scholarly journals Some spectral formulas for functions generated by differential and integral operators in Orlicz spaces

2021 ◽  
Vol 13 (2) ◽  
pp. 326-339
Author(s):  
H.H. Bang ◽  
V.N. Huy
Keyword(s):  
Fourier Transform ◽  
Orlicz Space ◽  
Differential Operators ◽  
Orlicz Spaces ◽  
Integral Operators ◽  
Young Function ◽  
The Fourier Transform

In this paper, we investigate the behavior of the sequence of $L^\Phi$-norm of functions, which are generated by differential and integral operators through their spectra (the support of the Fourier transform of a function $f$ is called its spectrum and denoted by sp$(f)$). With $Q$ being a polynomial, we introduce the notion of $Q$-primitives, which will return to the notion of primitives if ${Q}(x)= x$, and study the behavior of the sequence of norm of $Q$-primitives of functions in Orlicz space $L^\Phi(\mathbb R^n)$. We have the following main result: let $\Phi $ be an arbitrary Young function, ${Q}({\bf x} )$ be a polynomial and $(\mathcal{Q}^mf)_{m=0}^\infty \subset L^\Phi(\mathbb R^n)$ satisfies $\mathcal{Q}^0f=f, {Q}(D)\mathcal{Q}^{m+1}f=\mathcal{Q}^mf$ for $m\in\mathbb{Z}_+$. Assume that sp$(f)$ is compact and $sp(\mathcal{Q}^{m}f)= sp(f)$ for all $m\in \mathbb{Z}_+.$ Then $$ \lim\limits_{m\to \infty } \|\mathcal{Q}^m f\|_{\Phi}^{1/m}= \sup\limits_{{\bf x} \in sp(f)} \bigl|1/ {Q}({\bf x}) \bigl|. $$ The corresponding results for functions generated by differential operators and integral operators are also given.

scholarly journals A Sharp Rearrangement Principle in Fourier Space and Symmetry Results for PDEs with Arbitrary Order

2020 ◽  
Author(s):  
Enno Lenzmann ◽  
Jérémy Sok
Keyword(s):  
Fourier Transform ◽  
Phase Retrieval ◽  
Differential Operators ◽  
Arbitrary Order ◽  
Radial Symmetry ◽  
Fourier Space ◽  
Polyharmonic Operator ◽  
Pseudo Differential Operators ◽  
Retrieval Problem ◽  
The Fourier Transform

Abstract We prove sharp inequalities for the symmetric-decreasing rearrangement in Fourier space of functions in $\mathbb{R}^d$. Our main result can be applied to a general class of (pseudo-)differential operators in $\mathbb{R}^d$ of arbitrary order with radial Fourier multipliers. For example, we can take any positive power of the Laplacian $(-\Delta )^s$ with $s> 0$ and, in particular, any polyharmonic operator $(-\Delta )^m$ with integer $m \geqslant 1$. As applications, we prove radial symmetry and real-valuedness (up to trivial symmetries) of optimizers for (1) Gagliardo–Nirenberg inequalities with derivatives of arbitrary order, (2) ground states for bi- and polyharmonic nonlinear Schrödinger equations (NLS), and (3) Adams–Moser–Trudinger type inequalities for $H^{d/2}(\mathbb{R}^d)$ in any dimension $d \geqslant 1$. As a technical key result, we solve a phase retrieval problem for the Fourier transform in $\mathbb{R}^d$. To achieve this, we classify the case of equality in the corresponding Hardy–Littlewood majorant problem for the Fourier transform in $\mathbb{R}^d$.

Spectral theory for a class of pseudodifferential operators

1977 ◽  
Vol 76 (3) ◽  
pp. 197-214
Author(s):  
E. A. Catchpole
Keyword(s):  
Fourier Transform ◽  
Spectral Theory ◽  
Constant Coefficient ◽  
Eigenfunction Expansion ◽  
Differential Operators ◽  
Pseudodifferential Operators ◽  
Multiplication Operator ◽  
Expansion Theorem ◽  
Potential Term ◽  
The Fourier Transform

