scholarly journals Singular integrals in quantum Euclidean spaces

2021 ◽  
Vol 272 (1334) ◽  
Author(s):  
Adrían González-Pérez ◽  
Marius Junge ◽  
Javier Parcet
Keyword(s):  
Singular Integral ◽  
Singular Integrals ◽  
Integral Theory ◽  
Regularity Of Solutions ◽  
Elliptic Pdes ◽  
Euclidean Spaces ◽  
Content Type ◽  
The Core ◽  
Pseudodifferential Calculus ◽  
New Form

We shall establish the core of singular integral theory and pseudodifferential calculus over the archetypal algebras of noncommutative geometry: quantum forms of Euclidean spaces and tori. Our results go beyond Connes’ pseudodifferential calculus for rotation algebras, thanks to a new form of Calderón-Zygmund theory over these spaces which crucially incorporates nonconvolution kernels. We deduce L p L_p -boundedness and Sobolev p p -estimates for regular, exotic and forbidden symbols in the expected ranks. In the L 2 L_2 level both Calderón-Vaillancourt and Bourdaud theorems for exotic and forbidden symbols are also generalized to the quantum setting. As a basic application of our methods, we prove L p L_p -regularity of solutions for elliptic PDEs.

scholarly journals vector-valued singular integrals and maximal functions on spaces of homogeneous type

2009 ◽  
Vol 104 (2) ◽  
pp. 296 ◽  
Author(s):  
Loukas Grafakos ◽  
Liguang Liu ◽  
Dachun Yang
Keyword(s):  
Singular Integral ◽  
Maximal Function ◽  
Singular Integrals ◽  
Maximal Functions ◽  
Integral Theory ◽  
Homogeneous Type ◽  
Spaces Of Homogeneous Type ◽  
Vector Valued

The Fefferman-Stein vector-valued maximal function inequality is proved for spaces of homogeneous type. The approach taken here is based on the theory of vector-valued Calderón-Zygmund singular integral theory in this context, which is appropriately developed.

scholarly journals Existence via regularity of solutions for elliptic systems and saddle points of functionals of the calculus of variations

2017 ◽  
Vol 6 (2) ◽  
pp. 99-120 ◽  
Author(s):  
Lucio Boccardo ◽  
Luigi Orsina
Keyword(s):  
Calculus Of Variations ◽  
Weak Solutions ◽  
Saddle Points ◽  
Elliptic Systems ◽  
Regularity Of Solutions ◽  
Integral Functionals ◽  
Content Type ◽  
The Core ◽  
Regularity Of Weak Solutions

AbstractThe core of this paper concerns the existence (via regularity) of weak solutions in ${W_{0}^{1,2}}$ of a class of elliptic systems such as$\left\{\begin{aligned} \displaystyle-\operatorname{div}((A+\varphi)\nabla u)&% \displaystyle=f,\\ \displaystyle-\operatorname{div}(M(x)\nabla\varphi)&\displaystyle=\frac{1}{2}% \lvert\nabla u\rvert^{2},\end{aligned}\right.$deriving from saddle points of integral functionals of the type$J(v,\psi)=\frac{1}{2}\int_{\Omega}(A+\psi_{+})\lvert\nabla v\rvert^{2}-\frac{1% }{2}\int_{\Omega}M(x)\nabla\psi\nabla\psi-\int_{\Omega}fv.$

Multi-parameter Singular Integrals II: General Theory

2017 ◽  
Author(s):  
Brian Street
Keyword(s):  
General Theory ◽  
Singular Integral ◽  
Singular Integrals ◽  
Integral Operators ◽  
Singular Integral Operators ◽  
Main Issue

This chapter turns to a general theory which generalizes and unifies all of the examples in the preceding chapters. A main issue is that the first definition from the trichotomy does not generalize to the multi-parameter situation. To deal with this, strengthened cancellation conditions are introduced. This is done in two different ways, resulting in four total definitions for singular integral operators (the first two use the strengthened cancellation conditions, while the later two are generalizations of the later two parts of the trichotomy). Thus, we obtain four classes of singular integral operators, denoted by A1, A2, A3, and A4. The main theorem of the chapter is A1 = A2 = A3 = A4; i.e., all four of these definitions are equivalent. This leads to many nice properties of these singular integral operators.

