Khintchine Inequality for Sets of Small Measure

S. V. Astashkin

Funkcionalʹnyj analiz i ego priloženiâ, Tome 48 (2014) no. 4, pp. 1-8

Résumé

The following theorem is proved. Let $r_i$ be the Rademacher functions, i.e., $r_i(t):=\operatorname{sign}\sin(2^i\pi t)$, $t\in[0,1]$, $i\in\mathbb{N}$. If a set $E\subset [0,1]$ satisfies the condition $m(E\cap (a,b))>0$ for any interval $(a,b)\subset [0,1]$, then, for some constant $\gamma=\gamma(E)>0$ depending only on $E$ and for all sequences $a=(a_k)_{k=1}^\infty\in\ell^2$, $$ \int_E\bigg|\sum_{i=1}^\infty a_ir_i(t)\bigg|\,dt\ge \gamma \bigg(\sum_{i=1}^\infty a_i^2\bigg)^{1/2}. $$ As a consequence of this result, a version of the weighted Khintchine inequality is obtained.

Détail

Keywords: Rademacher functions, Khintchine inequality, Paley–Zygmund inequality.
Mots-clés : $L_p$-spaces