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1997
The theory of pseudo-differential operators is one of the most important toolsin modern mathematics. It has found important applications in many mathematicaldevelopments. It was used in a crucial way in the proof of the Atiyah-SingerIndex theorem in [AtSi] and in the regularity of elliptic differential equations.In the theory of several complex ...
Yuri V. Egorov, Bert-Wolfgang Schulze
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The theory of pseudo-differential operators is one of the most important toolsin modern mathematics. It has found important applications in many mathematicaldevelopments. It was used in a crucial way in the proof of the Atiyah-SingerIndex theorem in [AtSi] and in the regularity of elliptic differential equations.In the theory of several complex ...
Yuri V. Egorov, Bert-Wolfgang Schulze
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Theory of Pseudo-differential Operators
2004In this chapter we present a brief description of the basic concepts and results from the theory of pseudo-differential operators – a modern theory of potentials –which will be used in the subsequent chapters. The development of the theory of pseudo-differential operators has greatly advanced our understanding of partial differential equations, and the
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2010
The main purpose of this chapter is to obtain the boundedness on \(A_{p, q}^{s, \tau }({\mathbb{R}}^ n)\) of all pseudo-differential operators of type (1,1) with inhomogeneous symbols. The smooth molecular decomposition characterizations of \(A_{p, q}^{s, \tau }({\mathbb{R}}^ n)\) play an important role in this chapter.
Wen Yuan, Winfried Sickel, Dachun Yang
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The main purpose of this chapter is to obtain the boundedness on \(A_{p, q}^{s, \tau }({\mathbb{R}}^ n)\) of all pseudo-differential operators of type (1,1) with inhomogeneous symbols. The smooth molecular decomposition characterizations of \(A_{p, q}^{s, \tau }({\mathbb{R}}^ n)\) play an important role in this chapter.
Wen Yuan, Winfried Sickel, Dachun Yang
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Pseudo-Differential Operators of Principal Type
1981In Section 10.4 we saw that the strength of a differential operator with constant coefficients in ℝ n is determined by the principal part p if and only if p=0 implies dp≠0 in ℝ n \0. Such operators were said to be of principal type. The purpose of this chapter is to study general operators P∈Ψ phg m (X) on a manifold X assuming that the condition dp ≠0
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Gel'Fand Theory of Pseudo Differential Operators
American Journal of Mathematics, 1968Introduction. The conventional pseudo differential operators were introduced by Kohn and Nirenberg [8] and extensively studied in a generalized version by Hdrmander [7]. In either case they appear as operators acting on C, -functions of a differentiable manifold; their definition is designed to obtain a class of linear operators containing all linear ...
Cordes, H. O., Herman, E. A.
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Annali di Matematica Pura ed Applicata, 1972
We present here a number of results on some aspects of Kohn-Nirenberg's theory of pseudo-differential operators. We hope that some parts of Kohn-Nirenberg's paper[1] are presented here in a more detailed and explicit form; this could help a larger audience to understand their ideas and methods.
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We present here a number of results on some aspects of Kohn-Nirenberg's theory of pseudo-differential operators. We hope that some parts of Kohn-Nirenberg's paper[1] are presented here in a more detailed and explicit form; this could help a larger audience to understand their ideas and methods.
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A Characterization of Fredholm Pseudo-Differential Operators
Journal of the London Mathematical Society, 1997The authors consider a pseudo-differential operator \(p(x,D)\) in \(\mathbb{R}^n\) with symbol satisfying for \(m>0\): \[ |D^\alpha_x D^\beta_\eta p(x,\eta)|\leq C_{\alpha\beta}(1+ |x|)^{-|\alpha|}(1+ |\eta|)^{m-|\beta|} \] and \[ |p(x,\eta)|\geq C(1+ |\eta|)^m\quad\text{for }|\eta|> R.
Fan, Qihong, Wong, M. W.
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2013
Pseudo-differential operators are important generalization of differential operators. These operators were first introduced in 1960 by Friedrichs and Lax in the study of singular integral differential operators, mainly, for inverting differential operators to solve differential equations.
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Pseudo-differential operators are important generalization of differential operators. These operators were first introduced in 1960 by Friedrichs and Lax in the study of singular integral differential operators, mainly, for inverting differential operators to solve differential equations.
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An algebra of pseudo‐differential operators
Communications on Pure and Applied Mathematics, 1965Kohn, J. J., Nirenberg, Louis
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A problem of nirenberg on pseudo‐differential operators
Communications on Pure and Applied Mathematics, 1970Consider a \(C^\infty\)-function \(p(x, \xi)\) in \(R_x^n\times R_\xi^n\) which satisfies the conditions: I. \(\vert p(x, \xi)\vert \le C\) for a constant \(C\), and II. for any \(a\) and \(N>0\) there exists a constant \(M_{\alpha,N}>0\) such that, for any \(\beta\), \(\vert\beta\vert \ge M_{\alpha,N}\), \[ \left\vert \partial_x^\alpha \partial_\xi ...
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