Weak $L^{1}$ norms of random sums

Abstract:

Let $\left \{g_{j}\right \}$ denote a sequence of measurable functions on $\mathbf {R}^{n}$, and let $\left \|\cdot \right \|_{WL^{1}}$ denote the weak $L^{1}$ norm. It is shown that \[ \left \|\mathbb {E}\left (\left |\sum _{j=1}^{N} \epsilon _{j}g_{j}\right |\right )\right \|_{WL^{1}} \lesssim \sum _{j=1}^{N} \left \|g_{j}\right \|_{WL^{1}},\] where $\left \{\epsilon _{j}\right \}$ is a sequence of independent random variables taking on values $+1$ and $-1$ with equal probability. Moreover, it is shown that \[ \left \| \mathbb {E}\left (\left |\sum _{j=1}^{N} \epsilon _{j}g_{j}\right |\right )\right \|_{WL^{1}} \lesssim \mathbb {E}\left (\left \|\sum _{j=1}^{N} \epsilon _{j}g_{j}\right \|_{WL^{1}}\right ).\] The paper concludes by providing an example indicating that, if $\left \|g_{1}\right \|_{WL^{1}}$ $= \cdots = \left \|g_{N}\right \|_{WL^{1}} = 1$, then the estimate \[ \mathbb {E}\left (\left \|\sum _{j=1}^{N}\epsilon _{j}g_{j}\right \|_{WL^{1}}\right ) \lesssim N \log N\] is the best possible.