Traces of BMO-Sobolev spaces

Abstract:

The trace operator $RF(x) = F(x,0)$ where $F(x,t)$ is a function of $x \in {{\mathbf {R}}^n}$ and $t \in {{\mathbf {R}}^1}$ maps ${I_\alpha }(BMO)$, the $BMO$-Sobolev space of Riesz potentials of order $\alpha$ of functions of bounded mean oscillation on ${{\mathbf {R}}^{n + 1}}$, onto the homogeneous Besov space $\Lambda _\alpha ^0(\infty ,\infty )$ on ${{\mathbf {R}}^n}$, for $\alpha > 0$. A right inverse is given by the extension operator $Ef(x,t) = {\mathcal {F}^{ - 1}}({e^{ - {t^2}{{\left | \xi \right |}^2}}}\hat f(\xi ))$.