May 02, 2026
Inspired by the classical Besov \(p\)-spaces defined via higher-order derivatives on the upper half-plane, we introduce Besov-type spaces on simply connected domains. We first prove that on quasidisks, the first-order
Besov space is isomorphic to its higher-order counterparts, and that these higher-order spaces preserve conformal quasi-invariance. Based on this result, we characterize chord-arc domains in terms of the isomorphism between the first-order Besov space and
the boundary Besov space. This extends recent results for the Dirichlet space (\(p=2\)) to the general case \(1 < p < \infty\).
Keywords: Besov space, chord-arc curve, quasidisk, conformal invariance.
MSC 2020: 30C62, 30E25, 30H25, 42B35.
Recall that the Besov \(p\)-space \(I^{sec:1}_{p}(\mathbb{H})\) on the upper half plane \(\mathbb{H}\) is the set of all harmonic functions \(u\) such that \[I^1_p(u, \mathbb{H}) := \|u\|^p= \int_{\mathbb{H}} |\nabla u(z)|^p y^{p-2} dm(z) <\infty,\] where \(dm(z)\) denotes the two-dimensional Lebesgue measure \(dxdy\). The classical Möbius invariance of the Besov energy \(I^1_p\) provides the essential framework for developing composition operator and generalized Carleson measure theory on this space (see [1], [2]), and \(p\)-integrable Teichmüller spaces theory (see [3]–[6]). Furthermore, the Besov space admits equivalent norms involving higher-order derivatives. A classical result by Lizorkin et al. establishes that the \(n\)-th order Besov-type norm is equivalent to both the first-order energy and the boundary trace norm (see [7] or [8]). To be specific, the \(n\)-th order Besov space \(I^n_p(\mathbb{H})\) ( \(n\ge2\)) is defined as the set of all harmonic functions \(u\) on \(\mathbb{H}\) that satisfy the vanishing condition \(\nabla^k u(z) \to 0\) as \(y \to \infty\) for all \(1 \le k < n\), and \[I_p^n(u, \mathbb{H}) = \int_{\mathbb{H}} |\nabla^n u(z)|^p y^{np-2} dm(z).\] The corresponding boundary Besov space \(B_p(\mathbb{R})\) (or the trace space) on the real line \(\mathbb{R}\) is the set of all measurable functions \(f\) such that \[\|f\|^p_{B_p(\mathbb{R})} = \int_{\mathbb{R}}\int_{\mathbb{R}} \frac{|f(x)-f(y)|^p}{|x-y|^2} dx dy<\infty.\] Then we obtain the full chain of norm equivalences: \[\label{eq:equivalence95chain} I_p^1(u, \mathbb{H}) \asymp I_p^n(u, \mathbb{H}) \asymp \|u|_{\mathbb{R}}\|_{B_p},\tag{1}\] where the vanishing condition is implicitly assumed so that these norms can be compared with each other; otherwise, the equivalence fails (a polynomial, for instance, is a counterexample). Analogous results hold for the unit disk \(\mathbb{D}\) (see e.g. [2]), which have been extensively applied to characterize Hankel operators in the Schatten–von Neumann classes (see [9], [10]).
The definitions of the Besov spaces admit direct generalizations to a simply connected domain \(\Omega \subset \mathbb{C}\) with a locally rectifiable boundary \(\Gamma = \partial\Omega\). For a harmonic function \(u\) on \(\Omega\), the \(n\)-th order Besov norm is defined by \[I_p^n(u, \Omega) =\int_{\Omega} |\nabla^n u(z)|^p \delta(z)^{np-2} dm(z),\] where \(\delta(\cdot)\) denotes the distance function with respect to \(\Gamma\). Correspondingly, the trace Besov space \(B_p(\Gamma)\) on the boundary curve \(\Gamma\) is the set of all measurable functions \(f\) such that \[\|f\|^p_{B_p(\Gamma)} = \int_{\Gamma} \int_{\Gamma} \frac{|f(w)-f(z)|^p}{|w-z|^2} |dw| |dz| <\infty.\] Evidently, the distortion theorem implies that the first-order Besov energy is conformally invariant between \(\mathbb{H}\) and the domain \(\Omega\) (see 2 ). It is natural to ask to what extent these norm equivalences 1 and the conformal invariance of higher-order Besov spaces remain valid for general simply connected domains? A primitive form of this problem, which arose from the study of the Cauchy integral on quasicircles, was proposed by the authors in [11]: what class of curves \(\Gamma\) is characterized by the norm equivalence \(I_2^1(u, \Omega) \asymp \|u|_{\Gamma}\|_{B_2(\Gamma)}\)? Recently, Wei and Zinsmeister [12] resolved this problem for the classical Dirichlet space (\(p=2\)), proving that such an equivalence characterizes chord-arc curves. We will review their key arguments in Section 3. Our first purpose is to study the norm equivalence between higher-order and first-order Besov spaces on general domains, alongside the higher-order conformal quasi-invariance.The second is to extend the characterization of chord-arc curves to the general case \(1 < p < \infty\).
