Degenerate Pseudodifferential Operators Research Articles

On some degenerate pseudodifferential operators

In this paper, a new class of degenerate pseudo-differential operators is investigated, with a variable symbol depending on the complex parameter. Pseudodifferential operators are constructed by a special integral transform. Theorems on the composition and boundedness of these operators in special weighted spaces are proved. The behavior of these operators on hyperplanes of degeneration is investigated. The theorems on the commutation of these operators with differentiation operators are established. A adjoint operator is constructed and an analogue of Goring inequality for degenerate pseudodifferential operators is proved.

On Some Degenerate Pseudodifferential Operators

On Some Degenerate Pseudodifferential Operators

On some properties of a class of degenerate pseudodifferential operators

On some properties of a class of degenerate pseudodifferential operators

On some properties of a class of degenerate pseudodifferential operators

A new class of pseudodifferential operators with degeneration is considered. The operators are constructed using a special integral transform mapping a weighted differentiation operator to a multiplication operator. The composition and boundedness properties of such operators in special weighted spaces are examined. Theorems on commutation of such operators with differentiation operators are obtained. The behavior of these operators as t → 0and t → +∞ is investigated. The properties of adjoint operators are studied, and an analogue of Garding’s inequality is proved.

Some s‐numbers of embeddings of function spaces with weights of logarithmic type

AbstractIn this paper we deal with compact embeddings of weighted function spaces of Besov and Triebel‐Lizorkin type with weights of logarithmic growth near infinity. We obtain the exact estimates for the asymptotic behaviour of Gelfand, Kolmogorov and Weyl numbers of the embeddings. As an application we get eigenvalue estimates of some degenerate pseudodifferential operators of type. Finally we deal with the “negative spectrum” of operators Hγ = A − γV for γ → ∞, where the potential V is a positive function and A is a positive elliptic pseudodifferential operator. We concentrate on the case when the Weyl numbers and Pietsch's inequality give us better estimates then entropy numbers and Carl's inequality.

Spectral theory of some degenerate elliptic operators with local singularities

This paper is based on our previous results (Haroske and Skrzypczak (2008) [23], Haroske and Skrzypczak (in press) [25]) on compact embeddings of Muckenhoupt weighted function spaces of Besov and Triebel–Lizorkin type with example weights of polynomial growth near infinity and near some local singularity. Our approach also extends (Haroske and Triebel (1994) [21]) in various ways. We obtain eigenvalue estimates of degenerate pseudodifferential operators of type b 2 ○ p ( x , D ) ○ b 1 where b i ∈ L r i ( R n , w i ) , w i ∈ A ∞ , i = 1 , 2 , and p ( x , D ) ∈ Ψ 1 , 0 − ϰ , ϰ > 0 . Finally we deal with the ‘negative spectrum’ of some operator H γ = A − γ V for γ → ∞ , where the potential V may have singularities (in terms of Muckenhoupt weights), and A is a positive elliptic pseudodifferential operator of order ϰ > 0 , self-adjoint in L 2 ( R n ) . This part essentially relies on the Birman–Schwinger principle. We conclude this paper with a number of examples, also comparing our results with preceding ones.

Semilinear boundary value problems for degenerate pseudodifferential operators in spaces of Sobolev type

Boundary value problems for degenerate semilinear elliptic pseudodifferential operators are considered. Using the apparatus of the theory of pseudodifferential operators, function spaces introduced in [4–8], and the Rabinowitz construction [12] based on the Borsuk theorem (see [11]), we prove the existence of solutions of the problem in suitable function spaces.

Entropy Numbers in Weighted Function Spaces and Eigenvalue Distributions of Some Degenerate Pseudodifferential Operators II

AbstractThis paper is the continuation of [17]. We investigate mapping and spectral properties of pseudodifferential operators of type Ψ with χ χ ϵ ℝ and 0 ≤ γ ≤ 1 in the weighted function spaces B (ℝn, w(x)) and F (ℝn, w(x)) treated in [17]. Furthermore, we study the distribution of eigenvalues and the behaviour of corresponding root spaces for degenerate pseudodifferential operators preferably of type b2(x) b(x, D) b1(x), where b1(x) and b2(x) are appropriate functions and b(x, D) ϵ Ψ. Finally, on the basis of the Birman‐Schwinger principle, we deal with the “negative spectrum” (bound states) of related symmetric operators in L2.

Entropy Numbers in Weighted Function Spaces and Eigenvalue Distributions of Some Degenerate Pseudodifferential Operators I

AbstractIn this paper we study weighted function spaces of type B(ℝn, Q(x)) and F(ℝn, Q(x)), where Q(x) is a weight function of at most polynomial growth. Of special interest are the weight functions Q(x) = (1 + |x|2)α/2 with α ϵ ℝ. The main result deals with estimates for the entropy numbers of compact embeddings between spaces of this type.

On the index of degenerate pseudodifferential operators on a closed curve

We consider a class of degenerate classical pseudodifferential operators on a closed curve and compute their index in Sobolev spaces. The index is expressed as a winding number by means of the principal and the subprincipal symbol. Furthermore, applications to the smoothness of solutions and the degenerate oblique derivative problem in the plane are given.

BOUNDARY VALUE PROBLEMS IN A HALF-SPACE FOR QUASIELLIPTIC PSEUDODIFFERENTIAL OPERATORS DEGENERATING ON THE BOUNDARY

In this paper, a local regularizer is constructed for degenerate pseudodifferential operators. Bibliography: 6 titles.

On degenerate boundary value problems for elliptic differential operators of second order in the plane and the index of degenerate pseudo-differential operators on a closed curve

Mathematische NachrichtenVolume 102, Issue 1 p. 277-292 Article On degenerate boundary value problems for elliptic differential operators of second order in the plane and the index of degenerate pseudo-differential operators on a closed curve Johannes Elschner, Johannes Elschner Institut für Mathematik, der Akademie der Wissenschaften der DDR, DDR - 1080 Berlin, Mohrenstraße 39Search for more papers by this author Johannes Elschner, Johannes Elschner Institut für Mathematik, der Akademie der Wissenschaften der DDR, DDR - 1080 Berlin, Mohrenstraße 39Search for more papers by this author First published: 1981 https://doi.org/10.1002/mana.19811020123Citations: 3AboutPDF ToolsRequest permissionExport citationAdd to favoritesTrack citation ShareShare Give accessShare full text accessShare full-text accessPlease review our Terms and Conditions of Use and check box below to share full-text version of article.I have read and accept the Wiley Online Library Terms and Conditions of UseShareable LinkUse the link below to share a full-text version of this article with your friends and colleagues. Learn more.Copy URL Share a linkShare onFacebookTwitterLinked InRedditWechat Citing Literature Volume102, Issue11981Pages 277-292 RelatedInformation