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.

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.

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.

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).

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 TO A PAIR OF HANKEL–CLIFFORD TRANSFORMATIONS ON CERTAIN BEURLING TYPE FUNCTION SPACES

2013 ◽  
Vol 06 (03) ◽  
pp. 1350039 ◽  
Author(s):  
Akhilesh Prasad ◽  
V. K. Singh
Keyword(s):  
Differential Operators ◽  
Function Spaces ◽  
Type Function ◽  
Basic Properties ◽  
Pseudo Differential Operators

A brief introduction to the Hankel–Clifford transformations and its basic properties is given. The spaces [Formula: see text] and [Formula: see text] generalizing the spaces Hμ(I) and S(I), respectively are defined. It is given that the pseudo-differential operators h1,μ,a and h2,μ,a are automorphism of [Formula: see text] and [Formula: see text], respectively. Product and convolution on [Formula: see text] and [Formula: see text] are investigated. Some other spaces related to [Formula: see text] and [Formula: see text] are introduced and studied the continuity of the pseudo-differential operators h1,μ,a and h2,μ,a, respectively.

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.

The Mehler-Fock-Clifford transform and pseudo-differential operator on function spaces

Filomat ◽  
2019 ◽  
Vol 33 (8) ◽  
pp. 2457-2469
Author(s):  
Akhilesh Prasad ◽  
S.K. Verma
Keyword(s):  
Differential Operator ◽  
Function Spaces ◽  
Pseudo Differential Operator ◽  
Test Function ◽  
Basic Properties ◽  
Cone Function ◽  
Translation Operators

In this article, weintroduce a new index transform associated with the cone function Pi ??-1/2 (2?x), named as Mehler-Fock-Clifford transform and study its some basic properties. Convolution and translation operators are defined and obtained their estimates under Lp(I, x-1/2 dx) norm. The test function spaces G? and F? are introduced and discussed the continuity of the differential operator and MFC-transform on these spaces. Moreover, the pseudo-differential operator (p.d.o.) involving MFC-transform is defined and studied its continuity between G? and F?.

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