Boundedness and spectral invariance for standard pseudodifferential operators on anisotropically weighted LP-Sobolev spaces

1990 ◽  
Vol 13 (2) ◽  
pp. 271-284 ◽  
Author(s):  
Elmar Schrohe
Keyword(s):  
Sobolev Spaces ◽  
Pseudodifferential Operators ◽  
Spectral Invariance

scholarly journals SPECTRAL INVARIANCE FOR CERTAIN ALGEBRAS OF PSEUDODIFFERENTIAL OPERATORS

2005 ◽  
Vol 4 (3) ◽  
pp. 405-442 ◽  
Author(s):  
Robert Lauter ◽  
Bertrand Monthubert ◽  
Victor Nistor
Keyword(s):  
Pseudodifferential Operators ◽  
Spectral Invariance

scholarly journals The principal transmission condition

2021 ◽  
Vol 4 (4) ◽  
pp. 1-33
Author(s):  
Gerd Grubb ◽  
Keyword(s):  
Sobolev Spaces ◽  
Fractional Laplacian ◽  
Pseudodifferential Operators ◽  
Preceding Paper ◽  
Limited Range ◽  
Transmission Condition ◽  
Large Solutions ◽  
Homogeneous Dirichlet Problem ◽  
Integration By Parts Formula ◽  
Regularity Results

<abstract><p>The paper treats pseudodifferential operators $ P = \operatorname{Op}(p(\xi)) $ with homogeneous complex symbol $ p(\xi) $ of order $ 2a &gt; 0 $, generalizing the fractional Laplacian $ (-\Delta)^a $ but lacking its symmetries, and taken to act on the halfspace ${\mathbb R}^n_+$. The operators are seen to satisfy a principal $ \mu $-transmission condition relative to ${\mathbb R}^n_+$, but generally not the full $ \mu $-transmission condition satisfied by $ (-\Delta)^a $ and related operators (with $ \mu = a $). However, $ P $ acts well on the so-called $ \mu $-transmission spaces over ${\mathbb R}^n_+$ (defined in earlier works), and when $ P $ moreover is strongly elliptic, these spaces are the solution spaces for the homogeneous Dirichlet problem for $ P $, leading to regularity results with a factor $ x_n^\mu $ (in a limited range of Sobolev spaces). The information is then shown to be sufficient to establish an integration by parts formula over ${\mathbb R}^n_+$ for $ P $ acting on such functions. The formulation in Sobolev spaces, and the results on strongly elliptic operators going beyond certain operators with real kernels, are new. Furthermore, large solutions with nonzero Dirichlet traces are described, and a halfways Green's formula is established, as new results for these operators. Since the principal $ \mu $-transmission condition has weaker requirements than the full $ \mu $-transmission condition assumed in earlier papers, new arguments were needed, relying on work of Vishik and Eskin instead of the Boutet de Monvel theory. The results cover the case of nonsymmetric operators with real kernel that were only partially treated in a preceding paper.</p></abstract>

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