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 $\emph{a priori}$ control of $\sup_{r}\|u_{r}\|_{L^{p}(\mathbb{S}^{1})}$. Our main result shows that this assumption already forces $u\in H^{p}(\mathbb{D})$ and that the radial boundary trace $u^{*}$ belongs to $W^{s,p}(\mathbb{S}^{1})$, with $u_{r}\to u^{*}$ in $W^{s,p}(\mathbb{S}^{1})$ as $r\to1^{-}$. The key mechanism combines the mean-value property (which pins the constant mode at $u(0)$) with a fractional Poincar$\'e$ inequality on $\mathbb{S}^{1}$, recovering $L^{p}$ control from oscillation alone. As a consequence, the trace map $u\mapsto u^{*}$ is a surjective isomorphism $\mathcal{W}^{s,p}(\mathbb{D})\xrightarrow{\sim}B^{s}_{p,p,+}(\mathbb{S}^{1})$ with explicit norm equivalence.