English (United Kingdom)

https://curated-unify.zendy.io/wp-json/zendy-region/v1/featured_content/oa?rat=en

https://curated-unify.zendy.io/wp-json/zendy-region/v1/highlighted_journal/

Global: Zendy Plus annual

Presents the access of premium content as premium feature

Premium Content

Presents the keyphrase highlighting as premium feature

Keyphrase Highlighting

Presents the summarisation as premium feature

Summarisation

Insights

Presents the pdf analysis as premium feature

PDF Analysis

Presents the zaia usage as premium feature

ZAIA

Global: Zendy Tools annual

Global: Zendy Free Plan

Global: Zendy Tools monthly

Global: Zendy Plus monthly

It is a well known fact that the union of the Reverse H\"{o}lder classes coincides with the union of the Muckenhoupt classes $A_p$, but the $A_\infty$ constant of the weight $w$, which is a limit of its $A_p$ constants, is not a natural characterization for the weight in Reverse H\"{o}lder classes. We introduce the $RH_1$ condition as a limiting case of the $RH_p$ inequalities as $p$ tends to 1. Then we show sharp bound on $RH_1$ constant of the weight $w$ in terms of its $A_\infty$ constant (from above and from below). We also prove the sharp version of the Gehring theorem for the case $p=1$, completing the answer to the famous question of Bojarski in dimension one. We illustrate our results by two straight-forward applications: to the Dirichlet problem for elliptic PDE's.

Sharp estimates involving $A_\infty $ and $L\log L$ constants, and their applications to PDE