scholarly journals Inversion formulas for Riemann-Liouville transform and its dual associated with singular partial differential operators

2006 ◽  
Vol 2006 ◽  
pp. 1-26 ◽  
Author(s):  
C. Baccar ◽  
N. B. Hamadi ◽  
L. T. Rachdi
Keyword(s):  
Fourier Transform ◽  
Harmonic Analysis ◽  
Differential Operators ◽  
Partial Differential ◽  
Inversion Formulas ◽  
Plancherel Theorem ◽  
Partial Differential Operators ◽  
The Fourier Transform

We define Riemann-Liouville transformℛαand its dualtℛαassociated with two singular partial differential operators. We establish some results of harmonic analysis for the Fourier transform connected withℛα. Next, we prove inversion formulas for the operatorsℛα,tℛαand a Plancherel theorem fortℛα.

Uncertainty principle related to Flensted–Jensen partial differential operators

2019 ◽  
pp. 2150004
Author(s):  
Raoudha Laffi ◽  
Selma Negzaoui
Keyword(s):  
Fourier Transform ◽  
Uncertainty Principle ◽  
Differential Operators ◽  
Generalized Fourier Transform ◽  
Partial Differential ◽  
Hardy’S Theorem ◽  
Partial Differential Operators ◽  
Hörmander's Theorem

This paper deals with some formulations of the uncertainty principle associated to generalized Fourier transform [Formula: see text] related to Flensted–Jensen partial differential operators. The aim result is to prove the analogue of Bonami–Demange–Jaming’s theorem : A version of Beurling–Hörmander’s theorem which gives more precision in the form of nonzero functions verifying modified-Beurling’s condition. As application, we get analogous of Gelfand–Schilov’s theorem, Cowling–Price’s theorem and Hardy’s theorem for [Formula: see text].

The Calder´on-Zygmund Theory I: Ellipticity

2017 ◽  
Author(s):  
Brian Street
Keyword(s):  
Classical Theory ◽  
Singular Integral ◽  
Singular Integrals ◽  
Differential Operators ◽  
Integral Operators ◽  
Singular Integral Operators ◽  
Single Parameter ◽  
Partial Differential ◽  
Translation Invariant ◽  
Partial Differential Operators

This chapter discusses a case for single-parameter singular integral operators, where ρ‎ is the usual distance on ℝn. There, we obtain the most classical theory of singular integrals, which is useful for studying elliptic partial differential operators. The chapter defines singular integral operators in three equivalent ways. This trichotomy can be seen three times, in increasing generality: Theorems 1.1.23, 1.1.26, and 1.2.10. This trichotomy is developed even when the operators are not translation invariant (many authors discuss such ideas only for translation invariant, or nearly translation invariant operators). It also presents these ideas in a slightly different way than is usual, which helps to motivate later results and definitions.

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