scholarly journals Spaces of Pointwise Multipliers on Morrey Spaces and Weak Morrey Spaces

Mathematics ◽  
2021 ◽  
Vol 9 (21) ◽  
pp. 2754
Author(s):  
Eiichi Nakai ◽  
Yoshihiro Sawano
Keyword(s):  
Morrey Spaces ◽  
Banach Lattices ◽  
Complete Picture ◽  
Pointwise Multiplier ◽  
Pointwise Multipliers ◽  
Vector Valued ◽  
Weak Morrey Spaces

The spaces of pointwise multipliers on Morrey spaces are described in terms of Morrey spaces, their preduals, and vector-valued Morrey spaces introduced by Ho. This paper covers weak Morrey spaces as well. The result in the present paper completes the characterization of the earlier works of the first author’s papers written in 1997 and 2000, as well as Lemarié-Rieusset’s 2013 paper. As a corollary, the main result in the present paper shows that different quasi-Banach lattices can create the same vector-valued Morrey spaces. The goal of the present paper is to provide a complete picture of the pointwise multiplier spaces.

scholarly journals Pointwise Multipliers on Weak Morrey Spaces

2020 ◽  
Vol 8 (1) ◽  
pp. 363-381
Author(s):  
Ryota Kawasumi ◽  
Eiichi Nakai
Keyword(s):  
Growth Condition ◽  
Lebesgue Space ◽  
Morrey Space ◽  
Morrey Spaces ◽  
Homogeneous Type ◽  
Spaces Of Homogeneous Type ◽  
Pointwise Multipliers ◽  
Weak Lebesgue Space ◽  
Weak Morrey Spaces

Abstract We consider generalized weak Morrey spaces with variable growth condition on spaces of homogeneous type and characterize the pointwise multipliers from a generalized weak Morrey space to another one. The set of all pointwise multipliers from a weak Lebesgue space to another one is also a weak Lebesgue space. However, we point out that the weak Morrey spaces do not always have this property just as the Morrey spaces not always.

A macroscopic theory of microcrack growth in brittle materials

1991 ◽  
Vol 335 (1639) ◽  
pp. 455-485 ◽  
Keyword(s):  
Crack Growth ◽  
Brittle Materials ◽  
Macroscopic Theory ◽  
Rate Of Increase ◽  
Inelastic Behaviour ◽  
Constitutive Response ◽  
Macroscopic Plasticity ◽  
Microcrack Growth ◽  
Vector Valued

This paper is concerned with the development of a macroscopic theory of crack growth in fairly brittle materials. Average characteristics of the cracks are described in terms of an additional vector-valued variable in the macroscopic theory, which is determined by an additional momentum-like balance law associated with the rate of increase of the area of the cracks and includes the effects of forces maintaining the crack growth and the inertia of microscopic particles surrounding the cracks. The basic developments represent an idealized characterization of inelastic behaviour in the presence of crack growth, which accounts for energy dissipation without explicit use of macroscopic plasticity effects. A physically plausible constraint on the rate of crack growth is adopted to simplify the theory. To ensure that the results of the theory are physically reasonable, the constitutive response of the dependent variables are significantly restricted by consideration both of the energetic effects and of the microscopic processes that give rise to crack growth. These constitutive developments are in conformity with many of the standard results and observations reported in the literature on fracture mechanics. The predictive nature of the theory is illustrated with reference to two simple examples concerning uniform extensive and compressive straining.

scholarly journals Non-smooth atomic decomposition of variable 2-microlocal Besov-type and Triebel–Lizorkin-type spaces

2021 ◽  
Vol 15 (3) ◽  
Author(s):  
Helena F. Gonçalves
Keyword(s):  
Atomic Decomposition ◽  
Morrey Spaces ◽  
Maximal Functions ◽  
Variable Exponents ◽  
Atomic Decompositions ◽  
Local Means ◽  
Essential Tool ◽  
Pointwise Multipliers ◽  
Type Spaces

