scholarly journals Domains of pseudo-differential operators: a case for the Triebel-Lizorkin spaces

2005 ◽  
Vol 3 (3) ◽  
pp. 263-286 ◽  
Author(s):  
Jon Johnsen
Keyword(s):  
Differential Operator ◽  
Differential Operators ◽  
Direct Application ◽  
Pseudo Differential Operator ◽  
Lizorkin Space ◽  
Pseudo Differential Operators ◽  
Hörmander’S Condition ◽  
Compact Spectrum

The main result is that every pseudo-differential operator of type 1, 1 and orderdis continuous from the Triebel-Lizorkin spaceFp,1dtoLp,1≤p≺∞, and that this is optimal within the Besov and Triebel-Lizorkin scales. The proof also leads to the known continuity fors≻d, while for all real s the sufficiency of Hörmander's condition on the twisted diagonal is carried over to the Besov and Triebel-Lizorkin framework. To obtain this, type 1, 1-operators are extended to distributions with compact spectrum, and Fourier transformed operators of this type are on such distributions proved to satisfy a support rule, generalising the rule for convolutions. Thereby the use of reduced symbols, as introduced by Coifman and Meyer, is replaced by direct application of the paradifferential methods. A few flaws in the literature have been detected and corrected.

Pseudo-differential operators associated with the Jacobi differential operator and Fourier-cosine wavelet transform

2015 ◽  
Vol 08 (01) ◽  
pp. 1550010 ◽  
Author(s):  
Akhilesh Prasad ◽  
Manoj K. Singh
Keyword(s):  
Wavelet Transform ◽  
Differential Operator ◽  
Differential Operators ◽  
Sobolev Type ◽  
Reconstruction Formula ◽  
Pseudo Differential Operator ◽  
Type Space ◽  
Fourier Cosine ◽  
Pseudo Differential Operators ◽  
Jacobi Transform

Using the inverse of Fourier–Jacobi transform a symbol is defined, and the pseudo-differential operator (p.d.o.) 𝒫α, β (x,D) associated with Jacobi-differential operator in terms of this symbol is defined. It is shown that the p.d.o. is bounded in a certain Sobolev type space associated with the Fourier–Jacobi transform. Continuous Jacobi wavelet transform (JWT) and Fourier-cosine wavelet transform are defined and a reconstruction formula is obtained for Fourier-cosine wavelet transform. Properties of Fourier-cosine wavelet transform are investigated.

PSEUDO-DIFFERENTIAL OPERATORS INVOLVING HANKEL–CLIFFORD TRANSFORMATION

2012 ◽  
Vol 05 (03) ◽  
pp. 1250040 ◽  
Author(s):  
Akhilesh Prasad ◽  
V. K. Singh ◽  
M. M. Dixit
Keyword(s):  
Differential Operator ◽  
Integral Representation ◽  
Differential Operators ◽  
Norm Inequality ◽  
Sobolev Type ◽  
Continuous Linear ◽  
Pseudo Differential Operator ◽  
Pseudo Differential Operators ◽  
Hankel Convolution ◽  
Type Spaces

Pseudo-differential operator (p.d.o) associated with the symbol a(x, y) whose derivatives satisfy certain growth condition is defined and the Zemanian-type spaces Hμ(I) and S(I) are introduced. It is shown that the p.d.o is continuous linear map of the space Hμ(I) and S(I) into itself. An integral representation of p.d.o h1, μ, a is obtained. Using the Hankel convolution it is shown that p.d.o h1, μ, a satisfies a certain [Formula: see text]-norm inequality. Properties of Sobolev-type space [Formula: see text] are studied.

scholarly journals PSEUDO-DIFFERENTIAL OPERATORS AND EQUATIONS IN A DISCRETE HALF-SPACE

2018 ◽  
Vol 23 (3) ◽  
pp. 492-506 ◽  
Author(s):  
Vladimir B. Vasilyev ◽  
Alexander V. Vasilyev
Keyword(s):  
Differential Operator ◽  
General Solution ◽  
Differential Equations ◽  
Half Space ◽  
Differential Operators ◽  
Pseudo Differential Operator ◽  
Pseudo Differential Operators ◽  
Special Case

We introduce a digital pseudo-differential operator acting in discrete Sobolev--Slobodetskii spaces and consider pseudo-differential equations with such operators in a discrete half-space. The theorem on a general solution of such equations is proved for a special case.

