1 Introduction↩︎

Let \(\Omega\) be a measurable set of \(\mathbb{R}^{N}\), \(N\in\mathbb{N}-\{0\}\). For \(u\in L^{p}(\Omega)\) the Gagliardo seminorm of \(u\) is defined by the quantity

\[[u]_{W^{s,p}(\Omega)}:=\begin{cases} {\displaystyle \bigg(\int_{\Omega}\int_{\Omega}\frac{|u(x)-u(y)|^{p}}{|x-y|^{N+sp}}\;dx\,dy\bigg)^{\frac{1}{p}},} & 1\le p<\infty,\\[1.2em] {\displaystyle \text{ess}\sup_{x,y\in\Omega,\;x\neq y}\frac{|u(x)-u(y)|}{|x-y|^{s}},} & p=+\infty. \end{cases}\] The quantity \([\cdot]_{W^{s,p}(\Omega)}\) is a seminorm. It assigns the magnitude zero to constant functions for example.

In our setting, the (periodic) fractional Sobolev space is \[W^{s,p}(\mathbb{S}^{1})=\{\gamma\in L^{p}(\mathbb{S}^{1},\mathbb{R})\mid[\gamma]_{W^{s,p}}<+\infty\}\] where \[\begin{alignedat}{1}[\gamma]_{W^{s,p}(\mathbb{S}^{1})} & :=\left(\iint_{(-\pi,\pi)\times(-\pi,\pi)}{\textstyle \frac{\vert\gamma(e^{i\xi})-\gamma(e^{i\sigma})\vert^{p}}{\vert e^{i\xi}-e^{i\sigma}\vert^{1+sp}}}d\xi\,d\sigma\right)^{\frac{1}{p}}\\ & =\left(\frac{1}{2^{1+sp}}\iint_{(-\pi,\pi)\times(-\pi,\pi)}{\textstyle \frac{\vert\gamma(e^{i\xi})-\gamma(e^{i\sigma})\vert^{p}}{\vert\sin(\frac{\xi-\sigma}{2})\vert^{1+sp}}}d\xi\,d\sigma\right)^{\frac{1}{p}}. \end{alignedat}\]

More generally, if \[u(\theta)=\sum_{k\in\mathbb{Z}}\hat{u}(k)\,e^{ik\theta}\in\mathcal{D}'(\mathbb{S}^{1})\] (periodic distribution) then \[(-\Delta)^{s}u(\theta)=\sum_{k\in\mathbb{Z}}\vert k\vert^{2s}\hat{u}(k)\,e^{ik\theta}\] with the convention \((-\Delta)^{0}u=u\). In particular \[\Vert(-\Delta)^{\frac{s}{2}}u\Vert^{2}_{L^{2}}=\sum_{k\in\mathbb{Z}}\vert k\vert^{2s}\vert\hat{u}(k)\vert^{2}.\] The reader may notice that the kernel \[\varrho_{s}(\theta)=\frac{1}{2^{1+sp}}\frac{1}{\vert\sin(\theta)\vert^{1+sp}}\] can be viewed as the wrapping (periodization) of the one dimensional kernel \(\frac{1}{\vert x\vert^{1+sp}}\). In particular, the following estimate holds.

For \(0<r<1\), the Poisson kernel is defined by \[P_{r}(\theta)=\frac{1-r^{2}}{1-2r\cos\theta+r^{2}},\qquad\theta\in[-\pi,\pi],\] and satisfies \[\int^{\pi}_{-\pi}P_{r}(\theta)\,\frac{d\theta}{2\pi}=1,\qquad P_{r}(\theta)\ge0.\] For a distribution (or function) \(u\) on \(\mathbb{S}^{1}\) we define the Poisson mean \[\mathscr{P}_{r}u:=P_{r}*u=\int_{\mathbb{S}^{1}}P_{r}(t)u(\cdot-t)\,\frac{dt}{2\pi}.\] That is, \(\mathscr{P}_{r}u\) is a harmonic function in the unit disc, with boundary value \(u\) whenever \(u\in L^{1}(\mathbb{S}^{1})\) (see Rudin’s book for example).

