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Geometric Harmonic Analysis III - Dorina Mitrea & Irina Mitrea & Marius Mitrea
Description of Volume III
From the outset, the very formulation of our versions of the Divergence Theorem
from Volume I (cf. [112, §1.2–§1.12]) has been motivated and shaped by potential
applications to Harmonic Analysis, Partial Differential Equations, Function Space
Theory, and Complex Analysis. We have envisioned these versions of the Divergence
Theorem not as end-products, in and of themselves, but as effective tools to further
progress in these areas of mathematics. This has already become apparent in Volume
II ([113]), when dealing with function spaces measuring smoothness of Sobolev type
on the geometric measure theoretic boundaries of sets of locally finite perimeter.
In the opening chapter of the present volume (Chapter 1, titled “Integral Repre-
sentations and Integral Identities”), we further elaborate on this vision. We begin
in §1.1 by revisiting the classical Cauchy-Pompeiu integral representation formula
in open sets ⊆ C with a lower Ahlfors regular boundary and whose arc-length
σ := H1∂ is a doubling measure. Our Divergence Theorem specialized to this
setting then permits us to identify a very general class of functions for which the
Cauchy-Pompeiu integral representation formula is valid. By means of counterexam-
ples, we show that the analytic conditions imposed are in the nature of best possible.
A very general phiên bản of the Cauchy integral representation formula, allowing one
to recover a holomorphic function from its (nontangential) boundary trace via the
(boundary-to-domain) Cauchy integral operator, is then obtained as a corollary. In the
same spirit, generalizations of the classical Morera Theorem and Residue Theorem
are established. Variants with no explicit lower Ahlfors regularity assumptions made
on the topological boundary are also discussed.
This line of work continues in §1.2 where higher-dimensional versions of some
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Geometric Harmonic Analysis III - Dorina Mitrea & Irina Mitrea & Marius Mitrea
Description of Volume III
From the outset, the very formulation of our versions of the Divergence Theorem
from Volume I (cf. [112, §1.2–§1.12]) has been motivated and shaped by potential
applications to Harmonic Analysis, Partial Differential Equations, Function Space
Theory, and Complex Analysis. We have envisioned these versions of the Divergence
Theorem not as end-products, in and of themselves, but as effective tools to further
progress in these areas of mathematics. This has already become apparent in Volume
II ([113]), when dealing with function spaces measuring smoothness of Sobolev type
on the geometric measure theoretic boundaries of sets of locally finite perimeter.
In the opening chapter of the present volume (Chapter 1, titled “Integral Repre-
sentations and Integral Identities”), we further elaborate on this vision. We begin
in §1.1 by revisiting the classical Cauchy-Pompeiu integral representation formula
in open sets ⊆ C with a lower Ahlfors regular boundary and whose arc-length
σ := H1∂ is a doubling measure. Our Divergence Theorem specialized to this
setting then permits us to identify a very general class of functions for which the
Cauchy-Pompeiu integral representation formula is valid. By means of counterexam-
ples, we show that the analytic conditions imposed are in the nature of best possible.
A very general phiên bản of the Cauchy integral representation formula, allowing one
to recover a holomorphic function from its (nontangential) boundary trace via the
(boundary-to-domain) Cauchy integral operator, is then obtained as a corollary. In the
same spirit, generalizations of the classical Morera Theorem and Residue Theorem
are established. Variants with no explicit lower Ahlfors regularity assumptions made
on the topological boundary are also discussed.
This line of work continues in §1.2 where higher-dimensional versions of some
Do Drive thay đổi chính sách, nên một số link cũ yêu cầu duyệt download. các bạn chỉ cần làm theo hướng dẫn.
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