SynopsisWe investigate the spectral theory for a class of pseudodifferential operators which includes all constant coefficient differential operators, and also operators such as The operators considered are of the form Su(x) = Au(x)+q(x)u(x), where A is an operator which corresponds in the Fourier transform plane to a multiplication operator, and q(x) is a potential term. We prove an eigenfunction expansion theorem for S and derive some results concerning the spectrum of S.

scholarly journals Ray transform on Sobolev spaces of symmetric tensor fields, I: Higher order Reshetnyak formulas

2021 ◽  
Vol 0 (0) ◽  
pp. 0
Author(s):  
Venkateswaran P. Krishnan ◽  
Vladimir A. Sharafutdinov
Keyword(s):  
Fourier Transform ◽  
Sobolev Spaces ◽  
Tensor Field ◽  
Differential Operators ◽  
Symmetric Tensor ◽  
Higher Order ◽  
Tensor Fields ◽  
Ray Transform ◽  
The Fourier Transform ◽  
Derivatives Of

<p style='text-indent:20px;'>For an integer <inline-formula><tex-math id="M1">\begin{document}$ r\ge0 $\end{document}</tex-math></inline-formula>, we prove the <inline-formula><tex-math id="M2">\begin{document}$ r^{\mathrm{th}} $\end{document}</tex-math></inline-formula> order Reshetnyak formula for the ray transform of rank <inline-formula><tex-math id="M3">\begin{document}$ m $\end{document}</tex-math></inline-formula> symmetric tensor fields on <inline-formula><tex-math id="M4">\begin{document}$ {{\mathbb R}}^n $\end{document}</tex-math></inline-formula>. Roughly speaking, for a tensor field <inline-formula><tex-math id="M5">\begin{document}$ f $\end{document}</tex-math></inline-formula>, the order <inline-formula><tex-math id="M6">\begin{document}$ r $\end{document}</tex-math></inline-formula> refers to <inline-formula><tex-math id="M7">\begin{document}$ L^2 $\end{document}</tex-math></inline-formula>-integrability of higher order derivatives of the Fourier transform <inline-formula><tex-math id="M8">\begin{document}$ \widehat f $\end{document}</tex-math></inline-formula> over spheres centered at the origin. Certain differential operators <inline-formula><tex-math id="M9">\begin{document}$ A^{(m,r,l)}\ (0\le l\le r) $\end{document}</tex-math></inline-formula> on the sphere <inline-formula><tex-math id="M10">\begin{document}$ {{\mathbb S}}^{n-1} $\end{document}</tex-math></inline-formula> are main ingredients of the formula. The operators are defined by an algorithm that can be applied for any <inline-formula><tex-math id="M11">\begin{document}$ r $\end{document}</tex-math></inline-formula> although the volume of calculations grows fast with <inline-formula><tex-math id="M12">\begin{document}$ r $\end{document}</tex-math></inline-formula>. The algorithm is realized for small values of <inline-formula><tex-math id="M13">\begin{document}$ r $\end{document}</tex-math></inline-formula> and Reshetnyak formulas of orders <inline-formula><tex-math id="M14">\begin{document}$ 0,1,2 $\end{document}</tex-math></inline-formula> are presented in an explicit form.</p>

Fourier operators on weighted Hardy spaces

1987 ◽  
Vol 101 (1) ◽  
pp. 113-121
Author(s):  
Hans P. Heinig
Keyword(s):  
Fourier Transform ◽  
Hardy Spaces ◽  
Atomic Decomposition ◽  
Integral Operators ◽  
Muckenhoupt Weight ◽  
Hardy’S Inequality ◽  
Weighted Hardy Spaces ◽  
Hardy's Inequality ◽  
The Fourier Transform

AbstractIn this note we utilize the atomic decomposition of weighted Hardy spaces to prove weighted versions of Hardy's inequality for the Fourier transform with Muckenhoupt weight. The result extends to certain integral operators with homogeneous kernels of degree −1.