The Hardy space (of index \(p\)), denoted by \(\mathbf{H}^{p}(\mathbb{D})\) is the space of functions with finite \(p^{th}\)- Hardy norm. A crucial result about Hardy norms is that, if \(\Vert f\Vert_{\mathbf{H}^{p}(\mathbb{D})}\) is finite then \(f\) has a radial extension to the boundary \[f^{*}(\xi):=\lim_{r\rightarrow1}f(r\xi)\] exists a.e for all \(\xi\in\mathbb{S}^{1}\). Moreover \[\Vert f^{*}\Vert_{p}=\Vert f\Vert_{\mathbf{H}^{p}(\mathbb{D})}.\] We invite the reader to look at [1], [2] for a more in-depth comprehensive treatment of the theory of Hardy spaces. For the sake of simplicity, we shall keep writing \(f\) instead of \(f^{*}\). The Hardy space is also defined for harmonic functions, and often called real Hardy space (See [2]).

The classical Hardy-Besov spaces \(B^{s}_{p,p,+}\) of analytic functions on the disc are defined as the analytic subspace of \(B^{s}_{p,p}(\mathbb{S}^{1})\simeq W^{s,p}(\mathbb{S}^{1})\). They have been extensively studied, notably by Flett [3], Oswald [4], and Triebel [5]. In these works, boundary smoothness is either prescribed from the outset or built into the definition via Hardy norms. The present paper takes a different starting point: we ask how much can be deduced from the sole knowledge that the Gagliardo seminorms \([u_{r}]_{W^{s,p}(\mathbb{S}^{1})}\) are uniformly bounded, with no \(L^{p}\) or Hardy assumption. The answer, given by our main theorem below, is that this minimal hypothesis already forces full membership in \(\mathbf{H}^{p}(\mathbb{D})\) and \(W^{s,p}\) regularity of the boundary trace. The key tool is the interplay between the mean-value property of analytic functions and the fractional Poincaré inequality on \(\mathbb{S}^{1}\).

Main result and contribution. Given an analytic function \(u\) on \(\mathbb{D}\), write its radial restrictions \(u_r(\xi):=u(r\xi)\) for \(0<r<1\) and \(\xi\in\mathbb{S}^1\). We introduce the Banach space \[\mathcal{W}^{s,p}(\mathbb{D}) :=\left\{u \text{ analytic on }\mathbb{D}:\;\sup_{0<r<1}[u_r]_{W^{s,p}(\mathbb{S}^1)}<\infty\right\}, \qquad \|u\|_{\mathcal{W}^{s,p}}:=|u(0)|+\sup_{0<r<1}[u_r]_{W^{s,p}(\mathbb{S}^1)}.\] The distinctive feature is that the defining assumption controls only tangential oscillation of each slice and makes no a priori assumption on \(\sup_r\|u_r\|_{L^p(\mathbb{S}^1)}\). Theorem 6 shows that, for analytic functions, this seminorm-only hypothesis already forces \(u\in \mathbf{H}^p(\mathbb{D})\) and yields a boundary trace \(u^*\in W^{s,p}(\mathbb{S}^1)\) with \(u_r\to u^*\) in \(W^{s,p}(\mathbb{S}^1)\) as \(r\to1^{-}\). The key mechanism is the combination of the mean-value property \(\overline{u_r}=u(0)\) (which pins the constant mode) with a fractional Poincaré inequality on \(\mathbb{S}^1\), recovering \(L^p\) control from oscillation alone. As a consequence, the trace map \(u\mapsto u^*\) is identified as a surjective isomorphism \(\mathcal{W}^{s,p}(\mathbb{D})\xrightarrow{\sim}B^s_{p,p,+}(\mathbb{S}^1)\) with explicit norm equivalence constants (Theorem 12). This identification is a consequence of the seminorm criterion and is not used to prove Theorem 6.

Organization. Section 2 studies \(\mathcal{W}^{s,p}(\mathbb{D})\) as a Banach space and proves the trace isomorphism with \(B^s_{p,p,+}(\mathbb{S}^1)\). Section 3 contains applications (harmonic and semilinear problems, a variational characterization, and a stochastic interpretation), and Section 4 collects explicit examples. We first recall the strong continuity of the Poisson semigroup on \(W^{s,p}(\mathbb{S}^1)\), which is used in Step 3 of the proof of Theorem 6.

Theorem 5 is referred to as the continuity of the fractional Poisson operator. It says that one can construct a harmonic function inside the unit disc from boundary data in \(W^{s,p}(\mathbb{S}^{1})\), and that the construction is continuous. The natural converse question is the following: given an analytic function \(u\) in the unit disc whose radial restrictions \(u_{r}\) have uniformly controlled fractional oscillation, must \(u\) admit a boundary trace with the same regularity?