The paper is organized as follows. In section 2, we briefly revisit some basic facts related to quasiconformal mapping theory, and the boundary behaviors of the first-order Besov space on quasidisks. These results have previously appeared in [13]. Section 3 is primarily devoted proving the main results of this paper.
In this paper, the notation \(A\lesssim B\) \((A\gtrsim B)\) means that there is an independent constant \(C\) such that \(A\le CB\) \((A\ge CB)\). The notation \(A \asymp B\) means both \(A\lesssim B\) and \(A\gtrsim B\). Also, \(B(z,r)\) denotes a disk of radius \(r\) centered at the point \(z\in\mathbb{C}\). Finally, regarding the differential operators, we note that for a harmonic function \(u\), all mixed partial derivatives vanish since \(\partial \bar{\partial} u = 0\). Consequently, the \(n\)-th order gradient reduces to its purely holomorphic and anti-holomorphic derivatives, with its norm defined by\[|\nabla^n u(z)| = \sqrt{ \left| \frac{\partial^n u}{\partial z^n} \right|^2 + \left| \frac{\partial^n u}{\partial \bar{z}^n} \right|^2}.\]
A sense-preserving homeomorphism \(\rho\) of the complex plane \(\mathbb{C}\) is called quasiconformal if it has locally square integrable distributional derivatives \(\overline{\partial}\rho\), \({\partial}\rho\) which satisfy the Beltrami equation \(\overline{\partial}\rho=\mu{\partial}\rho,\) where \(\mu\in L^{\infty}(\mathbb{C})\) with \(\|\mu\|_{\infty}<1\) is called the Beltrami coefficient or complex dilatation of \(\rho\). The image of \(\mathbb{R}\) under a global quasiconformal mapping is called a quasicircle. A Jordan domain is called a quasidisk if it is bounded by a quasicircle. A sense-preserving homeomorphism \(h\) of \(\mathbb{R}\) is said to be quasisymmetric and belongs to the class \(\text{QS}(\mathbb{R})\) if there exists a positive constant \(C\), called the quasisymmetric constant of \(h\), such that \[{C^{-1}}\le{|h(I_1)|}/{|h(I_2)|}\le C\] for all pairs of adjacent arcs \(I_1\) and \(I_2\) on \(\mathbb{R}\) with the same arc-length \(|I_1|=|I_2|\). Beurling-Ahlfors proved that a sense-preserving homeomorphism \(h\) of \(\mathbb{R}\) is quasisymmetric if and only if there exists some quasiconformal mapping of \(\mathbb{H}\) onto itself which has boundary values \(h\) (see [14]). Throughout this paper, we always assume that any conformal mapping defined on the half-planes \(\mathbb{H}\) or \(\mathbb{L}\) fixes the point at \(\infty\). Let \(\Gamma\) be a Jordan curve with complementary domains \(\Omega^+\) and \(\Omega^-\), and let \(\phi\) and \(\psi\) map \(\mathbb{H}\) and \(\mathbb{L}\) conformally onto \(\Omega^+\) and \(\Omega^-\), respectively. Since \(\phi\) and \(\psi\) can be continuously extended to \(\mathbb{R}\), we can form \(h_{\Gamma}=\psi^{-1}\circ \phi\), which is known to be a conformal sewing for \(\Gamma\). It is well known that \(h_{\Gamma}\) is quasisymmetric if and only if \(\Gamma\) is a quasicircle (see [15]).
