Khintchine inequality for sets of small measure

Abstract

The following theorem is proved. Let r _i be the Rademacher functions, i.e., r _i (t):= sign sin(2ⁱ πt), t ∈ [0, 1], i ∈ ℕ. If a set E ⊂ [0, 1] satisfies the condition m(E ∩ (a, b)) > 0 for any interval (a, b) ⊂ [0, 1], then, for some constant γ = γ(E) > 0 depending only on E and for all sequences $a = (a_k )_{k = 1}^\infty \in \ell ^2$,

$\int_E {\left| {\sum\limits_{i = 1}^\infty {a_i r_i (t)} } \right|dt} \geqslant \gamma \left( {\sum\limits_{i = 1}^\infty {a_i^2 } } \right)^{1/2} $

. As a consequence of this result, a version of the weighted Khintchine inequality is obtained.