Rearrangementsof the Haar system that preserve BMO

Abstract

Let $\cal D $ denote the collection of dyadic intervals in the unit interval. Let $\tau$ be a rearrangement of the dyadic intervals. We study the induced operator

$$ Th_I = h_{\tau(I)}$$

where $h_I$ is the $L^{\infty}$ normalized Haar function. We find geometric conditions on $\tau$ which are necessary and sufficient for $T$ to be bounded on $BMO$. We also characterize the rearrangements of the $L^p$ normalized Haar system in $L^p.$

1991 Mathematics Subject Classification: 42C20, 43A17, 47B38.