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.

scholarly journals Smoothness and Function Spaces Generated by Homogeneous Multipliers

2012 ◽  
Vol 2012 ◽  
pp. 1-22 ◽  
Author(s):  
Konstantin Runovski ◽  
Hans-Jürgen Schmeisser
Keyword(s):  
Fourier Transform ◽  
Approximation Theory ◽  
Differential Operators ◽  
Function Spaces ◽  
Explicit Representation ◽  
Periodic Functions ◽  
Homogeneous Functions ◽  
Representation Formulas ◽  
The Fourier Transform ◽  
And Function

Differential operators generated by homogeneous functionsψof an arbitrary real orders>0(ψ-derivatives) and related spaces ofs-smooth periodic functions ofdvariables are introduced and systematically studied. The obtained scale is compared with the scales of Besov and Triebel-Lizorkin spaces. Explicit representation formulas forψ-derivatives are obtained in terms of the Fourier transform of their generators. Some applications to approximation theory are discussed.

scholarly journals GLUING OF ABELIAN CATEGORIES AND DIFFERENTIAL OPERATORS ON THE BASIC AFFINE SPACE

2002 ◽  
Vol 1 (4) ◽  
pp. 543-557 ◽  
Author(s):  
Roman Bezrukavnikov ◽  
Alexander Braverman ◽  
Leonid Positselskii
Keyword(s):  
Fourier Transform ◽  
Affine Space ◽  
Differential Operators ◽  
General Notion ◽  
Unipotent Subgroup ◽  
D Modules ◽  
Abelian Categories ◽  
The Fourier Transform ◽  
Simply Connected ◽  
Basic Affine

The notion of gluing of abelian categories was introduced in a paper by Kazhdan and Laumon in 1988 and studied further by Polishchuk. We observe that this notion is a particular case of a general categorical construction.We then apply this general notion to the study of the ring of global differential operators $\mathcal{D}$ on the basic affine space $G/U$ (here $G$ is a semi-simple simply connected algebraic group over $\mathbb{C}$ and $U\subset G$ is a maximal unipotent subgroup).We show that the category of $\mathcal{D}$-modules is glued from $|W|$ copies of the category of $D$-modules on $G/U$ where $W$ is the Weyl group, and the Fourier transform is used to define the gluing data. As an application we prove that the algebra $\mathcal{D}$ is Noetherian, and get some information on its homological properties.AMS 2000 Mathematics subject classification: Primary 13N10; 16S32; 17B10; 18C20

Fourier transforms of convolution operators on orlicz spaces

2021 ◽  
Vol 71 (2) ◽  
pp. 369-382
Author(s):  
Seyyed Mohammad Tabatabaie ◽  
AliReza Bagheri Salec
Keyword(s):  
Fourier Transform ◽  
Convolution Operator ◽  
Fourier Transforms ◽  
Orlicz Spaces ◽  
Locally Compact Group ◽  
Convolution Operators ◽  
Locally Compact ◽  
Locally Compact Abelian Groups ◽  
Compact Abelian Groups ◽  
The Fourier Transform

Abstract In this paper, we study convolution operators on an Orlicz space L Φ(G) commuting with left translations, where Φ is an N-function and G is a locally compact group. We also present some basic properties of the Fourier transform of a Φ-convolution operator in the context of locally compact abelian groups.

scholarly journals Complete transformation of Radon-Kipriyanov. Some properties

2019 ◽  
Vol 489 (2) ◽  
pp. 125-130
Author(s):  
L. N. Lyakhov ◽  
M. G. Lapshina ◽  
S. A. Roshchupkin
Keyword(s):  
Fourier Transform ◽  
Differential Operator ◽  
Differential Operators ◽  
Support Theorem ◽  
Singular Differential Operator ◽  
Wiener Theorem ◽  
Bessel Transform ◽  
Bessel Transforms ◽  
The Fourier Transform ◽  
Complete Transformation

The even Radon-Kipriyanov transform (Kg-transform) is suitable for investigating problems with the Bessel singular differential operator Bi = 2i2+iii,i 0. In this paper, we introduce the odd Radon-Kipriyanov transform and complete Radon-Kipriyanov transform to investigation more general equations containing odd B‑derivativesiBik, k = 0, 1, 2, ... (in particular, gradients of functions). Formulas of K-transforms of singular differential operators are given. Based on the Bessel transforms introduced by B. M. Levitan and the odd Bessel transform introduced by I. A. Kipriyanov and V. V. Katrakhov, a connection was obtained between the complete Radon-Kipriyanov transform with the Fourier transform and the mixed Fourier-Levitan-Kipriyanov-Katrakhov transform. An analogue of Helgasons support theorem and an analogue of the Paley-Wiener theorem are presented.

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