Dense weakly lacunary subsystems of orthogonal systems

Let $\Phi=\{\varphi_i\}_{i=1}^N$ be an orthogonal system of functions defined on a probability space $(X,\mu)$. Let $p>2$. A deep result by J. Bourgain states: Under the additional assumption $\|\varphi_i\|_{\infty}\leq M$, $1\leq i\leq N$, we can choose a subsystem $\{\varphi_j\}_{j\in\Lambda}$ in $\Phi$ with $|\Lambda|\geq N^{2/p}$ such that $\|\sum_{k\in\Lambda}{a_k\varphi_k}\|_p~\leq~C(M,p)\|\sum_{k\in\Lambda}{a_k\varphi_k}\|_2$. We establish analogs of this theorem for the class of Orlicz spaces that are close to $L_2$. As a consequence we obtain the existence of a large subsystem of $\Phi$ with the norm of the maximal partial sum operator being estimated better than the classical Menshov–Rademacher theorem guarantees for general systems.