𝐵𝑀𝑂(𝜌) and Carleson measures

Abstract

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 > y > l ( Q ) } \{ (x,y) \in {\mathbf {R}}_ + ^{m + 1}:x \in Q,\;0 > y > 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.