Year: 2026
Type: preprint
Abstract: For $β\ge0$, let $A_1^{(β)}$ be the class of weights satisfying $$ [w]_{A_1^{(β)}}:= \sup_{0<\varepsilon\le1} \varepsilon^β[w]_{A_{1+\varepsilon}}<\infty. $$ These classes satisfy $A_1\subsetneq A_1^{(β)}\subsetneq\bigcap_{p>1}A_p$ for $β>0$. For every $α\ge0$, we prove that $w\in A_1^{(β)}$ if and only if $$ w\left(\left\{M_{L(\log L)^α}f>λ\right\}\right) \le C\int_{\mathbb{R}^n} \frac{|f(x)|}λ \left[ \log\left(e+\frac{|f(x)|}λ\right) \right]^{α+β} w(x)\,\mathrm{d}x. $$ The least constant $C$ is comparable to $[w]_{A_1^{(β)}}$. Also, we obtain corresponding characterizations for Calderón--Zygmund operators and their commutators.
Source: arXiv (Cornell University)
Author: Wang, Dinghuai
Language: English
Subfield: Applied Mathematics
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
Awards: G1808468159, G4799366558
OpenAlex ID: W7223709449