scholarly journals Flett Potential Spaces and a Weighted Wavelet-Like Transform

2020 ◽  
Vol 10 (3) ◽  
pp. 112-122
Author(s):  
Sinem Sezer Evcan ◽  
◽  
Sevda Barut
Keyword(s):  
Finite Differences ◽  
Bessel Potentials ◽  
Poisson Semigroup ◽  
Classical Potential

In this study, the Flett potential spaces are defined and a characterization of these potential spaces is given. Most of the known characterizations of classical potential spaces such as Riesz, Bessel potentials spaces and their generalizations are given in terms of finite differences. Here, by taking wavelet measure instead of finite differences, a weighted wavelet-like transform associated with Poisson semigroup is defined. And, by making use of this weighted wavelet-like transform, a new “truncated" integrals are defined, then using these integrals a characterization of the Flett potential spaces is given.

Carleson measure characterizations of the Campanato type space associated with Schrödinger operators on stratified Lie groups

2020 ◽  
Vol 32 (5) ◽  
pp. 1337-1373 ◽  
Author(s):  
Yixin Wang ◽  
Yu Liu ◽  
Chuanhong Sun ◽  
Pengtao Li
Keyword(s):  
Carleson Measures ◽  
Heat Semigroup ◽  
Stratified Lie Group ◽  
Poisson Semigroup ◽  
Homogeneous Dimension ◽  
Type Spaces ◽  
Stratified Lie Groups ◽  
Lusin Area Function ◽  
Reverse Hölder

AbstractLet {\mathcal{L}=-{\Delta}_{\mathbb{G}}+V} be a Schrödinger operator on the stratified Lie group {\mathbb{G}}, where {{\Delta}_{\mathbb{G}}} is the sub-Laplacian and the nonnegative potential V belongs to the reverse Hölder class {B_{q_{0}}} with {q_{0}>\mathcal{Q}/2} and {\mathcal{Q}} is the homogeneous dimension of {\mathbb{G}}. In this article, by Campanato type spaces {\Lambda^{\alpha}_{\mathcal{L}}(\mathbb{G})}, we introduce Hardy type spaces associated with {\mathcal{L}} denoted by {H^{{p}}_{\vphantom{\varepsilon}{\mathcal{L}}}(\mathbb{G})} and prove the atomic characterization of {H^{{p}}_{\vphantom{\varepsilon}{\mathcal{L}}}(\mathbb{G})}. Further, we obtain the following duality relation:\Lambda_{\mathcal{L}}^{\mathcal{Q}(1/p-1)}(\mathbb{G})=(H^{{p}}_{\vphantom{% \varepsilon}{\mathcal{L}}}(\mathbb{G}))^{\ast},\quad\mathcal{Q}/(\mathcal{Q}+% \delta)<p<1\quad\text{for}\ \delta=\min\{1,2-\mathcal{Q}/q_{0}\}.The above relation enables us to characterize {\Lambda^{\alpha}_{\mathcal{L}}(\mathbb{G})} via two families of Carleson measures generated by the heat semigroup and the Poisson semigroup, respectively. Also, we obtain two classes of perturbation formulas associated with the semigroups related to {\mathcal{L}}. As applications, we obtain the boundedness of the Littlewood–Paley function and the Lusin area function on {\Lambda^{\alpha}_{\mathcal{L}}(\mathbb{G})}.

scholarly journals Characterization of Riesz and Bessel potentials on variable Lebesgue spaces

2006 ◽  
Vol 4 (2) ◽  
pp. 113-144 ◽  
Author(s):  
Alexandre Almeida ◽  
Stefan Samko
Keyword(s):  
Sobolev Spaces ◽  
Variable Exponent ◽  
Regularity Conditions ◽  
Bessel Potential ◽  
Lebesgue Spaces ◽  
Hypersingular Integrals ◽  
Variable Lebesgue Spaces ◽  
Bessel Potentials ◽  
Bessel Potential Spaces

Riesz and Bessel potential spaces are studied within the framework of the Lebesgue spaces with variable exponent. It is shown that the spaces of these potentials can be characterized in terms of convergence of hypersingular integrals, if one assumes that the exponent satisfies natural regularity conditions. As a consequence of this characterization, we describe a relation between the spaces of Riesz or Bessel potentials and the variable Sobolev spaces.

scholarly journals Fractional vector-valued Littlewood–Paley–Stein theory for semigroups

