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Transactions of the American Mathematical Society

Published by the American Mathematical Society, the Transactions of the American Mathematical Society (TRAN) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6850 (online) ISSN 0002-9947 (print)

The 2020 MCQ for Transactions of the American Mathematical Society is 1.43.

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$L^ p$ theory of differential forms on manifolds
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by Chad Scott PDF
Trans. Amer. Math. Soc. 347 (1995), 2075-2096 Request permission

Abstract:

In this paper, we establish a Hodge-type decomposition for the ${L^p}$ space of differential forms on closed (i.e., compact, oriented, smooth) Riemannian manifolds. Critical to the proof of this result is establishing an ${L^p}$ estimate which contains, as a special case, the ${L^2}$ result referred to by Morrey as Gaffney’s inequality. This inequality helps us show the equivalence of the usual definition of Sobolev space with a more geometric formulation which we provide in the case of differential forms on manifolds. We also prove the ${L^p}$ boundedness of Green’s operator which we use in developing the ${L^p}$ theory of the Hodge decomposition. For the calculus of variations, we rigorously verify that the spaces of exact and coexact forms are closed in the ${L^p}$ norm. For nonlinear analysis, we demonstrate the existence and uniqueness of a solution to the $A$-harmonic equation.
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Additional Information
  • © Copyright 1995 American Mathematical Society
  • Journal: Trans. Amer. Math. Soc. 347 (1995), 2075-2096
  • MSC: Primary 58A14; Secondary 58G03
  • DOI: https://doi.org/10.1090/S0002-9947-1995-1297538-7
  • MathSciNet review: 1297538