Let be the space of all analytic functions on the unit disk . For , let and let be the Green function of with logarithmic singularity at . Let , and let . Then if
and , if
where is the Lebesgue area measure. The Bloch space consists of all such that
and the little Bloch space is the space of functions for which as . It is known that for , and . Further, and , which are called -, and -spaces, respectively. For , we have and . We know that, for , and are Möbius invariant function spaces, and for , and are subspaces of BMOA and VMOA, respectively. For , a positive measure defined on is an -Carleson measure provided for all subarcs of , where denotes the arc length of and denotes the Carleson box based on . For a subspace of , the distance from a function to the space is denoted by .
The well-known distance formula, known as Jones’ theorem, gives the distance when . The author’s interesting main theorem generalizes this theorem to the case where in the following way: Let , and let . Then the following quantities are equivalent:
is an -Carleson measure};
}, where .
The corresponding “little oh”-theorem, where instead of there is , is also proved, and several corollaries are represented.