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On the dimensional weak-type $(1,1)$ bound for Riesz transforms
- Publication Year :
- 2020
-
Abstract
- Let $R_j$ denote the $j^{\text{th}}$ Riesz transform on $\mathbb{R}^n$. We prove that there exists an absolute constant $C>0$ such that \begin{align*} |\{|R_jf|>\lambda\}|\leq C\left(\frac{1}{\lambda}\|f\|_{L^1(\mathbb{R}^n)}+\sup_{\nu} |\{|R_j\nu|>\lambda\}|\right) \end{align*} for any $\lambda>0$ and $f \in L^1(\mathbb{R}^n)$, where the above supremum is taken over measures of the form $\nu=\sum_{k=1}^Na_k\delta_{c_k}$ for $N \in \mathbb{N}$, $c_k \in \mathbb{R}^n$, and $a_k \in \mathbb{R}^+$ with $\sum_{k=1}^N a_k \leq 16\|f\|_{L^1(\mathbb{R}^n)}$. This shows that to establish dimensional estimates for the weak-type $(1,1)$ inequality for the Riesz tranforms it suffices to study the corresponding weak-type inequality for Riesz transforms applied to a finite linear combination of Dirac masses. We use this fact to give a new proof of the best known dimensional upper bound, while our reduction result also applies to a more general class of Calder\'on-Zygmund operators.<br />Comment: 17 pages
- Subjects :
- dimensional dependence
Pure mathematics
Riesz transforms
Computer Science::Information Retrieval
Applied Mathematics
General Mathematics
Astrophysics::Instrumentation and Methods for Astrophysics
Mathematics::Classical Analysis and ODEs
Computer Science::Computation and Language (Computational Linguistics and Natural Language and Speech Processing)
Weak type
Functional Analysis (math.FA)
Mathematics - Functional Analysis
Riesz transform
Mathematics - Analysis of PDEs
Mathematics - Classical Analysis and ODEs
Classical Analysis and ODEs (math.CA)
FOS: Mathematics
Computer Science::General Literature
weak-type estimates
Absolute constant
Analysis of PDEs (math.AP)
Mathematics
Subjects
Details
- Language :
- English
- Database :
- OpenAIRE
- Accession number :
- edsair.doi.dedup.....9e9dfbf63337836fb66bfd8c51f209be