scholarly journals Remarks on the Unimodular Fourier Multipliers onα-Modulation Spaces

2014 ◽  
Vol 2014 ◽  
pp. 1-8 ◽  
Author(s):  
Guoping Zhao ◽  
Jiecheng Chen ◽  
Weichao Guo
Keyword(s):  
Besov Spaces ◽  
Fourier Multiplier ◽  
Radial Function ◽  
Necessary Conditions ◽  
Fourier Multipliers ◽  
Modulation Spaces ◽  
Multiplier Operator ◽  
Size Condition ◽  
Sufficient And Necessary Conditions ◽  
Fourier Multiplier Operator

We study the boundedness properties of the Fourier multiplier operatoreiμ(D)onα-modulation spacesMp,qs,α  (0≤α<1)and Besov spacesBp,qs(Mp,qs,1). We improve the conditions for the boundedness of Fourier multipliers with compact supports and for the boundedness ofeiμ(D)onMp,qs,α. Ifμis a radial functionϕ(|ξ|)andϕsatisfies some size condition, we obtain the sufficient and necessary conditions for the boundedness ofeiϕ(|D|)betweenMp1,q1s1,αandMp2,q2s2,α.

Embedding properties of weighted Besov-type spaces

2015 ◽  
Vol 13 (05) ◽  
pp. 507-553 ◽  
Author(s):  
Wen Yuan ◽  
Dorothee D. Haroske ◽  
Leszek Skrzypczak ◽  
Dachun Yang
Keyword(s):  
Besov Spaces ◽  
Necessary Conditions ◽  
Muckenhoupt Weight ◽  
Exponential Weights ◽  
Special Cases ◽  
Type Spaces ◽  
Sufficient And Necessary Conditions ◽  
Weighted Besov Spaces ◽  
Dyadic Cubes ◽  
Besov Type Spaces

In this paper, we consider the embeddings of weighted Besov spaces [Formula: see text] into Besov-type spaces [Formula: see text] with w being a (local) Muckenhoupt weight, and give sufficient and necessary conditions on the continuity and the compactness of these embeddings. As special cases, we characterize the continuity and the compactness of embeddings in case of some polynomial or exponential weights. The proofs of these conclusions strongly depend on the geometric properties of dyadic cubes.

Endpoint estimates for a trilinear pseudo-differential operator with flag symbols

2021 ◽  
Vol 33 (4) ◽  
pp. 1015-1032
Author(s):  
Jiao Chen ◽  
Liang Huang ◽  
Guozhen Lu
Keyword(s):  
Differential Operator ◽  
Hardy Space ◽  
Hardy Spaces ◽  
Lebesgue Space ◽  
Fourier Multiplier ◽  
Pseudo Differential Operator ◽  
Multiplier Operator ◽  
Local Hardy Spaces ◽  
Fourier Multiplier Operator

Abstract In this paper, we establish the endpoint estimate ( 0 < p ≤ 1 {0<p\leq 1} ) for a trilinear pseudo-differential operator, where the symbol involved is given by the product of two standard symbols from the bilinear Hörmander class B ⁢ S 1 , 0 0 {BS^{0}_{1,0}} . The study of this operator is motivated from the L p {L^{p}} ( 1 < p < ∞ {1<p<\infty} ) estimates for the trilinear Fourier multiplier operator with flag singularities considered in [C. Muscalu, Paraproducts with flag singularities. I. A case study, Rev. Mat. Iberoam. 23 2007, 2, 705–742] and Hardy space estimates in [A. Miyachi and N. Tomita, Estimates for trilinear flag paraproducts on L ∞ L^{\infty} and Hardy spaces, Math. Z. 282 2016, 1–2, 577–613], and the L p {L^{p}} ( 1 < p < ∞ {1<p<\infty} ) estimates for the trilinear pseudo-differential operator with flag symbols in [G. Lu and L. Zhang, L p L^{p} -estimates for a trilinear pseudo-differential operator with flag symbols, Indiana Univ. Math. J. 66 2017, 3, 877–900]. More precisely, we will show that the trilinear pseudo-differential operator with flag symbols defined in (1.3) maps from the product of local Hardy spaces to the Lebesgue space, i.e., h p 1 × h p 2 × h p 3 → L p {h^{p_{1}}\times h^{p_{2}}\times h^{p_{3}}\rightarrow L^{p}} with 1 p 1 + 1 p 2 + 1 p 3 = 1 p {\frac{1}{p_{1}}+\frac{1}{p_{2}}+\frac{1}{p_{3}}=\frac{1}{p}} with 0 < p < ∞ {0<p<\infty} (see Theorem 1.6 and Theorem 1.7).

Stability Analysis of Rectangular Plates in Incompressible Flow With Fourier Multiplier Operators

2013 ◽  
Author(s):  
Kensuke Hara ◽  
Masahiro Watanabe
Keyword(s):  
Integral Equation ◽  
Stability Analysis ◽  
Fourier Multiplier ◽  
Fourier Transforms ◽  
Mixed Boundary ◽  
Computation Time ◽  
Rectangular Plates ◽  
Multiplier Operator ◽  
Fourier Multiplier Operator ◽  
Interaction Problem

This paper describes a development of a method which improves the computational efficiency for a linear stability analysis of a plate in an uniform incompressible and irrotational flow. We introduce the Fourier multiplier operator to formulate the fluid and plate interaction problem with the mixed boundary condition. In previous typical approaches, a singular integral equation often appears in the formulation of a pressure distribution on the plate. The computation time for solving the integral equation is one of the problem encountered in the stability analysis. Applying the Fourier multiplier operator to this system, the equation of the plate-fluid interaction problem can be formulated with a pair of the Fourier and the inverse Fourier transforms. Moreover, the integration to derive the equations of motion can be efficiently carried out by using the discrete Fourier transform.