In the classical Hardy space theory, membership in \(\mathbf{H}^{p}(\mathbb{D})\) is established by controlling the full \(L^{p}\) means \(\sup_{r}\|u_{r}\|_{L^{p}}\). The distinctive feature of the following theorem is that no such \(L^{p}\) control is assumed. Only the homogeneous Gagliardo seminorm \([u_{r}]_{W^{s,p}(\mathbb{S}^{1})}\), which measures oscillation but discards mean values, is required to be uniformly bounded. The \(L^{p}\) control, and hence Hardy space membership, is then recovered as a consequence, via the mean-value property of analytic functions and a fractional Poincaré inequality on \(\mathbb{S}^{1}\). This recovery of size from oscillation is the core contribution of the result.

By abuse of language, we say that a complex-valued function is harmonic in the unit disc if both its real and imaginary parts are harmonic. In what follows, \(0<s<1\) and \(1<p<+\infty\) unless otherwise mentioned.

The proof proceeds in three steps:

1. Hardy membership. We show \(u\in\mathbf{H}^{p}(\mathbb{D})\) using the fractional Poincaré inequality on \(\mathbb{S}^{1}\) together with the mean-value property \(\overline{u_{r}}=u(0)\). This is the key step: the constant mode is pinned by \(u(0)\), allowing the seminorm bound to control the full \(L^{p}\) norm of \(u_{r}\).

2. Boundary regularity. Once \(u\in\mathbf{H}^{p}(\mathbb{D})\), the boundary trace \(u^{*}\in L^{p}(\mathbb{S}^{1})\) exists. We promote this to \(u^{*}\in W^{s,p}(\mathbb{S}^{1})\) using reflexivity of \(W^{s,p}\) and a weak compactness argument.

3. Seminorm convergence. We use the strong continuity of the Poisson semigroup on \[B^{s}_{p,p}(\mathbb{S}^{1})\simeq W^{s,p}(\mathbb{S}^{1})\] to conclude that \[[u_{r}-u^{*}]_{W^{s,p}}\to0.\]

Step 1: Case \(p=2\) (Hilbertian argument)↩︎

The case \(p=2\) is presented separately because the Fourier series structure of \(H^{2}(\mathbb{D})\) makes the argument completely explicit and self-contained, with no appeal to reflexivity or weak compactness. The general case below subsumes this one, but the \(p=2\) argument serves as transparent motivation for all three steps.

For each \(0<r<1\), write the Taylor expansion of \(u\) as \[u_{r}(e^{\theta i})=\sum^{+\infty}_{k=0}a_{k}r^{k}e^{ik\theta},\qquad\theta\in(-\pi,\pi),\] In particular, the Fourier coefficients of \(u_{r}\) on \(\mathbb{S}^{1}\) are \[(u_{r})_{k}=a_{k}r^{k}\quad\text{for }k\ge0,\qquad(u_{r})_{k}=0\quad\text{for }k<0.\] In particular, by ?? we obtain \[\begin{align} [u_{r}]^{2}_{W^{s,2}(\mathbb{S}^{1})} & \propto\sum_{k\in\mathbb{Z}}|k|^{2s}\,|(u_{r})_{k}|^{2}\\ & =\underset{=S(r)}{\underbrace{\sum^{\infty}_{k=1}k^{2s}|a_{k}|^{2}r^{2k}}}. \end{align}\] By hypothesis ?? with \(p=2\), we have \[\sup_{0<r<1}S(r)<+\infty.\] Observe that for each fixed \(k\ge1\) the function \(r\mapsto r^{2k}\) is increasing on \((0,1)\), and all terms in the series for \(S(r)\) are nonnegative. Hence \(S(r)\) is monotone nondecreasing in \(r\), and by monotone convergence, \[\lim_{r\to1^{-}}S(r)=\sum^{\infty}_{k=1}k^{2s}|a_{k}|^{2}\le\sup_{0<r<1}S(r)<+\infty.\label{eq:limit-Sr}\tag{2}\] In particular, \[\sum^{\infty}_{k=1}k^{2s}|a_{k}|^{2}<\infty.\label{eq:sum-k2s-ak2}\tag{3}\] Since \(k^{2s}\ge1\) for all \(k\ge1\), 3 implies \[\sum^{+\infty}_{k=0}|a_{k}|^{2}=|a_{0}|^{2}+\sum^{\infty}_{k=1}|a_{k}|^{2}\le|a_{0}|^{2}+\sum^{\infty}_{k=1}k^{2s}|a_{k}|^{2}<\infty.\] Thus \(u\in\mathbf{H}^{2}(\mathbb{D})\), and moreover \[\Vert u\Vert^{2}_{\mathbf{H}^{2}(\mathbb{D})}<+\infty.\] By the classical theory (see, e.g., [1], [2]) there exists a boundary function \(u^{*}\in L^{2}(\mathbb{S}^{1})\) such that \[\lim_{r\to1^{-}}u(re^{i\theta})=u^{*}(e^{i\theta})\quad\text{for almost every }\theta\in(-\pi,\pi),\] and the convergence also holds in \(L^{2}(\mathbb{S}^{1})\). This proves conclusion (1) of the theorem for \(p=2\). To prove (2) and (3), remark that the Fourier coefficients of \(u^{*}\) are exactly \(a_{k}\) for \(k\ge0\) and \(0\) for \(k<0\), and therefore, by Cauchy integral formula and ?? , \[\begin{align} [u^{*}]^{2}_{W^{s,2}(\mathbb{S}^{1})} & \propto\sum_{k\in\mathbb{Z}}|k|^{2s}|(u^{*})_{k}|^{2}\\ & \propto\sum^{\infty}_{k=1}k^{2s}|a_{k}|^{2}<+\infty. \end{align}\] Thus \(u^{*}\in W^{s,2}(\mathbb{S}^{1})\). In particular, by comparing with 2 , we obtain