AbstractIn this paper we provide non-smooth atomic decompositions of 2-microlocal Besov-type and Triebel–Lizorkin-type spaces with variable exponents $$B^{\varvec{w}, \phi }_{p(\cdot ),q(\cdot )}({\mathbb {R}}^n)$$ B p ( · ) , q ( · ) w , ϕ ( R n ) and $$F^{\varvec{w}, \phi }_{p(\cdot ),q(\cdot )}({\mathbb {R}}^n)$$ F p ( · ) , q ( · ) w , ϕ ( R n ) . Of big importance in general, and an essential tool here, are the characterizations of the spaces via maximal functions and local means, that we also present. These spaces were recently introduced by Wu et al. and cover not only variable 2-microlocal Besov and Triebel–Lizorkin spaces $$B^{\varvec{w}}_{p(\cdot ),q(\cdot )}({\mathbb {R}}^n)$$ B p ( · ) , q ( · ) w ( R n ) and $$F^{\varvec{w}}_{p(\cdot ),q(\cdot )}({\mathbb {R}}^n)$$ F p ( · ) , q ( · ) w ( R n ) , but also the more classical smoothness Morrey spaces $$B^{s, \tau }_{p,q}({\mathbb {R}}^n)$$ B p , q s , τ ( R n ) and $$F^{s,\tau }_{p,q}({\mathbb {R}}^n)$$ F p , q s , τ ( R n ) . Afterwards, we state a pointwise multipliers assertion for this scale.

scholarly journals Weak Type Estimates of Variable Kernel Fractional Integral and Their Commutators on Variable Exponent Morrey Spaces

2019 ◽  
Vol 2019 ◽  
pp. 1-11
Author(s):  
Xukui Shao ◽  
Shuangping Tao
Keyword(s):  
Lebesgue Space ◽  
Variable Exponent ◽  
Fractional Integral ◽  
Weak Type ◽  
Integral Operators ◽  
Morrey Spaces ◽  
Variable Kernel ◽  
Fractional Integral Operators ◽  
Weak Type Estimates ◽  
Weak Morrey Spaces

In this paper, the authors obtain the boundedness of the fractional integral operators with variable kernels on the variable exponent weak Morrey spaces based on the results of Lebesgue space with variable exponent as the infimum of exponent function p(·) equals 1. The corresponding commutators generated by BMO and Lipschitz functions are considered, respectively.

scholarly journals Applications of Littlewood-Paley Theory forB˙σ-Morrey Spaces to the Boundedness of Integral Operators

2013 ◽  
Vol 2013 ◽  
pp. 1-21 ◽  
Author(s):  
Yasuo Komori-Furuya ◽  
Katsuo Matsuoka ◽  
Eiichi Nakai ◽  
Yoshihiro Sawano
Keyword(s):  
Riesz Potential ◽  
Integral Operators ◽  
Morrey Spaces ◽  
Lipschitz Spaces ◽  
Potential Operators

The boundedness of the various operators onB˙σ-Morrey spaces is considered in the framework of the Littlewood-Paley decompositions. First, the Littlewood-Paley characterization ofB˙σ-Morrey-Campanato spaces is established. As an application, the boundedness of Riesz potential operators is revisted. Also, a characterization ofB˙σ-Lipschitz spaces is obtained: and, as an application, the boundedness of Riesz potential operators onB˙σ-Lipschitz spaces is discussed.

scholarly journals A note on bi-contractive projections on spaces of vector valued continuous functions

2018 ◽  
Vol 5 (1) ◽  
pp. 42-49
Author(s):  
Fernanda Botelho ◽  
T.S.S.R.K. Rao
Keyword(s):  
Hilbert Space ◽  
Continuous Functions ◽  
Choquet Simplex ◽  
Contractive Projections ◽  
Vector Valued

Abstract This paper concerns the analysis of the structure of bi-contractive projections on spaces of vector valued continuous functions and presents results that extend the characterization of bi-contractive projections given by the first author. It also includes a partial generalization of these results to affine and vector valued continuous functions from a Choquet simplex into a Hilbert space.

scholarly journals Polarimetry for Photonic Integrated Circuits

2019 ◽  
Vol 9 (15) ◽  
pp. 2987
Author(s):  
Moritz Baier ◽  
Axel Schoenau ◽  
Francisco M. Soares ◽  
Martin Schell
Keyword(s):  
Integrated Circuits ◽  
Photonic Integrated Circuits ◽  
The State ◽  
Polarization Dependence ◽  
Complete Picture ◽  
Wide Range ◽  
State Of Polarization ◽  
Matrix Yielding ◽  
Polarization Of Light

Photonic integrated circuits (PICs) play a key role in a wide range of applications. Very often, the performance of PICs depends strongly on the state of polarization of light. Classically, this is regarded as undesirable, but more and more applications emerge that make explicit use of polarization dependence. In either case, the characterization of the polarization properties of a PIC can be a nontrivial task. We present a way of characterizing PICs in terms of their full Müller matrix, yielding a complete picture of their polarization properties. The approach is demonstrated by carrying out measurements of fabricated PICs.

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