PDE, and/or Morrey Space Theory.
Additional text
“This book gives a comparatively systematic discussion on the theory of Morrey spaces. … This book puts a lot of emphasis on the predual theory of the Morrey spaces as well as their applications. … the book has an impressive level of generality on the modern theory of Morrey spaces … . In addition, the related theories for Morrey spaces promise developments of this field in the near future.” (Liguang Liu, Mathematical Reviews, May, 2017)
“This book contains the latest results obtained by the author. It is a useful reference to mathematicians working in potential theory, harmonic analysis and partial differential equations, in particular, in Morrey spaces theory.” (Sibei Yang, zbMATH 1339.42001, 2016)
Report
"This book gives a comparatively systematic discussion on the theory of Morrey spaces. ... This book puts a lot of emphasis on the predual theory of the Morrey spaces as well as their applications. ... the book has an impressive level of generality on the modern theory of Morrey spaces ... . In addition, the related theories for Morrey spaces promise developments of this field in the near future." (Liguang Liu, Mathematical Reviews, May, 2017)
"This book contains the latest results obtained by the author. It is a useful reference to mathematicians working in potential theory, harmonic analysis and partial differential equations, in particular, in Morrey spaces theory." (Sibei Yang, zbMATH 1339.42001, 2016)
