$\textrm {BMO}(\rho )$ and Carleson measures

Abstract:

This paper concerns certain generalizations of ${\text {BMO}}$, the space of functions of bounded mean oscillation. Let $\rho$ be a positive nondecreasing function on $(0,\infty )$ with $\rho (0 + ) = 0$. A locally integrable function on ${{\mathbf {R}}^m}$ is said to belong to ${\text {BMO}}(\rho )$ if its mean oscillation over any cube $Q$ is $O(\rho (l(Q)))$, where $l(Q)$ is the edge length of $Q$. Carleson measures are known to be closely related to ${\text {BMO}}$. Generalizations of these measures are shown to be similarly related to the spaces ${\text {BMO}}(\rho )$. For a cube $Q$ in ${{\mathbf {R}}^m},\;|Q|$ denotes its volume and $R(Q)$ is the set $\{ (x,y) \in {\mathbf {R}}_ + ^{m + 1}:x \in Q,\;0 < y < l(Q)\}$. A measure $\mu$ on ${\mathbf {R}}_ + ^{m + 1}$ is called a $\rho$-Carleson measure if $|\mu |(R(Q)) = O(\rho (l(Q))|Q|)$, for all cubes $Q$. L. Carleson proved that a compactly supported function in ${\text {BMO}}$ can be represented as the sum of a bounded function and the balyage, or sweep, of some Carleson measure. A generalization of this theorem involving ${\text {BMO}}(\rho )$ and $\rho$-Carleson measures is proved for a broad class of growth functions, and this is used to represent ${\text {BMO}}(\rho )$ as a dual space. The proof of the theorem is based on a proof of J. Garnett and P. Jones of Carleson’s theorem. Another characterization of ${\text {BMO}}(\rho )$ using $\rho$-Carleson measures is a corollary. This result generalizes a characterization of ${\text {BMO}}$ due to C. Fefferman. Finally, an atomic decomposition of the predual of ${\text {BMO}}(\rho )$ is given.