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This paper concerns certain generalizations of      BMO {\text {BMO}}      , the space of functions of bounded mean oscillation. Let      ρ  \rho       be a positive nondecreasing function on      ( 0 , ∞  ) (0,\infty )       with      ρ  ( 0 + ) = 0 \rho (0 + ) = 0      . A locally integrable function on      R m {{\mathbf {R}}^m}       is said to belong to      BMO ( ρ  ) {\text {BMO}}(\rho )       if its mean oscillation over any cube      Q Q       is      O ( ρ  ( l ( Q ) ) ) O(\rho (l(Q)))      , where      l ( Q ) l(Q)       is the edge length of      Q Q      . Carleson measures are known to be closely related to      BMO {\text {BMO}}      . Generalizations of these measures are shown to be similarly related to the spaces      BMO ( ρ  ) {\text {BMO}}(\rho )      . For a cube      Q Q       in      R m , | Q | {{\mathbf {R}}^m},\;|Q|       denotes its volume and      R ( Q ) R(Q)       is the set      { ( x , y ) ∈  R + m + 1 : x ∈  Q , 0 &gt; y &gt; l ( Q ) } \{ (x,y) \in {\mathbf {R}}_ + ^{m + 1}:x \in Q,\;0 &gt; y &gt; l(Q)\}      . A measure      μ  \mu       on      R + m + 1 {\mathbf {R}}_ + ^{m + 1}       is called a      ρ  \rho      -Carleson measure if      | μ  | ( R ( Q ) ) = O ( ρ  ( l ( Q ) ) | Q | ) |\mu |(R(Q)) = O(\rho (l(Q))|Q|)      , for all cubes      Q Q      . L. Carleson proved that a compactly supported function in      BMO {\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      BMO ( ρ  ) {\text {BMO}}(\rho )       and      ρ  \rho      -Carleson measures is proved for a broad class of growth functions, and this is used to represent      BMO ( ρ  ) {\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      BMO ( ρ  ) {\text {BMO}}(\rho )       using      ρ  \rho      -Carleson measures is a corollary. This result generalizes a characterization of      BMO {\text {BMO}}       due to C. Fefferman. Finally, an atomic decomposition of the predual of      BMO ( ρ  ) {\text {BMO}}(\rho )       is given.

𝐵𝑀𝑂(𝜌) and Carleson measures