\[\begin{align} [u_{r}-u^{*}]^{2}_{W^{s,2}(\mathbb{S}^{1})} & \propto\sum^{\infty}_{k=1}k^{2s}|a_{k}|^{2}(1-r^{k})^{2}\\ & \leq\sum^{\infty}_{k=1}k^{2s}|a_{k}|^{2}(1-r^{2k})\\ & =S(1)-S(r^{2})\\ & \underset{r\rightarrow1^{-}}{\longrightarrow}0 \end{align}\] which is the desired convergence of the fractional seminorms. This establishes (2) and (3) for \(p=2\) and completes Step 1.

General case\(\boldsymbol{ 1<p<\infty.}\)↩︎

We now explain how to adapt the argument to general \(1<p<\infty\). In this regime, the space \(W^{s,p}(\mathbb{S}^{1})\) can be identified, up to equivalent norms, with the Besov space \(B^{s}_{p,p}(\mathbb{S}^{1})\) (see, for example, [5], [8]), and the Poisson semigroup acts as a strongly continuous semigroup on \(B^{s}_{p,p}(\mathbb{S}^{1})\).
As sketched in the proof road-map, we begin by establishing the Hardy space membership and so the existence of the trace on the unit circle. By assumption ?? , \[\sup_{0<r<1}[u_{r}]_{W^{s,p}(\mathbb{S}^{1})}<\infty.\] On \(\mathbb{S}^{1}\) a fractional Poincaré inequality holds. Since \(\mathbb{S}^{1}\) is a compact Riemannian manifold without boundary, the inequality follows from the general theory for bounded Lipschitz domains applied to the universal cover \([-\pi,\pi]\) with periodic boundary conditions; see, e.g., [9] for the flat case, which transfers directly to \(\mathbb{S}^{1}\) via the parametrization \(\theta\mapsto e^{i\theta}\) and the equivalence of the kernels noted in the introduction. More precisely, there exists a constant \(\kappa=\kappa(s,p)>0\) such that for every \(\xi\in W^{s,p}(\mathbb{S}^{1})\), \[\|\xi-\overline{\xi}\|_{L^{p}(\mathbb{S}^{1})}\le\kappa[\xi]_{W^{s,p}(\mathbb{S}^{1})},\label{eq:frac-Poincare}\tag{4}\] where \(\overline{\xi}\) is the mean value, i.e. \[\overline{\xi}:=\frac{1}{2\pi}\int^{2\pi}_{0}\xi(e^{\theta i})d\theta.\] Note that 4 holds for all \(0<s<1\) and \(1<p<+\infty\) on the compact manifold \(\mathbb{S}^{1}\); no condition such as \(sp>1\) is required once the mean is subtracted. We apply 4 with \(\xi=u_{r}\) for each \(0<r<1\). Analyticity of \(u\) implies that the mean value of \(u\) over inner circles is constant. More precisely, for every \(0<r<1\), \[\overline{u_{r}}=\frac{1}{2\pi}\int^{2\pi}_{0}u(re^{i\theta})d\theta=u(0).\] Thus 4 yields \[\|u_{r}-u(0)\|_{L^{p}(\mathbb{S}^{1})}\le\kappa\,[u_{r}]_{W^{s,p}(\mathbb{S}^{1})}.\] Combining this with the trivial bound \[\|u_{r}\|_{L^{p}}\le\|u_{r}-u(0)\|_{L^{p}}+(2\pi)^{\frac{1}{p}}|u(0)|,\] we obtain \[\|u_{r}\|_{L^{p}(\mathbb{S}^{1})}\le\kappa[u_{r}]_{W^{s,p}(\mathbb{S}^{1})}+(2\pi)^{\frac{1}{p}}|u(0)|\le\kappa\sup_{0<r<1}[u_{r}]_{W^{s,p}}+(2\pi)^{\frac{1}{p}}|u(0)|\] for all \(0<r<1\). Therefore, \(u\in\mathbf{H}^{p}(\mathbb{D})\). More precisely, \[\Vert u\Vert_{\mathbf{H}^{p}(\mathbb{D})}=\sup_{0<r<1}\left(\frac{1}{2\pi}\int^{2\pi}_{0}|u(re^{i\theta})|^{p}\,d\theta\right)^{\frac{1}{p}}<+\infty.\label{eq:Hp-bound}\tag{5}\] In particular, \(u\) admits a boundary function \(u^{*}\in L^{p}(\mathbb{S}^{1})\) such that \[\lim_{r\to1^{-}}u(re^{i\theta})=u^{*}(e^{i\theta})\quad\text{for almost every }\theta\in(-\pi,\pi),\] and the convergence also holds in \(L^{p}(\mathbb{S}^{1})\).