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.

scholarly journals A note on singular integrals with angular integrability

2019 ◽  
Vol 2019 (1) ◽  
Author(s):  
Feng Liu
Keyword(s):  
Singular Integral ◽  
Singular Integrals ◽  
Function Spaces ◽  
Riesz Transforms ◽  
Hilbert Transforms ◽  
Rotation Method ◽  
Vector Valued Function ◽  
Vector Valued ◽  
L Log L

Abstract In this note we study the rough singular integral $$ T_{\varOmega }f(x)=\mathrm{p.v.} \int _{\mathbb{R}^{n}}f(x-y)\frac{\varOmega (y/ \vert y \vert )}{ \vert y \vert ^{n}}\,dy, $$ T Ω f ( x ) = p . v . ∫ R n f ( x − y ) Ω ( y / | y | ) | y | n d y , where $n\geq 2$ n ≥ 2 and Ω is a function in $L\log L(\mathrm{S} ^{n-1})$ L log L ( S n − 1 ) with vanishing integral. We prove that $T_{\varOmega }$ T Ω is bounded on the mixed radial-angular spaces $L_{|x|}^{p}L_{\theta }^{\tilde{p}}( \mathbb{R}^{n})$ L | x | p L θ p ˜ ( R n ) , on the vector-valued mixed radial-angular spaces $L_{|x|}^{p}L_{\theta }^{\tilde{p}}(\mathbb{R}^{n},\ell ^{\tilde{p}})$ L | x | p L θ p ˜ ( R n , ℓ p ˜ ) and on the vector-valued function spaces $L^{p}(\mathbb{R}^{n}, \ell ^{\tilde{p}})$ L p ( R n , ℓ p ˜ ) if $1<\tilde{p}\leq p<\tilde{p}n/(n-1)$ 1 < p ˜ ≤ p < p ˜ n / ( n − 1 ) or $\tilde{p}n/(\tilde{p}+n-1)< p\leq \tilde{p}<\infty $ p ˜ n / ( p ˜ + n − 1 ) < p ≤ p ˜ < ∞ . The same conclusions hold for the well-known Riesz transforms and directional Hilbert transforms. It should be pointed out that our proof is based on the Calderón–Zygmund’s rotation method.

scholarly journals A note on maximal singular integrals with rough kernels

2020 ◽  
Vol 2020 (1) ◽  
Author(s):  
Xiao Zhang ◽  
Feng Liu
Keyword(s):  
Singular Integral ◽  
Singular Integrals ◽  
Integral Operators ◽  
Maximal Operators ◽  
Rough Kernels ◽  
Lebesgue Spaces ◽  
Homogeneous Mapping ◽  
Recent Developments ◽  
Maximal Singular Integrals ◽  
Maximal Singular Integral

Abstract In this note we study the maximal singular integral operators associated with a homogeneous mapping with rough kernels as well as the corresponding maximal operators. The boundedness and continuity on the Lebesgue spaces, Triebel–Lizorkin spaces, and Besov spaces are established for the above operators with rough kernels in $H^{1}({\mathrm{S}}^{n-1})$ H 1 ( S n − 1 ) , which complement some recent developments related to rough maximal singular integrals.

