Remarks on some subspaces of BMO(\({\mathbb R}^n\)) and of bmo(\({\mathbb R}^n\)).
On the \(n\)-dimensional Euclidean space, the author introduces the function space VMO (resp., CMO) as the BMO-closure of all infinitely differentiable functions of bounded mean oscillation (resp., of all infinitely differentiable functions with compact support). Various characterizations of these spaces are provided. For example, \(f\in \text{VMO}\) iff \(f\) is the limit in BMO of a sequence of uniformly continuous BMO-functions iff \(f=f_0 + R_1 f_1+\cdots+R_n f_n\) where the \(f_j\) are bounded uniformly continuous functions and the \(R_j\) are the Riesz transformations iff \(\lim_{t\to 0}\sup_{| B| \leq t}| B| ^{-1}\int_B | f-f_B| =0\) (the last condition is the usual definition of VMO; throughout, \(B\) is a ball and \(f_B\) is the average of \(f\) over \(B\)), etc. Next, \(f\in \text{CMO}\) iff \(f\) is in the BMO-closure of the set of continuous functions vanishing at infinity iff \(f\) is in the intersection of VMO and the BMO-closure of the set of compactly supported continuous functions iff \(f=f_0+R_1 f_1 +\cdots+R_n f_n\) with all \(f_j\) continuous and vanishing at infinity, etc. The BMO-closure of the set of compactly supported BMO-functions is characterized in the same spirit. Also, a similar story is told about the closures of various natural function classes in the space bmo determined by the norm \[ \sup_{| B| <1} | B| ^{-1}\int_B | f-f_B| + \sup_{| B| =1}\int_B | f| . \] Various isolated facts of a similar nature can be found dispersed in the literature, but the paper under review seems to present the first systematic treatment of this range of problems. At the end, some mistakes of forerunners are indicated and the corresponding counterexamples are given.
scientific article; zbMATH DE number 1794828
scientific article