We say that a locally rectifiable curve \(\Gamma\) is Ahlfors regular (also known as Ahlfors-David regular) if its arc-length satisfies \(\ell(\Gamma \cap B(z,r)) \le Cr\) for all \(z \in \mathbb{C}\) and \(r>0\). A locally rectifiable Jordan curve \(\Gamma\) passing through \(\infty\) is a chord-arc curve if the arc-length between any two finite points \(\zeta, z \in \Gamma\) satisfies \(\ell(\zeta,z) \le C|\zeta-z|\). Every chord-arc curve is regular, but the converse is false (e.g., a parabola). It is known that a Jordan curve is a chord-arc curve if and only if it is an Ahlfors regular quasicircle (see [16]).
It is well-known that the Dirichlet problem has a unique solution for the Besov space. Specifically, every \(F\in I^1_p(\mathbb{H})\) has non-tangential limit values almost everywhere in \(\mathbb{R}\) (w.r.t. the arc-length measure) such that \(f:= F|_{\mathbb{R}}\in B_p(\mathbb{R})\) satisfies \(\|f\|_{B_p}\asymp I^1_p(F,\mathbb{H})\). Conversely, the usual Poisson extension operator \(P\) takes each element \(f\in B_p(\mathbb{R})\) to \(F:=Pf\in I^1_p(\mathbb{H})\) such that \(I^1_p(F,\mathbb{H}) \asymp \|f\|_{B_p(\mathbb{R})}\).
We now extend the Dirichlet problem to a locally rectifiable curve \(\Gamma\) with complementary domains \(\Omega^+\) and \(\Omega^-\). Let \(\phi:\mathbb{H} \to \Omega^+\) and \(\psi:\mathbb{L} \to \Omega^-\) be the two corresponding Riemann maps that fix \(\infty\). We define \(B_p^{\phi}(\Gamma)\) to be the set of all functions \(f\) on \(\Gamma\) such that \[\|f\|^p_{B^{\phi}_p(\Gamma)}:=\int_{\mathbb{R}}\int_{\mathbb{R}}\frac{|f\circ \phi(x)-f\circ \phi(y)|^p}{|x-y|^2} dxdy <\infty.\] By the conformal invariance of harmonic measure (see [16]), \(f\) is defined almost everywhere in \(\Gamma\) w.r.t. the harmonic measure. Since \(\Gamma\) is a locally rectifiable curve, \(\phi\) is locally absolutely continuous on \(\mathbb{R}\) by F. and M. Riesz theorem (see [17]). Consequently, \(\phi\) maps sets of zero arc-length measure on \(\mathbb{R}\) to sets of zero arc-length measure on \(\Gamma\), and \(f\) is also defined almost everywhere in \(\Gamma\) w.r.t. the arc-length measure. Similarly, let \(B_p^{\psi}(\Gamma)\) denote all functions \(f\) on \(\Gamma\) such that \(f\circ \psi \in B_p(\mathbb{R})\). Denoting by \(\lambda_{\Omega}\) the Poincaré metric on a simply connected domain \(\Omega\subset\mathbb{C}\), we recall the well-known fact that \(\lambda_{\Omega}(z) \asymp \delta(z)^{-1}\) for any \(z \in \Omega\). Then for \(F \in I^1_p(\Omega^+)\), \[\begin{align}\label{Eq:32transfer} \int_{\Omega} |\nabla F(w)|^p \delta(w)^{p-2} dm(w)&\simeq \int_{\Omega} |\nabla F(w)|^p \lambda^{2-p} _{\Omega}(w) dm(w) \\ &= \int_{\mathbb{H}} |\nabla( F\circ \phi )(z)|^p y^{p-2} dm(z), \end{align}\tag{2}\] which yields \(\widetilde{F}:=F \circ \phi \in I^1_p(\mathbb{H})\) and hence has non-tangential limit almost everywhere on \(\mathbb{R}\setminus E\), where \(E\) is a set with zero arc-length measure. Furthermore, the rectifiability of \(\Gamma\) implies that \(\phi\) preserves non-tangential regions (see [16]). Therefore, for any \(\zeta \in \Gamma \setminus \phi(E)\), the non-tangential limit of \(F\) exists and is determined by pushing forward the limit of pullback of \(F\): \[\lim_{w\stackrel{N.T}{\longrightarrow}\zeta} F(w) = \lim_{z\stackrel{N.T}{\longrightarrow}\phi^{-1}(\zeta)} \widetilde{F}(z), \quad z=\phi^{-1}(w).