Singular convolution operators on Wiener amalgam spaces

2015 ◽  
Vol 13 (01) ◽  
pp. 1550003
Author(s):  
Meifang Cheng
Keyword(s):  
Modulation Spaces ◽  
Convolution Operators ◽  
Lebesgue Spaces ◽  
Wiener Amalgam Spaces ◽  
Amalgam Spaces

The purpose of this paper is to investigate some properties of the Wiener amalgam spaces. As applications, we study the boundedness properties of some singular convolution operators on such spaces. From our results, we will see that, besides modulation spaces, Wiener amalgam spaces are another good substitutions for Lebesgue spaces.

scholarly journals Norm Inflation for Benjamin–Bona–Mahony Equation in Fourier Amalgam and Wiener Amalgam Spaces with Negative Regularity

Mathematics ◽  
2021 ◽  
Vol 9 (23) ◽  
pp. 3145
Author(s):  
Divyang G. Bhimani ◽  
Saikatul Haque
Keyword(s):  
Initial Data ◽  
Modulation Spaces ◽  
Well Posedness ◽  
Wiener Amalgam Spaces ◽  
General Initial Data ◽  
Amalgam Spaces ◽  
Loss Of Regularity ◽  
Bbm Equation

We consider the Benjamin–Bona–Mahony (BBM) equation of the form ut+ux+uux−uxxt=0,(x,t)∈M×R where M=T or R. We establish norm inflation (NI) with infinite loss of regularity at general initial data in Fourier amalgam and Wiener amalgam spaces with negative regularity. This strengthens several known NI results at zero initial data in Hs(T) established by Bona–Dai (2017) and the ill-posedness result established by Bona–Tzvetkov (2008) and Panthee (2011) in Hs(R). Our result is sharp with respect to the local well-posedness result of Banquet–Villamizar–Roa (2021) in modulation spaces Ms2,1(R) for s≥0.

OPERATING FUNCTIONS ON MODULATION AND WIENER AMALGAM SPACES

2017 ◽  
Vol 230 ◽  
pp. 72-82 ◽  
Author(s):  
MASAHARU KOBAYASHI ◽  
ENJI SATO
Keyword(s):  
Open Problem ◽  
Composition Operators ◽  
Affirmative Answer ◽  
Modulation Spaces ◽  
Wiener Amalgam Spaces ◽  
Amalgam Spaces ◽  
Funct Anal

The goal of this paper is to characterize the operating functions on modulation spaces$M^{p,1}(\mathbb{R})$and Wiener amalgam spaces$W^{p,1}(\mathbb{R})$. This characterization gives an affirmative answer to the open problem proposed by Bhimani (Composition Operators on Wiener amalgam Spaces, arXiv: 1503.01606) and Bhimani and Ratnakumar (J. Funct. Anal.270(2016), pp. 621–648).

scholarly journals On Variable Exponent Amalgam Spaces

2012 ◽  
Vol 20 (3) ◽  
pp. 5-20 ◽  
Author(s):  
İsmail Aydin
Keyword(s):  
Lebesgue Space ◽  
Variable Exponent ◽  
Weighted Lebesgue Space ◽  
Lebesgue Spaces ◽  
Wiener Amalgam Spaces ◽  
Local Component ◽  
Variable Exponent Lebesgue Space ◽  
New Family ◽  
Amalgam Spaces ◽  
Weighted Variable

Abstract We derive some of the basic properties of weighted variable exponent Lebesgue spaces Lp(.)w (ℝn) and investigate embeddings of these spaces under some conditions. Also a new family of Wiener amalgam spaces W(Lp(.)w ;Lqv) is defined, where the local component is a weighted variable exponent Lebesgue space Lp(.)w (ℝn) and the global component is a weighted Lebesgue space Lqv (ℝn) : We investigate the properties of the spaces W(Lp(.)w ;Lqv): We also present new Hölder-type inequalities and embeddings for these spaces.

scholarly journals SHARPNESS OF SOME PROPERTIES OF WIENER AMALGAM AND MODULATION SPACES

2009 ◽  
Vol 80 (1) ◽  
pp. 105-116 ◽  
Author(s):  
ELENA CORDERO ◽  
FABIO NICOLA
Keyword(s):  
Modulation Spaces ◽  
Wiener Amalgam Spaces ◽  
Dilation Operator ◽  
Sharp Estimates ◽  
Amalgam Spaces ◽  
Scaling Arguments

AbstractWe prove sharp estimates for the dilation operator f(x)⟼f(λx), when acting on Wiener amalgam spaces W(Lp,Lq). Scaling arguments are also used to prove the sharpness of the known convolution and pointwise relations for modulation spaces Mp,q, as well as the optimality of an estimate for the Schrödinger propagator on modulation spaces.

scholarly journals COMPOSITION OPERATORS ON WIENER AMALGAM SPACES

2019 ◽  
Vol 240 ◽  
pp. 257-274
Author(s):  
DIVYANG G. BHIMANI
Keyword(s):  
Composition Operator ◽  
Composition Operators ◽  
Complex Function ◽  
Affirmative Answer ◽  
Modulation Spaces ◽  
Wiener Amalgam Spaces ◽  
Real Analytic ◽  
Amalgam Spaces ◽  
Open Question ◽  
Funct Anal

For a complex function $F$ on $\mathbb{C}$, we study the associated composition operator $T_{F}(f):=F\circ f=F(f)$ on Wiener amalgam $W^{p,q}(\mathbb{R}^{d})\;(1\leqslant p<\infty ,1\leqslant q<2)$. We have shown $T_{F}$ maps $W^{p,1}(\mathbb{R}^{d})$ to $W^{p,q}(\mathbb{R}^{d})$ if and only if $F$ is real analytic on $\mathbb{R}^{2}$ and $F(0)=0$. Similar result is proved in the case of modulation spaces $M^{p,q}(\mathbb{R}^{d})$. In particular, this gives an affirmative answer to the open question proposed in Bhimani and Ratnakumar (J. Funct. Anal. 270(2) (2016), 621–648).

STRONGLY SINGULAR CONVOLUTION OPERATORS ON MODULATION SPACES

2011 ◽  
Vol 09 (05) ◽  
pp. 759-772 ◽  
Author(s):  
MEIFANG CHENG ◽  
ZHENQIU ZHANG
Keyword(s):  
Modulation Spaces ◽  
Convolution Operators ◽  
Lebesgue Spaces

The purpose of this paper is to investigate the mapping properties of the strongly singular convolution operators on general weighted modulation spaces [Formula: see text] for 0 < p ≤ ∞, 0 < q ≤ ∞ and s ∈ ℝ. Our results show that modulation spaces are good substitutions for Lebesgue spaces.

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