A remark on Carleson’s characterization of BMO

Abstract:

L. Carleson showed that if $\varphi \in \text {BMO}(R ^n)$ and supp $\varphi$ is compact, then $\varphi$ can be written in the form $\varphi (x) = \Sigma _{k = 1}^\infty \smallint {P_{{t_k}(y)}}(x - y){b_k}(y)dy + {b_0}(x)$ where $\Sigma _{k = 0}^\infty {\left \| {{b_k}} \right \|_\infty } \leqslant C{\left \| \varphi \right \|_{{\text {BMO}}}},{t_k}(y) > 0$ and ${P_t}(x) = {C_n}t{(|x{|^2} + {t^2})^{ - (n + 1)/2}}$ is the Poisson kernel. We show that we can take ${b_2} = {b_3} = \cdots = 0$. This can be generalized on the space of homogeneous type with certain assumptions.