A seminorm-only characterization of analytic Besov spaces on the disc

Abstract

We introduce the space ${\mathcal{W}}^{s,p}(\mathbb{D})$ of analytic functions $u$ on the unit disc such that the radial restrictions $u_r(\xi ):=u(r\xi )$ satisfy the Gagliardo seminorm-only bound \[ \sup_{0<r<1}[u_{r}]_{W^{s,p}(\mathbb{S}^{1})}<\infty, \] with no $\text{a priori}$ control of ${\sup }_r\Vert u_r{\Vert }_{L^p({\mathbb{S}}^1)}$. Our main result shows that this assumption already forces $u\in H^p(\mathbb{D})$ and that the radial boundary trace $u^{\ast }$ belongs to $W^{s,p}({\mathbb{S}}^1)$, with $u_r\to u^{\ast }$ in $W^{s,p}({\mathbb{S}}^1)$ as $r\to 1^{-}$. The key mechanism combines the mean-value property (which pins the constant mode at $u(0)$) with a fractional Poincar$\acute{e}$ inequality on ${\mathbb{S}}^1$, recovering $L^p$ control from oscillation alone. As a consequence, the trace map $u\mapsto u^{\ast }$ is a surjective isomorphism ${\mathcal{W}}^{s,p}(\mathbb{D})\stackrel{\sim }{\to }{B}_{p,p,+}^s({\mathbb{S}}^1)$ with explicit norm equivalence.