Now, we are going to show that the trace \(u^{*}\in W^{s,p}(\mathbb{S}^{1})\). First, recall that for each \(0<r<1\) we can write \[u_{r}(e^{i\theta})=u(re^{i\theta})=(P_{r}*u^{*})(\theta),\qquad\theta\in(-\pi,\pi),\] where \(P_{r}\) is the Poisson kernel on \(\mathbb{S}^{1}\). In fact, this is the standard Poisson representation of \(\mathbf{H}^{p}(\mathbb{D})\) functions (see, e.g., [2]). Thus the family \((u_{r})_{0<r<1}\) is exactly the Poisson semigroup applied to \(u^{*}\).

By assumption ?? and the bound 5 , the family \((u_{r})_{0<r<1}\) is bounded in \(W^{s,p}(\mathbb{S}^{1})\). Since \(1<p<\infty\), the space \(W^{s,p}(\mathbb{S}^{1})\) is reflexive and continuously embedded in \(L^{p}(\mathbb{S}^{1})\) (see, for instance, [10], [11]). Fix any sequence \((r_{n})_{n\ge1}\) with \(r_{n}\to1^{-}\) as \(n\to\infty\). Then \((u_{r_{n}})_{n\ge1}\) is a bounded sequence in \(W^{s,p}(\mathbb{S}^{1})\), so by reflexivity and the Eberlein–Šmulian theorem there exist a subsequence \((r_{n_{k}})_{k\ge1}\) and a function \(\widetilde{u}\in W^{s,p}(\mathbb{S}^{1})\) such that \[u_{r_{n_{k}}}\rightharpoonup\widetilde{u}\quad\text{weakly in }W^{s,p}(\mathbb{S}^{1})\quad\text{as }k\to\infty.\] Since the embedding \(W^{s,p}(\mathbb{S}^{1})\hookrightarrow L^{p}(\mathbb{S}^{1})\) is continuous, the same subsequence converges weakly in \(L^{p}\), that is, \[u_{r_{n_{k}}}\rightharpoonup\widetilde{u}\quad\text{weakly in }L^{p}(\mathbb{S}^{1}).\] On the other hand, the full family \((u_{r})_{0<r<1}\) converges to \(u^{*}\) in \(L^{p}(\mathbb{S}^{1})\) as \(r\to1^{-}\), hence in particular \[u_{r_{n_{k}}}\to u^{*}\quad\text{strongly in }L^{p}(\mathbb{S}^{1})\quad\text{as }k\to\infty.\] In a Banach space, strong convergence implies weak convergence with the same limit. Therefore the weak limit of \(u_{r_{n_{k}}}\) in \(L^{p}(\mathbb{S}^{1})\) must be \(u^{*}\). Comparing with the previous weak convergence, we conclude that \(\widetilde{u}=u^{*}\) almost everywhere on \(\mathbb{S}^{1}\). Since \(\widetilde{u}\in W^{s,p}(\mathbb{S}^{1})\), this shows that \(u^{*}\) coincides (as an \(L^{p}\)-function) with an element of \(W^{s,p}(\mathbb{S}^{1})\). Hence \(u^{*}\in W^{s,p}(\mathbb{S}^{1})\), as desired.

