${\Lambda }_p$ Style Bounds in Orlicz Spaces Close to $L^2$

Abstract

Let $({\phi }_i{)}_{i=1}^n$ be mutually orthogonal functions on a probability space such that $\Vert {\phi }_i{\Vert }_{\infty }\le 1$ for all $i\in [n]$. Let $\alpha >0$. Let $\Phi (u)=u^2{\log }^{\alpha }(u)$ for $u\ge u_0$, and $\Phi (u)=c(\alpha )u^2$ otherwise. $u_0\ge e$ 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: $|I|\ge \frac{n}{e{\log }^{\alpha +1}(n)}$ and for any $a\in {\mathbb{C}}^n$, $${\Vert \sum _{i\in I}{a}_i{\phi }_i\Vert }_{\Phi }\le K(\alpha ){\log }^{\frac{\alpha }{2}}(\log n)\cdot \Vert a{\Vert }_2.$$ $K(\alpha )$ 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.

Added probabilistic lower bounds and sharpened main result from ${\log }^{\alpha }$ to ${\log }^{\frac{\alpha }{2}}$