Dyadic weights on $\mathbb {R}^n$ and reverse Hölder inequalities

Eleftherios N. Nikolidakis ¹   ; Antonios D. Melas ¹

¹ Department of Mathematics National and Kapodistrian University of Athens Zografou, GR-15784, Athens, Greece

Studia Mathematica, Tome 234 (2016) no. 3, pp. 281-290

Résumé

We prove that for any weight $\phi $ defined on $[0,1]^n$ that satisfies a reverse Hölder inequality with exponent $p \gt 1$ and constant $c\ge 1$ on all dyadic subcubes of $[0,1]^n$, its non-increasing rearrangement $\phi ^\ast $ satisfies a reverse Hölder inequality with the same exponent and constant not more than $2^nc-2^n+1$ on all subintervals of the form $[0,t]$, $0 \lt t\le 1$. As a consequence, there is an interval $[p,p_0(p,c))=I_{p,c}$ such that $\phi \in L^q$ for any $q\in I_{p,c}$.

Détail

DOI : 10.4064/sm8621-6-2016

Keywords: prove weight phi defined satisfies reverse lder inequality exponent constant dyadic subcubes its non increasing rearrangement phi ast satisfies reverse lder inequality exponent constant nc subintervals form consequence there interval phi