The final step now is showing the convergence of the Gagliardo norms. Recall that on \(\mathbb{S}^{1}\), the fractional Sobolev space \(W^{s,p}(\mathbb{S}^{1})\) coincides with the Besov space \(B^{s}_{p,p}(\mathbb{S}^{1})\) up to equivalent norms (see [5], [8]); we use this identification freely below. The Poisson semigroup is strongly continuous on \(B^{s}_{p,p}(\mathbb{S}^{1})\): for every \(u\in B^{s}_{p,p}(\mathbb{S}^{1})\), \[\|P_{r}*u-u\|_{B^{s}_{p,p}(\mathbb{S}^{1})}\longrightarrow0\quad\text{as }r\to1^{-}.\] For this strong continuity we refer to [6], [7] (see [5] for a unified exposition). Since \(u^{*}\in W^{s,p}(\mathbb{S}^{1})\simeq B^{s}_{p,p}(\mathbb{S}^{1})\) and \(u_{r}=P_{r}*u^{*}\), applying the above with \(u=u^{*}\) gives \[\|u_{r}-u^{*}\|_{W^{s,p}(\mathbb{S}^{1})}\longrightarrow0\quad\text{as }r\to1^{-}.\] By the definition of the \(W^{s,p}\)-norm (equal to \(\|\cdot\|_{L^{p}}+[\cdot]_{W^{s,p}}\)) this yields both \[\|u_{r}-u^{*}\|_{L^{p}(\mathbb{S}^{1})}\to0\quad\text{and}\quad[u_{r}-u^{*}]_{W^{s,p}(\mathbb{S}^{1})}\to0\] as \(r\to1^{-}\). Using the triangle inequality for the seminorm we obtain \[\bigl|[u_{r}]_{W^{s,p}(\mathbb{S}^{1})}-[u^{*}]_{W^{s,p}(\mathbb{S}^{1})}\bigr|\le[u_{r}-u^{*}]_{W^{s,p}(\mathbb{S}^{1})}\longrightarrow0,\] i.e.\([u_{r}]_{W^{s,p}(\mathbb{S}^{1})}\to[u^{*}]_{W^{s,p}(\mathbb{S}^{1})}\) as \(r\to1^{-}\). This proves conclusions (2) and (3) of the theorem for \(1<p<+\infty\). Together with step (1), the proof is complete.

2 The space \(\mathcal{W}^{s,p}(\mathbb{D})\) and its norm↩︎

We begin with a lemma that is used repeatedly in the sequel.

Theorem 6 identifies a natural subspace of the Hardy space \(\mathbf{H}^{p}(\mathbb{D})\). We now study this space as a Banach space in its own right, describe its norm, and clarify its relationship to the classical analytic Besov space.

Several remarks are in order. The quantity 6 is indeed a norm: it is homogeneous and satisfies the triangle inequality, and \(\|u\|_{\mathcal{W}^{s,p}}=0\) implies \([u_{r}]_{W^{s,p}}=0\) for all \(r\), hence each \(u_{r}\) is constant; combined with \(u(0)=0\) this forces \(u\equiv0\) by analyticity. The two terms play distinct roles: \(|u(0)|\) anchors the constant mode (the mean value), while \(\sup_{r}[u_{r}]_{W^{s,p}}\) controls oscillation. Together they recover full \(L^{p}\) control via the argument of Theorem 6.

The choice of \(u(0)\) as the anchor point is canonical, not arbitrary. By the mean-value property of analytic functions, \(u(0)\) is precisely the mean of \(u_{r}\) over every circle \(r\mathbb{S}^{1}\), for every \(0<r<1\). It is therefore the unique interior point whose value is determined by the boundary oscillation data alone (any other interior point \(z_{0}\) would require knowing \(u(z_{0})\) separately). In particular, \(\|u\|_{\mathcal{W}^{s,p}}\) depends on the center of the disc, and the space \(\mathcal{W}^{s,p}(\mathbb{D})\) is naturally associated with the disc centered at the origin.

