Subspaces of $\textrm {BMO}(\textbf {R}^ n)$

Abstract:

We consider subspaces of ${\text {BMO}}({{\mathbf {R}}^n})$ generated by one singular integral transform. We show that the averages along ${x_j}$-lines of the $j$ th Riesz transform of $g \in {\text {BMO}} \cap {L^2}({{\mathbf {R}}^n})$ or $g \in {L^\infty }({{\mathbf {R}}^n})$ satisfy a certain strong regularity property. One consquence of this result is that such functions satisfy a uniform doubling condition on a.e. ${x_j}$-line. We give an example to show, however, that the restrictions to ${x_j}$-lines of the Riesz transform of $g \in {\text {BMO}} \cap {L^2}({{\mathbf {R}}^n})$ do not necessarily have uniformly bounded ${\text {BMO}}$ norm. Also, for a Calderón-Zygmund singular integral operator $K$ with real and odd kernel, we show that $K({\text {BMO}_c}) \subseteq \overline {{L^\infty } + K(L_c^\infty )}$, where $L_c^\infty$ and ${\text {BMO}_c}$ are the spaces of ${L^\infty }$ or ${\text {BMO}}$ functions of compact support, respectively, and the closure is taken in ${\text {BMO}}$ norm.