Constructive decomposition of functions of finite central mean oscillation

Abstract:

The space CMO of functions of finite central mean oscillation is an analogue of BMO where the condition that the sharp maximal function is bounded is replaced by the convergence of the sharp function at the origin. In this paper it is shown that each element of CMO is a singular integral image of an element of the Beurling space $B^{2}$ of functions whose Hardy-Littlewood maximal function converges at zero. This result is an analogue of Uchiyama’s constructive decomposition of BMO in terms of singular integral images of bounded functions. The argument shows, in fact, that to each element of CMO one can construct a vector Calderón-Zygmund operator that maps that element into the proper subspace $B^{2}$.