ON THE BILINEAR SQUARE FOURIER MULTIPLIER OPERATORS ASSOCIATED WITH FUNCTION

2018 ◽  
Vol 239 ◽  
pp. 123-152
Author(s):  
ZHENGYANG LI ◽  
QINGYING XUE
Keyword(s):  
Fourier Multiplier ◽  
Square Function ◽  
Type Estimate ◽  
Multiplier Operator ◽  
Image Position ◽  
Fourier Multiplier Operators ◽  
Multiplier Operators ◽  
Fourier Multiplier Operator ◽  
Formula Type ◽  
L Log L

This paper will be devoted to study a class of bilinear square-function Fourier multiplier operator associated with a symbol $m$ defined by $$\begin{eqnarray}\displaystyle & & \displaystyle \mathfrak{T}_{\unicode[STIX]{x1D706},m}(f_{1},f_{2})(x)\nonumber\\ \displaystyle & & \displaystyle \quad =\Big(\iint _{\mathbb{R}_{+}^{n+1}}\Big(\frac{t}{|x-z|+t}\Big)^{n\unicode[STIX]{x1D706}}\nonumber\\ \displaystyle & & \displaystyle \qquad \times \,\bigg|\int _{(\mathbb{R}^{n})^{2}}e^{2\unicode[STIX]{x1D70B}ix\cdot (\unicode[STIX]{x1D709}_{1}+\unicode[STIX]{x1D709}_{2})}m(t\unicode[STIX]{x1D709}_{1},t\unicode[STIX]{x1D709}_{2})\hat{f}_{1}(\unicode[STIX]{x1D709}_{1})\hat{f}_{2}(\unicode[STIX]{x1D709}_{2})\,d\unicode[STIX]{x1D709}_{1}\,d\unicode[STIX]{x1D709}_{2}\bigg|^{2}\frac{dz\,dt}{t^{n+1}}\Big)^{1/2}.\nonumber\end{eqnarray}$$ A basic fact about $\mathfrak{T}_{\unicode[STIX]{x1D706},m}$ is that it is closely associated with the multilinear Littlewood–Paley $g_{\unicode[STIX]{x1D706}}^{\ast }$ function. In this paper we first investigate the boundedness of $\mathfrak{T}_{\unicode[STIX]{x1D706},m}$ on products of weighted Lebesgue spaces. Then, the weighted endpoint $L\log L$ type estimate and strong estimate for the commutators of $\mathfrak{T}_{\unicode[STIX]{x1D706},m}$ will be demonstrated.

scholarly journals OPERATOR-VALUED FOURIER MULTIPLIERS ON PERIODIC BESOV SPACES AND APPLICATIONS

2004 ◽  
Vol 47 (1) ◽  
pp. 15-33 ◽  
Author(s):  
Wolfgang Arendt ◽  
Shangquan Bu
Keyword(s):  
Banach Space ◽  
Besov Space ◽  
Real Line ◽  
Besov Spaces ◽  
Fourier Multiplier ◽  
Maximal Regularity ◽  
Fourier Multipliers ◽  
Subject Classification ◽  
Cauchy Problems ◽  
Multiplier Theorem

AbstractLet $1\leq p,q\leq\infty$, $s\in\mathbb{R}$ and let $X$ be a Banach space. We show that the analogue of Marcinkiewicz’s Fourier multiplier theorem on $L^p(\mathbb{T})$ holds for the Besov space $B_{p,q}^s(\mathbb{T};X)$ if and only if $1\ltp\lt\infty$ and $X$ is a UMD-space. Introducing stronger conditions we obtain a periodic Fourier multiplier theorem which is valid without restriction on the indices or the space (which is analogous to Amann’s result (Math. Nachr.186 (1997), 5–56) on the real line). It is used to characterize maximal regularity of periodic Cauchy problems.AMS 2000 Mathematics subject classification: Primary 47D06; 42A45

scholarly journals Oscillation of Second-Order Differential Equations with Multiple and Mixed Delays under a Canonical Operator

Mathematics ◽  
2021 ◽  
Vol 9 (12) ◽  
pp. 1323
Author(s):  
Shyam Sundar Santra ◽  
Rami Ahmad El-Nabulsi ◽  
Khaled Mohamed Khedher
Keyword(s):  
Differential Equations ◽  
Necessary Conditions ◽  
Second Order ◽  
Multiple Delays ◽  
Mixed Delays ◽  
Neutral Differential Equations ◽  
Canonical Operator ◽  
Second Order Differential Equations ◽  
The Future ◽  
Sufficient And Necessary Conditions

In this work, we obtained new sufficient and necessary conditions for the oscillation of second-order differential equations with mixed and multiple delays under a canonical operator. Our methods could be applicable to find the sufficient and necessary conditions for any neutral differential equations. Furthermore, we proved the validity of the obtained results via particular examples. At the end of the paper, we provide the future scope of this study.

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