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This volume presents advances that have been made over recent
decades in areas of research featuring Hardy's inequality and
related topics. The inequality and its extensions and refinements
are not only of intrinsic interest but are indispensable tools in
many areas of mathematics and mathematical physics. Hardy
inequalities on domains have a substantial role and this
necessitates a detailed investigation of significant geometric
properties of a domain and its boundary. Other topics covered in
this volume are Hardy- Sobolev-Maz'ya inequalities; inequalities of
Hardy-type involving magnetic fields; Hardy, Sobolev and
Cwikel-Lieb-Rosenbljum inequalities for Pauli operators; the
Rellich inequality. The Analysis and Geometry of Hardy's Inequality
provides an up-to-date account of research in areas of contemporary
interest and would be suitable for a graduate course in mathematics
or physics. A good basic knowledge of real and complex analysis is
a prerequisite.
The book deals with the representation in series form of compact
linear operators acting between Banach spaces, and provides an
analogue of the classical Hilbert space results of this nature that
have their roots in the work of D. Hilbert, F. Riesz and E.
Schmidt. The representation involves a recursively obtained
sequence of points on the unit sphere of the initial space and a
corresponding sequence of positive numbers that correspond to the
eigenvectors and eigenvalues of the map in the Hilbert space case.
The lack of orthogonality is partially compensated by the
systematic use of polar sets. There are applications to the
p-Laplacian and similar nonlinear partial differential equations.
Preliminary material is presented in the first chapter, the main
results being established in Chapter 2. The final chapter is
devoted to the problems encountered when trying to represent
non-compact maps.
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