RECORD
Quarkonial analysis
Functional Analysis · Submitted 31 Aug 2022
abstract compiled · 2026-09-23 · no language-model review
The authors' main claim It is the main aim of these notes to raise quarkonial decompositions to the same level as related representations of the spaces $A^s_{p,q}({\mathbb R}^n)$ in terms of atoms or wavelets.
The compiled record
The paper's own sentences, arranged. Each label is drawn from a closed vocabulary; the number beside it is the probability the classifier gave that label. Numbers in the tables are the authors', checked by code. No sentence in this record was written by a model.
Open the compiled record
The abstract, sentence by sentence
The spaces $A^s_{p,q}({\mathbb R}^n)$ with $A \in \{B,F \}$, $s\in {\mathbb R}$ and $0<p,q \le \infty$ are usually introduced in terms of Fourier--analytical decompositions. Related characterizations based on atoms and wavelets are known nowadays in a rather final way. Quarks atomize the atoms into constructive building blocks. It is the main aim of these notes to raise quarkonial decompositions to the same level as related representations of the spaces $A^s_{p,q}({\mathbb R}^n)$ in terms of atoms or wavelets. This will be complemented by some applications. In addition we deal also with quarks in domains and their relations to so--called refined localization spaces.
background contribution method result interpretation main claim
| # | sentence | role | the authors' own work | states the main conclusion | strength |
|---|---|---|---|---|---|
| s0 | The spaces $A^s_{p,q}({\mathbb R}^n)$ with $A \in \{B,F \}$, $s\in {\mathbb R}$ and $0<p,q \le \infty$ are usually introduced in terms of Fourier--analytical decompositions. | background 1.00 | 0.04 | 0.02 | asserted 0.99 |
| s1 | Related characterizations based on atoms and wavelets are known nowadays in a rather final way. | background 0.98 | 0.04 | 0.02 | asserted 1.00 |
| s2 | Quarks atomize the atoms into constructive building blocks. | contribution 0.49 | 0.22 | 0.04 | asserted 0.99 |
| s3 | It is the main aim of these notes to raise quarkonial decompositions to the same level as related representations of the spaces $A^s_{p,q}({\mathbb R}^n)$ in terms of atoms or wavelets. | contribution 0.99 | 0.35 | 0.29 | asserted 0.95 |
| s4 | This will be complemented by some applications. | contribution 0.87 | 0.35 | 0.04 | asserted 0.89 |
| s5 | In addition we deal also with quarks in domains and their relations to so--called refined localization spaces. | contribution 1.00 | 0.80 | 0.04 | asserted 0.93 |
What was examined, and what was not
- arXiv listing abstract v1
- full text
- figures and tables
- references and citing sentences
- relations to other papers
- P abstract
- S sections
- T body sentences
- C citations
- R cross-paper relations
The paper
Original abstract
The spaces $A^s_{p,q}({\mathbb R}^n)$ with $A \in \{B,F \}$, $s\in {\mathbb R}$ and $0<p,q \le \infty$ are usually introduced in terms of Fourier--analytical decompositions. Related characterizations based on atoms and wavelets are known nowadays in a rather final way. Quarks atomize the atoms into constructive building blocks. It is the main aim of these notes to raise quarkonial decompositions to the same level as related representations of the spaces $A^s_{p,q}({\mathbb R}^n)$ in terms of atoms or wavelets. This will be complemented by some applications. In addition we deal also with quarks in domains and their relations to so--called refined localization spaces.
Human review
How the literature uses it
What this paper does with the arXiv papers it cites
What it cites · 6 sentences without a confident label
Cites as background 1 paper
-
This applies especially to the spaces \begin Λ^ () and Λ_ () with A∈{B,F } \end and \begin s∈, 0<p<∞, 0<q ≤∞, -n <<0, \end using the notation as introduced in [ref] .
Morrey smoothness spaces: A new approach · Representations: negative smoothness · 0.90
Discussion
Know something about this paper? Leave a comment, or highlight any passage above to comment on it.