A note on the Besov spaces ${\rm B}_1^{Φ_2}$

Abstract

Regarding the Besov space ${\rm B}_1^{Φ_2}$, defined by S. Geiss and Ylinen (Memoirs AMS 1335, 2021) by a coupling method, we prove for $ξ\in{\rm B}_1^{Φ_2} \cap L_2$ a distributional estimate in terms of the Lorentz-Zygmund space $L_{1,\infty}(\ln L)^{-1}$ for the Malliavin derivative ${\rm D}(T_uξ)$ uniformly in $u>0$, where $(T_u)_{u\ge }$ is the Ornstein-Uhlenbeck semi-group. Here $ξ$ is not necessarily Malliavin differentiable in the ${\rm D}_{1,1}$-sense. This provides for Theorem 4.22 of the aforementioned reference an extension to the borderline case $p=1$.