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Morrey spaces were introduced by Charles Morrey to investigate the
local behaviour of solutions to second order elliptic partial
differential equations. The technique is very useful in many areas
in mathematics, in particular in harmonic analysis, potential
theory, partial differential equations and mathematical physics.
Across two volumes, the authors of Morrey Spaces: Introduction and
Applications to Integral Operators and PDE's discuss the current
state of art and perspectives of developments of this theory of
Morrey spaces, with the emphasis in Volume II focused mainly
generalizations and interpolation of Morrey spaces. Features
Provides a 'from-scratch' overview of the topic readable by anyone
with an understanding of integration theory Suitable for graduate
students, masters course students, and researchers in PDE's or
Geometry Replete with exercises and examples to aid the reader's
understanding
Morrey spaces were introduced by Charles Morrey to investigate the
local behaviour of solutions to second order elliptic partial
differential equations. The technique is very useful in many areas
in mathematics, in particular in harmonic analysis, potential
theory, partial differential equations and mathematical physics.
Across two volumes, the authors of Morrey Spaces: Introduction and
Applications to Integral Operators and PDE's discuss the current
state of art and perspectives of developments of this theory of
Morrey spaces, with the emphasis in Volume I focused mainly on
harmonic analysis. Features Provides a 'from-scratch' overview of
the topic readable by anyone with an understanding of integration
theory Suitable for graduate students, masters course students, and
researchers in PDE's or Geometry Replete with exercises and
examples to aid the reader's understanding
Morrey spaces were introduced by Charles Morrey to investigate the
local behaviour of solutions to second order elliptic partial
differential equations. The technique is very useful in many areas
in mathematics, in particular in harmonic analysis, potential
theory, partial differential equations and mathematical physics.
Across two volumes, the authors of Morrey Spaces: Introduction and
Applications to Integral Operators and PDE's discuss the current
state of art and perspectives of developments of this theory of
Morrey spaces, with focus on harmonic analysis in volume I and
generalizations and interpolation of Morrey spaces in volume II.
Features Provides a 'from-scratch' overview of the topic readable
by anyone with an understanding of integration theory Suitable for
graduate students, masters course students, and researchers in
PDE's or Geometry Replete with exercises and examples to aid the
reader's understanding
This is a self-contained textbook of the theory of Besov spaces and
Triebel-Lizorkin spaces oriented toward applications to partial
differential equations and problems of harmonic analysis. These
include a priori estimates of elliptic differential equations, the
T1 theorem, pseudo-differential operators, the generator of
semi-group and spaces on domains, and the Kato problem. Various
function spaces are introduced to overcome the shortcomings of
Besov spaces and Triebel-Lizorkin spaces as well. The only prior
knowledge required of readers is familiarity with integration
theory and some elementary functional analysis.Illustrations are
included to show the complicated way in which spaces are defined.
Owing to that complexity, many definitions are required. The
necessary terminology is provided at the outset, and the theory of
distributions, L^p spaces, the Hardy-Littlewood maximal operator,
and the singular integral operators are called upon. One of the
highlights is that the proof of the Sobolev embedding theorem is
extremely simple. There are two types for each function space: a
homogeneous one and an inhomogeneous one. The theory of function
spaces, which readers usually learn in a standard course, can be
readily applied to the inhomogeneous one. However, that theory is
not sufficient for a homogeneous space; it needs to be reinforced
with some knowledge of the theory of distributions. This topic,
however subtle, is also covered within this volume. Additionally,
related function spaces-Hardy spaces, bounded mean oscillation
spaces, and Hoelder continuous spaces-are defined and discussed,
and it is shown that they are special cases of Besov spaces and
Triebel-Lizorkin spaces.
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