Some results on hypoelliptic pseudo-differential operators

1970 ◽  
Vol 68 (3) ◽  
pp. 685-695
Author(s):  
Robert J. Elliott
Keyword(s):  
Differential Operator ◽  
Differential Operators ◽  
Normal Operator ◽  
A Priori ◽  
Adjoint Operator ◽  
Elliptic Operators ◽  
Pseudo Differential Operator ◽  
Principal Type ◽  
Subelliptic Operators ◽  
Pseudo Differential Operators

In this paper, by extending the results of Yoshikawa (8), we obtain local a priori inequalities for hypoelliptic pseudo-differential operators. Using these inequalities we then show how the results of Hormander ((3), Theorem 8·7·2), on the solvability of the adjoint operator of a principally normal operator can be extended to the adjoint operator of a hypoelliptic pseudo-differential operator. Finally, we consider a class of operators which satisfy more particular a priori inequalities and we show that these operators are hypoelliptic. This class of operators was studied by Egorov (1), and he shows them to be ‘of principal type’. They include elliptic operators and also the subelliptic operators of Hormander (4).

Pseudo-differential operator involving generalized Hankel–Clifford transformation

2016 ◽  
Vol 09 (01) ◽  
pp. 1650016
Author(s):  
P. D. Pansare ◽  
B. B. Waphare
Keyword(s):  
Differential Operator ◽  
Integral Representation ◽  
Growth Condition ◽  
Differential Operators ◽  
Function Spaces ◽  
Continuous Linear ◽  
Pseudo Differential Operator ◽  
Type Function ◽  
Linear Map ◽  
Pseudo Differential Operators

Pseudo-differential operators (p.d.os) involving generalized Hankel–Clifford transformation associated with the symbol [Formula: see text] whose derivatives satisfy certain growth condition are defined and the Zemanian type function spaces [Formula: see text] and [Formula: see text] are introduced. It is shown that p.d.o’s are continuous linear map of the space [Formula: see text] and [Formula: see text] into itself. Also an Integral representation of p.d.o is obtained.

GENERATORS OF MEHLER-TYPE SEMIGROUPS AS PSEUDO-DIFFERENTIAL OPERATORS

2002 ◽  
Vol 05 (03) ◽  
pp. 297-315 ◽  
Author(s):  
PAUL LESCOT ◽  
MICHAEL RÖCKNER
Keyword(s):  
Hilbert Space ◽  
Differential Operator ◽  
Invariant Measure ◽  
Differential Operators ◽  
Pseudo Differential Operator ◽  
Type Formula ◽  
Pseudo Differential Operators

We study semigroups (Pt)t ≥ 0 on a Hilbert space E, given by a Mehler-type formula: [Formula: see text] Under reasonable assumptions, the Lp(E,μ)-generator [Formula: see text] of (Pt)t ≥ 0 turns out to be expressible as a pseudo-differential operator, provided μ is an invariant measure for (Pt)t ≥ 0. The question of Lp-uniqueness is also answered positively.

Modeling of pure acoustic wave in tilted transversely isotropic media using optimized pseudo-differential operators

Geophysics ◽  
2016 ◽  
Vol 81 (3) ◽  
pp. T91-T106 ◽  
Author(s):  
Jun Yan ◽  
Hong Liu
Keyword(s):  
Spectral Method ◽  
Differential Operators ◽  
Transversely Isotropic ◽  
Low Rank ◽  
Pseudo Differential Operator ◽  
Time Stepping ◽  
Pseudo Differential Operators ◽  
Tti Media ◽  
Pseudo Spectral Method ◽  
Pseudo Spectral

Conventional modeling and imaging for tilted transversely isotropic (TTI) media may suffer from numerical instabilities and shear artifacts due to the coupling of the P- and SV-wave modes in the coupled equations. On the contrary, the decoupled equations for TTI media provide a more stable solution due to the separated P- and S-wave modes. Because the decoupled equations involve complicated pseudo-differential operators in space, it is more convenient to apply the pseudo-spectral method to these equations. However, the second-order time-stepping scheme of the pseudo-spectral method may suffer from time-stepping errors and instabilities for a large time step. We have developed an optimized pseudo-differential operator (OPO) that incorporates not only the spectral evaluation of the pseudo-differential operator but also a temporal correction term that would effectively compensate the time-stepping errors associated with the time-wavenumber domain extrapolation of the wave equation. The OPO was constructed through multiplying the symbol of the OPO by the normalized pseudo-Laplacian operator, which contains a variable compensation velocity. It was efficiently solved through the low-rank decomposition. We have applied OPOs to solve the TTI decoupled equation to simulate the pure acoustic wave. Our 2D and 3D synthetic results demonstrate that the proposed method has high accuracy in time and space with relaxed stability conditions compared with the conventional pseudo-spectral method. The low rank of symbols of OPOs makes the proposed method more efficient than the dispersion relation-based low-rank wave extrapolation and pseudo-analytical methods.

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