2014 ◽  
Vol 144 (3) ◽  
pp. 637-667 ◽  
Author(s):  
José L. Torrea ◽  
Chao Zhang
Keyword(s):  
Banach Space ◽  
Fractional Derivative ◽  
Banach Spaces ◽  
Martingale Difference ◽  
Area Function ◽  
Poisson Semigroup ◽  
Relationship Of ◽  
The Relationship ◽  
Vector Valued

We consider the fractional derivative of a general Poisson semigroup. With this fractional derivative, we define the generalized fractional Littlewood–Paley g-function for semigroups acting on Lp-spaces of functions with values in Banach spaces. We give a characterization of the classes of Banach spaces for which the fractional Littlewood–Paley g-function is bounded on Lp-spaces. We show that the class of Banach spaces is independent of the order of derivation and coincides with the classical (Lusin-type/-cotype) case. We also show that the same kind of results exist for the case of the fractional area function and the fractional gλ*-function on ℝn. Finally, we consider the relationship of the almost sure finiteness of the fractional Littlewood–Paley g-function, the area function and the gλ*-function with the Lusin-cotype property of the underlying Banach space. As a byproduct of the techniques developed, one can find some results of independent interest for vector-valued Calderón–Zygmund operators. For example, one can find the following characterization: a Banach space is the unconditional martingale difference if and only if, for some (or, equivalently, for every) p ∈ [1, ∞), dy exists for almost every x ∈ ℝ and every .

scholarly journals Higher Order Sobolev-Type Spaces on the Real Line

2014 ◽  
Vol 2014 ◽  
pp. 1-13
Author(s):  
Bogdan Bojarski ◽  
Juha Kinnunen ◽  
Thomas Zürcher
Keyword(s):  
Sobolev Spaces ◽  
Real Line ◽  
Finite Differences ◽  
Higher Order ◽  
Sobolev Type ◽  
The Real ◽  
Sobolev Functions ◽  
Sobolev Type Spaces ◽  
Type Spaces

This paper gives a characterization of Sobolev functions on the real line by means of pointwise inequalities involving finite differences. This is also shown to apply to more general Orlicz-Sobolev, Lorentz-Sobolev, and Lorentz-Karamata-Sobolev spaces.

Morphological Characterization of Mycobacteriophage R1

1974 ◽  
Vol 32 ◽  
pp. 254-255
Author(s):  
B. L. Soloff ◽  
T. A. Rado
Keyword(s):  
Carbon Source ◽  
Stationary Phase ◽  
Growth Phase ◽  
Minimal Medium ◽  
Morphological Characterization ◽  
Logarithmic Growth Phase ◽  
Complete Lysis ◽  
Lysogenic Culture ◽  
Logarithmic Growth

Mycobacteriophage R1 was originally isolated from a lysogenic culture of M. butyricum. The virus was propagated on a leucine-requiring derivative of M. smegmatis, 607 leu−, isolated by nitrosoguanidine mutagenesis of typestrain ATCC 607. Growth was accomplished in a minimal medium containing glycerol and glucose as carbon source and enriched by the addition of 80 μg/ ml L-leucine. Bacteria in early logarithmic growth phase were infected with virus at a multiplicity of 5, and incubated with aeration for 8 hours. The partially lysed suspension was diluted 1:10 in growth medium and incubated for a further 8 hours. This permitted stationary phase cells to re-enter logarithmic growth and resulted in complete lysis of the culture.

AEM analysis of crystallization in Ti-based amorphous alloys

1983 ◽  
Vol 41 ◽  
pp. 270-271
Author(s):  
A.R. Pelton ◽  
A.F. Marshall ◽  
Y.S. Lee
Keyword(s):  
Magnetic Properties ◽  
Amorphous Alloys ◽  
Amorphous Materials ◽  
Isothermal Annealing ◽  
Amorphous Matrix ◽  
Nucleation And Growth ◽  
Current Interest ◽  
Amorphous Films ◽  
Electrical And Magnetic Properties

Amorphous materials are of current interest due to their desirable mechanical, electrical and magnetic properties. Furthermore, crystallizing amorphous alloys provides an avenue for discerning sequential and competitive phases thus allowing access to otherwise inaccessible crystalline structures. Previous studies have shown the benefits of using AEM to determine crystal structures and compositions of partially crystallized alloys. The present paper will discuss the AEM characterization of crystallized Cu-Ti and Ni-Ti amorphous films.Cu60Ti40: The amorphous alloy Cu60Ti40, when continuously heated, forms a simple intermediate, macrocrystalline phase which then transforms to the ordered, equilibrium Cu3Ti2 phase. However, contrary to what one would expect from kinetic considerations, isothermal annealing below the isochronal crystallization temperature results in direct nucleation and growth of Cu3Ti2 from the amorphous matrix.

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