3 Applications↩︎

We present five applications of the main theorem. The first two are direct corollaries. The third gives a new sufficient condition for boundary regularity of semilinear equations. The fourth is the most structurally significant: for analytic functions, it provides an alternative seminorm-only hypothesis, different in character from the classical trace theorem, yielding the same boundary conclusion. The fifth gives a regularity criterion for value functions in stochastic control. Explicit examples verifying the seminorm condition are collected in Section 4.

Throughout this section, \(1<p<\infty\) and \(0<s<1\).

3.1 Boundary regularity for harmonic functions↩︎

3.2 Comparison with the classical trace theorem↩︎

The classical trace theorem states that \(u\in W^{s+\frac{1}{p},\,p}(\mathbb{D})\) implies \(u|_{\mathbb{S}^{1}}\in W^{s,p}(\mathbb{S}^{1})\), with no analyticity required. The following proposition records that condition (B) of the main theorem yields the same trace conclusion for analytic functions, from a hypothesis that is genuinely different in character: it controls only the boundary oscillation of the radial restrictions, with no reference to interior fractional derivatives.

3.3 Boundary regularity for semilinear equations↩︎

3.4 Variational characterization of \(W^{s,p}(\mathbb{S}^{1})\)↩︎

3.5 Regularity of value functions in stochastic control↩︎

Let \((\Omega,\mathcal{F},\mathbb{P})\) be a probability space carrying a standard two-dimensional Brownian motion \((B_{t})_{t\ge0}\). For \(z\in\mathbb{D}\), let \(\tau_{z}:=\inf\{t\ge0:|B^{z}_{t}|=1\}\) be the first exit time of \(B^{z}\) (Brownian motion started at \(z\)) from the unit disc \(\mathbb{D}\).

4 Explicit Examples↩︎

We illustrate the main theorem on four explicit families of functions. The first three are chosen so that the seminorm condition can be verified by a direct computation. The fourth — conformal maps of Hölder domains — is the most instructive: the conclusion is reachable by classical methods, but our criterion reduces the verification to a single geometric estimate, bypassing both the interior Sobolev norm and the Fourier coefficients of the map.

Throughout, we work with \(p=2\) where Fourier series are available, and note which computations extend to general \(p\).

Comparison with the classical route. To reach the same conclusions directly, one would verify two separate series: \[H_{2}(u)^{2}\le\sum^{\infty}_{k=1}|a_{k}|^{2}=\sum^{\infty}_{k=1}k^{-2s-1-2\varepsilon}<\infty\quad\text{(Hardy membership)}\] and \[[u^{*}]^{2}_{W^{s,2}}\propto\sum^{\infty}_{k=1}k^{2s}|a_{k}|^{2}=\sum^{\infty}_{k=1}k^{-1-2\varepsilon}<\infty\quad\text{(boundary regularity)}.\] In this example all three series are equally explicit, so the advantage of our criterion is that it collapses two checks into one. The deeper advantage — collapsing two hard checks into one geometric estimate — appears in Example 4 below.

Case \(s<\tfrac{1}{2}\): The exponent \(2s-2<-1\), so \(\sum^{\infty}_{k=1}k^{2s-2}<\infty\). Hence \[\sup_{0<r<1}[u_{r}]_{W^{s,2}}<\infty,\] and Theorem 6 gives \(u^{*}\in W^{s,2}(\mathbb{S}^{1})\).

Case \(s=\tfrac{1}{2}\): The series becomes \(\sum^{\infty}_{k=1}k^{-1}r^{2k}\), which diverges as \(r\to1^{-}\) (harmonic series). Hence \(\sup_{r}[u_{r}]_{W^{1/2,2}}=+\infty\) and the hypothesis of Theorem 6 fails. Indeed \(u^{*}(\theta)=\log|1-e^{i\theta}|^{-1}\notin W^{1/2,2}(\mathbb{S}^{1})\), which is consistent.

Case \(s>\tfrac{1}{2}\): Similarly \(\sum^{\infty}_{k=1}k^{2s-2}=\infty\) and the supremum is infinite.

This example shows that the threshold \(s=1/2\) is sharp for \(\log\frac{1}{1-z}\): the seminorm criterion correctly identifies the exact range of \(s\) for which \(u^{*}\in W^{s,2}(\mathbb{S}^{1})\), and the criterion fails precisely where the conclusion fails.

