$\Lambda_p$ Style Bounds in Orlicz Spaces Close to $L^2$

Let $(\varphi_i)_{i=1}^n$ be mutually orthogonal functions on a probability space such that $\|\varphi_i\|_\infty \leq 1$ for all $i \in [n]$. Let $\alpha > 0$. Let $\Phi(u) = u^2 \log^{\alpha}(u)$ for $u \geq u_{0}$, and $\Phi(u) = c(\alpha) u^2$ otherwise. $u_0 \geq e$ and $c(\alpha)$ are constants chosen so that $\Phi$ is a Young function, depending only on $\alpha$. Our main result shows that with probability at least $1/4$ over subsets $I$ of $[n]$, where $I$ is constructed by choosing each index of $[n]$ independently from a Bernoulli distribution, the following holds: $|I| \geq \frac{n}{e \log^{\alpha+1}(n)}$ and for any $a \in \mathbb{C}^n$, $$\left \|\sum_{i \in I} a_i \varphi_i \right \|_{\Phi} \leq K(\alpha) \log^{\frac{\alpha}{2}}(\log n) \cdot \|a\|_2.$$ $K(\alpha)$ is a constant depending only on $\alpha$. In the main Theorem of \cite{Ryou22}, Ryou proved the result above to a constant factor, depending on $p$ and $\alpha$, when the Orlicz space is a $L^p(\log L)^{p\alpha}$ space for $p > 2$ where $|I| \sim \frac{n^{2/p}}{\log^{2 \alpha /p}(n)}$. However, their work did not extend to the case where $p=2$, an open question in \cite{Iosevich25}. Our result resolves the latter question up to $\log \log n$ factors. Moreover, our result sharpens the constants of Limonova's main result in \cite{Limonova23} from a factor of $\log n$ to a factor of $\log \log n$, if the orthogonal functions are bounded by a constant. In addition, our proof is much shorter and simpler than the latter's. Finally, to complement our main result, we give a probabilistic lower bound (subsets of $[n]$ are selected by a Bernoulli distribution over $[n]$'s indices) that matches our main result's upper bound.