Non-smooth atomic decomposition of 2-microlocal spaces and application to pointwise multipliers

Gonçalves, Helena F.; Kempka, Henning

doi:10.1016/j.jmaa.2015.10.007

Cited by 11 publications

(31 citation statements)

References 16 publications

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“…Scharf [, Theorem 3.12] considered the non‐smooth atomic decomposition for Besov and Triebel–Lizorkin spaces. Recently, Gonçaleves and Kempka [, Theorem 3.14] have stated the non‐smooth atomic decomposition for 2‐microlocal Besov and Triebel–Lizorkin spaces with variable exponents by slightly adapting the definition of non‐smooth atom from . To introduce a non‐smooth atom considered by Gonçalves and Kempka , we use the Hölder space

{scriptC}^{s} false(R^{n} false)

.…”

Section: Preliminariesmentioning

confidence: 99%

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Generalized Besov–Morrey spaces and generalized Triebel–Lizorkin–Morrey spaces on domains

Izuki

Noi

2019

Mathematische Nachrichten

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In this paper, we consider a non‐smooth atomic decomposition by using a smooth atomic decomposition. Applying the non‐smooth atomic decomposition, a local means characterization and a quarkonical decomposition, we obtain a pointwise multiplier and a trace operator for generalized Besov–Morrey spaces and generalized Triebel–Lizorkin–Morrey spaces on the whole space. We also develop the theory of those spaces on domains. We consider an extension operator and a trace operator on the upper half space and on compact oriented Riemannian manifolds.

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{scriptC}^{s} false(R^{n} false)

.…”

Section: Preliminariesmentioning

confidence: 99%

“…We rely on the proof of [, Theorem 3.12] and [, Theorem 3.14]. Let

K^{'} \in {double-struckN}_{0}

and

L^{'} \in {double-struckN}_{0} \cup false{0 false}

satisfy

K^{'} \geq {[1 + K]}_{+} \geq K

and

L^{'} \geq \max false(- 1, L false) \geq L

.…”

Section: Preliminariesmentioning

confidence: 99%

Generalized Besov–Morrey spaces and generalized Triebel–Lizorkin–Morrey spaces on domains

Izuki

Noi

2019

Mathematische Nachrichten

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“…For the next two auxiliary results we refer to [, Lemmas 3.6,3.7]. Lemma Let

k_{j}

be the local means according to Corollary with

d = 3

.…”

Section: Non‐smooth Atomic Decompositionmentioning

confidence: 99%

“…Remark We would like to highlight that the results in are not properly written. There the authors used a sequence

{true(ψ_{j} true)}_{j \in N_{0}}

built upon a pair of functions defined as in Theorem , with

k = 2

.…”

Section: Non‐smooth Atomic Decompositionmentioning

confidence: 99%

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Characterization of Triebel–Lizorkin type spaces with variable exponents via maximal functions, local means and non‐smooth atomic decompositions

Gonçalves

Moura

2018

Mathematische Nachrichten

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In this paper we study the maximal function and local means characterizations and the non‐smooth atomic decomposition of the Triebel–Lizorkin type spaces with variable exponents Fp(·),q(·)sfalse(·false),ϕfalse(Rnfalse). These spaces were recently introduced by Yang et al. and cover the Triebel–Lizorkin spaces with variable exponents Fp(·),q(·)s(·)false(Rnfalse) as well as the classical Triebel–Lizorkin spaces Fp,qsfalse(Rnfalse), even the case when p=∞. Moreover, covered by this scale are also the Triebel–Lizorkin‐type spaces Fp,qs,τfalse(Rnfalse) with constant exponents which, in turn cover the Triebel–Lizorkin–Morrey spaces. As an application we obtain a pointwise multiplier assertion for those spaces.

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Intrinsic Characterization and the Extension Operator in Variable Exponent Function Spaces on Special Lipschitz Domains

Kempka

2017

Springer Proceedings in Mathematics &Amp; Statistics

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We study 2-microlocal Besov and Triebel-Lizorkin spaces with variable exponents on special Lipschitz domains Ω. These spaces are as usual defined by restriction of the corresponding spaces on R n . In this paper we give two intrinsic characterizations of these spaces using local means and the Peetre maximal operator. Further we construct a linear and bounded extension operator following the approach done by Rychkov in [12], which at the end also turns out to be universal.

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Non-smooth atomic decomposition of 2-microlocal spaces and application to pointwise multipliers

Cited by 11 publications

References 16 publications

Generalized Besov–Morrey spaces and generalized Triebel–Lizorkin–Morrey spaces on domains

Generalized Besov–Morrey spaces and generalized Triebel–Lizorkin–Morrey spaces on domains

Characterization of Triebel–Lizorkin type spaces with variable exponents via maximal functions, local means and non‐smooth atomic decompositions

Intrinsic Characterization and the Extension Operator in Variable Exponent Function Spaces on Special Lipschitz Domains

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