Classical routes are harder. To verify \(u^{*}\in W^{s,p}(\mathbb{S}^{1})\) by existing methods one would proceed via one of:

1. Via Hölder regularity and embedding: Warschawski’s theorem gives \(u\in C^{0,\alpha}(\overline{\mathbb{D}})\), hence \(u^{*}\in C^{0,\alpha}(\mathbb{S}^{1})\), and the embedding \(C^{0,\alpha}(\mathbb{S}^{1})\hookrightarrow W^{s,p}(\mathbb{S}^{1})\) for \(s<\alpha\) then gives the conclusion. This requires invoking two separate non-trivial results: Warschawski’s boundary correspondence theorem and the Hölder-to-Sobolev embedding.

2. Via interior Sobolev norm: Verify \(u\in W^{s+\frac{1}{p},p}(\mathbb{D})\) and apply the classical trace theorem. Controlling the full interior fractional Sobolev norm of a conformal map requires detailed estimates on \(u'\) in weighted spaces, going through Muckenhoupt weight theory and the Jacobian of \(u\) — substantially more machinery.

Our route reduces everything to one geometric estimate. The Kellogg–Warschawski theorem in its classical form applies to \(C^{1,\alpha}\) boundaries; for the general \(C^{0,\alpha}\) Jordan curve case, the uniform Hölder continuity of \(u\) up to the boundary follows from Warschawski’s theorem on the modulus of continuity of the boundary correspondence (see [13], Theorem 3.5). More precisely, Pommerenke’s result applies when the arc-length parametrization \(\gamma:[-\pi,\pi]\to\partial\Omega\) satisfies \(|\gamma(\xi)-\gamma(\sigma)|\le C|\xi-\sigma|^{\alpha}\) (i.e.is \(C^{0,\alpha}\) as a map between intervals); under this hypothesis the boundary correspondence and hence \(u\) itself is \(C^{0,\alpha}\)-continuous up to \(\partial\mathbb{D}\), giving a constant \(C=C(\Omega)>0\) such that \[|u(z_{1})-u(z_{2})|\le C|z_{1}-z_{2}|^{\alpha}\qquad\text{for all }z_{1},z_{2}\in\overline{\mathbb{D}}.\label{eq:holder-u}\tag{9}\] Applying 9 with \(z_{1}=r\xi\) and \(z_{2}=r\eta\) for \(\xi,\eta\in\mathbb{S}^{1}\): \[\begin{alignedat}{1}[u_{r}]^{p}_{W^{s,p}(\mathbb{S}^{1})} & =\iint_{\mathbb{S}^{1}\times\mathbb{S}^{1}}\frac{|u(r\xi)-u(r\eta)|^{p}}{|\xi-\eta|^{1+sp}}\,d\xi\,d\eta\\ & \le C^{p}\iint_{\mathbb{S}^{1}\times\mathbb{S}^{1}}\frac{|\xi-\eta|^{\alpha p}}{|\xi-\eta|^{1+sp}}\,d\xi\,d\eta\\ & =C^{p}\iint_{\mathbb{S}^{1}\times\mathbb{S}^{1}}\frac{d\xi\,d\eta}{|\xi-\eta|^{1+sp-\alpha p}}. \end{alignedat}\]

The last integral is finite if and only if \(1+sp-\alpha p<1\), i.e.\(s<\alpha\). Therefore, for every \(s<\alpha\), \[\sup_{0<r<1}[u_{r}]_{W^{s,p}(\mathbb{S}^{1})}\le C\left(\iint_{\mathbb{S}^{1}\times\mathbb{S}^{1}}\frac{d\xi\,d\eta}{|\xi-\eta|^{1+(s-\alpha)p}}\right)^{\frac{1}{p}}<+\infty.\] Theorem 6 now applies immediately and gives \(u\in\mathbf{H}^{p}(\mathbb{D})\) and \(u^{*}\in W^{s,p}(\mathbb{S}^{1})\) for all \(s<\alpha\).

The advantage is not that the conclusion is unreachable classically — it is. The advantage is that our theorem shortens the proof by one non-trivial step and avoids the need to control the interior Sobolev norm of \(u\). Once \(C^{0,\alpha}\) regularity of \(u\) is known, our criterion converts it to \(W^{s,p}\) boundary regularity in a single line, via a direct estimate on the Gagliardo double integral. The classical route requires either an additional embedding theorem or interior Sobolev estimates on the conformal map — both of which involve further machinery. The Gagliardo seminorm has direct geometric content that can be estimated from the boundary geometry without knowing the Taylor coefficients of \(u\), and this is precisely the advantage our theorem exploits.

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