𝐵𝑀𝑂(𝜌) and Carleson measures

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 )