A criterion on oscillation and variation for the commutators of singular integral operators

2015 ◽  
Vol 27 (1) ◽  
Author(s):  
Feng Liu ◽  
Huoxiong Wu
Keyword(s):  
Singular Integral ◽  
Singular Integrals ◽  
Integral Operators ◽  
Singular Integral Operators ◽  
Weighted Norm ◽  
Norm Estimates ◽  
Variation Operators

AbstractThis paper gives a criterion on the weighted norm estimates of the oscillatory and variation operators for the commutators of Calderón–Zygmund singular integrals in dimension 1. As applications, the weighted

A quantitative investigation of the effects of new generation vs traditional upper secondary schools in Cambodia through the lens of STEM transfer model

2021 ◽  
Vol ahead-of-print (ahead-of-print) ◽  
Author(s):  
Sovansophal Kao
Keyword(s):  
Secondary Schools ◽  
Attitudes Toward Science ◽  
Content Type ◽  
Math Teachers ◽  
Upper Secondary ◽  
New Form ◽  
K 12 ◽  
New Generation ◽  
Upper Secondary Schools ◽  
Practical Implications

PurposeThis study aimed to examine the effects of Cambodia's New Generation Schools (NGS), as compared to their traditional counterparts, in enhancing the seven constructs: (1) science and math achievement, (2) science and math self-efficacy, (3) science and math outcome expectations, (4) attitudes toward science, (5) interactive science and math lessons, (6) support from science and math teachers and (7) encouragement and support in science from family.Design/methodology/approachThe two observations data was collected using self-rated questionnaire from 301 11th graders from five upper secondary schools located in three provinces of Cambodia. Independent sample t-test, Analysis of Covariance (ANCOVA) and dependent sample t-test were used to analyze the repeated measures data.FindingsThe study revealed that students from the NGS exhibited statistically higher scores for most constructs, especially for attitudes toward science. However, when controlling for the differences in the first observation data, the significant effects of the three constructs have been neutralized. Moreover, though there was an increase in science activities outside school, there was a negative trajectory in the other two sub-constructs of attitudes toward science and support from science and math teachers, both in NGS and traditional schools.Practical implicationsThese findings point some practical implications for enhancing the effectiveness of the two school types and further research.Originality/valueThere is heavy investment in new form of schools across the world to enhance students' learning and academic achievement in science and math in K-12 and to promote their interest in STEM in higher education. However, there is little document on the effectiveness of this new form of school, particularly in the Cambodian context.

Who is a “Senior”? Looking for a definition in the healthcare system

2021 ◽  
Vol ahead-of-print (ahead-of-print) ◽  
Author(s):  
Monica Consolandi
Keyword(s):  
Quality Of Life ◽  
Health Care ◽  
Healthcare System ◽  
Health Care System ◽  
Design Methodology ◽  
Social Role ◽  
Content Type ◽  
The Core ◽  
Starting Point

Purpose Seniors are nowadays at the core of important reflections to understand both how to ensure them a proper quality of life and better recognize their social role, providing them services and proper health care to value them as persons and resources. This paper aims to find a through definition about who is a senior, in the author’s opinion the starting point to help them flourishing. Design/methodology/approach As an example of definitions, an online dictionary and two geriatric text-books are quoted, highlighting qualities and rights referred to seniors especially in the delicate context of the health-care system. Findings The lack of a commonly shared perspective on this delicate kind of patient entails the difficulty to reach a coherent and satisfying definition about who a senior is. Originality/value The lack of a commonly shared definition leads to inevitable misunderstandings and could explain the arduousness of considering seniors in all their aspects. Further investigations are suggested.

Election delay will not solve Libya's problems

2021 ◽  
Keyword(s):  
House Of Representatives ◽  
Political Elite ◽  
The Political ◽  
Parliamentary Election ◽  
Electoral Process ◽  
Content Type ◽  
Parliamentary Elections ◽  
Armed Group ◽  
The Core

Significance This is expected to be followed by the first parliamentary election since 2014, at some point in early 2022. It now looks increasingly likely that both elections will be delayed. The electoral process lacks the elements it would need to be truly transformative, but it is prompting shifts in the political elite which will dictate developments for at least the next year. Impacts Khalifa Haftar will keep pushing for his armed group to form the core of Libya’s future army Seif al-Islam Qadhafi’s candidacy in the elections is unlikely to result in him becoming president. Aguileh Saleh looks set to stay on as House of Representatives speaker with no clear date for parliamentary elections.

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