I am working with $BMO$ and $BMOA$ spaces (Let us assume in the disk).
I am asking myself if it is true that $BMO=BMOA + \bar{BMOA}$, where with $\bar{BMOA}$ , I mean functions in $BMOA(\mathbb{C}\setminus \bar{\mathbb{D}})$.
I think yes, but I am not sure. Indeed $$ BMO:=\left\lbrace f \in BMO(T) \text{ such that } f \text{ admits an harmonic extension in } \mathbb{D}\right\rbrace $$ while $$ BMOA= BMOA\cap H^1_{\text{hol}}(D)$$ I know that $L^1(T)\neq H^1_{\text{hol}}(T)+\bar{H^1_{\text{hol}}(T)}$, but since the Cauchy transform is bounded in $BMO$ I think that one could say that $BMO\subset H^1_{\text{real}}(T)$ where $$ H^1_{\text{real}}(T):=\left\lbrace f \in L^1(T) \text{ such that cauchy transform of } f \in L^1(T)\right\rbrace$$ and $H^1_{\text{real}}(T)=H^1_{\text{hol}}(T)+\bar{H^1_{\text{hol}}(T)}$.
Thanks a lot.