Bounded Mean Oscillation on the Bidisk and Operator BMO

We identify several of the spaces in the inclusion chain of BMO spaces in two variables with certain BMO spaces of operator-valued functions in one variable and discuss various interesting consequences of this identification. Our main result is a previously unknown strict inclusion in the chain of BMO spaces in two variables that translates into the fact that there exists a function b on the bidisk such that the associated little Hankel operator γb=P⊥1P⊥2bP2P1 is bounded on products of normalized Szegő kernels kz1⊗kz2, z1, z2∈D, but does not extend to a bounded linear operator on H2(T2). In other words, the well-known result that boundedness of Hankel operators in one variable can be tested on normalized Szegő kernels does not extend to little Hankel operators in two variables. In the framework of operator BMO functions, this can be expressed as a new result about BMO spaces of Hankel operator-valued functions. We also study an interesting link between the celebrated Carleson counterexample (L. Carleson, Mittag–Leffler Report No. 7, 1974) for the bidisk and counterexamples to the Operator Carleson Embedding Theorem.