Sharp square-function inequalities for conditionally symmetric martingales

Abstract:

Let $f$ be a conditionally symmetric martingale taking values in a Hilbert space $\mathbb {H}$ and let $S(f)$ be its square function. If ${\nu _p}$ is the smallest positive zero of the confluent hypergeometric function and ${\mu _p}$ is the largest positive zero of the parabolic cylinder function of parameter $p$, then the following inequalities are sharp: \[ \| f \|_{p} \leq \nu _{p}\| S(f)\|_{p}\qquad \text {if}\;0 < p \leq 2,\] \[ \|f \|_{p} \leq \mu _{p} \| S(f)\|_{p}\qquad \text {if}\;p \geq 3,\] \[ \nu _{p}\| S(f)\|_{p}\; \leq \; \|f\|_p \qquad \text {if}\; p \geq 2.\] Moreover, the constants $\nu _p$ and $\mu _p$ for the cases mentioned above are also best possible for the Marcinkiewicz-Paley inequalities for Haar functions.