Pointwise multipliers for functions of bounded mean oscillation
Eiichi Nakai, Kôzô Yabuta · Journal of the Mathematical Society of Japan · 1985
The purpose of this paper is to characterize the set of pointwise multipliers on $bmo_{\phi}(R^{n})$ , which is the function space defined using the mean oscillation and a growth function $\phi$ .Janson [2] has characterized pointwise multipliers on $bmo_{\phi}(T^{n})$ on the n- dimensional torus $T^{n}$ .We extend his result to the case of the n-dimensional Euclidean space $R^{n}$ .To define $bmo_{\phi}(R^{n})$ , let $I(a, r)$ be the cube { $x\in R^{n}$ ; $|x_{i}-a_{i}|\leqq r/2,$ $i=1,2$, .. , $n$ } whose edges have length $r$ and are parallel to the coordinate axes.For a cube $I$ , we denote by $|I|$ the Lebesgue measure of $I$ , by $M(f, I)$ or $f_{I}$ the mean value of a function $f$ on $I,$ $i$ .$e$ .$|I|^{-1} \int_{I}f(x)dx$ , and by $MO(f, I)$ the mean oscillation of $f$ on $I,$ $i.e$ .$|I|^{-1}!_{I}|f(x)-f_{I}|dx$ .We now define $bmo_{\phi}(R^{n})=\{f\in L_{1oc}^{1}(R^{n})$ ; $\sup_{I(a,r)}\frac{MO(f,I(a,r))}{\phi(r)}0$ such that $C^{-1}\phi_{1}(r)\leqq\phi_{2}(r)\leqq C\phi_{1}(r)$ , then $bmo_{\phi_{1}}(R^{n})=bmo_{\phi_{2}}(R^{n})$ .A function $g$ on $R^{n}$ is called a pointwise multiplier on $bmo_{\phi}(R^{n})$ , if the pointwise multiplication $fg$ belongs to $bmo_{\phi}(R^{n})$ for all $f$ belonging to $bmo_{\phi}(R^{n})$ .Janson's characterization is the following.If $\phi$ is a growth function and $\phi(r)/r$ is almost decreasing, then a function $g$ is a pointwise multiplier on $bmo_{\phi}(T^{n})$if and only if $g$ belongs to $bmo_{\psi}(T^{n})\cap L^{\infty}(T^{n})$ where $\psi(r)=$ $\phi(r)/\int_{r}^{1}\phi(t)t^{-1}dt$ . (A positive function $h(t)$ is said to be almost decreasing if there