\] In other words, \(F\) has non-tangential limit values almost everywhere on \(\Gamma\) (w.r.t. the arc-length measure). Let \(f := F|_{\Gamma}\). We then have \(f \circ \phi = \widetilde{F}|_{\mathbb{R}}\) almost everywhere on \(\mathbb{R}\) and \[\|f\|_{B^{\phi}_p(\Gamma)} = \|\widetilde{F}|_{\mathbb{R}}\|_{B_p(\mathbb{R})} \asymp {I_p^1(\widetilde{F},\mathbb{H})}\asymp I_p^1(F,\Omega^+).\] Similarly, for any \(G\in I^1_p(\Omega^-)\), letting \(g:=G|_{\Gamma}\), we can deduce that \(\|g\|_{B^{\psi}_p(\Gamma)} \asymp I_p^1(G,\Omega^-)\).
Conversely, let \(f\in B_p^\phi (\Gamma)\) so that \(\widetilde{f} :=f\circ\phi \in B_p(\mathbb{R})\). Then \(\widetilde{F}:=P{\widetilde{f}}\in I^1_p(\mathbb{H})\) and \(F:=\widetilde{F}\circ\phi ^{-1} \in I^1_p(\Omega^+)\) with \(I_p^1(F,\Omega^+) \asymp {I_p^1(\widetilde{F},\mathbb{H})} \asymp\| \widetilde{f}\|_{B_p(\mathbb{R})}=\|f\|_{B^{\phi}_p(\Gamma)}\). It follows that \(F\) has non-tangential limit almost everywhere on \(\Gamma\) (w.r.t. the arc-length measure) such that \(f=F|_{\Gamma}\). Thus, the extension map \(f \mapsto F\) induces a bounded isomorphism \(P_+\) from \(B_p^\phi(\Gamma)\) onto \(I^1_p(\Omega^+)\), which we denote by \(I^1_p(\Omega^+) \simeq B_p^\phi(\Gamma)\). By a similar argument, one can establish the bounded isomorphism \(I^1_p(\Omega^-) \simeq B_p^\psi(\Gamma)\).
To establish the equivalence between \(B_p^\phi(\Gamma)\) and \(B_p^\psi(\Gamma)\), we rely on the following result, which indicates that \(B_p(\mathbb{R})\) can be used to characterize the quasi-symmetry of a homeomorphism.
Lemma 1. [18], [19]Let \(h\) be an oriented homeomorphism on \(\mathbb{R}\). Then the pull-back operator \(P_h\) defined by \(P_hu=u\circ h\) is a bounded operator on \(B_p(\mathbb{R})\) \((1<p<\infty)\) if and only if \(h\) is quasisymmetric.
Lemma 2. Suppose \(\Gamma\) is a locally rectifiable Jordan curve. Then the identity map induces a bounded isomorphism between \(B_p^\phi(\Gamma)\) and \(B_p^\psi(\Gamma)\) if and only if \(\Gamma\) is a quasicircle. Consequently, \(I^1_p(\Omega^+) \simeq I^1_p(\Omega^-)\) if and only if \(\Gamma\) is a quasicircle.
Proof. It is worth noting that for \(p=2\), this result was first proved in [20]. We now proceed to prove the general case within our descriptive framework.
Suppose now \(B_p^\phi(\Gamma)\simeq B_p^\psi(\Gamma)\). For any \(f \in B_p^\phi(\Gamma)\), it follows that \(f\circ \psi \in B_p(\mathbb{R})\) and \(\|f \circ \psi \|_{B_p} \asymp \|f \circ \phi\|_{B_p}\). Note that \[\begin{align} f \circ \phi=f\circ \psi \circ \psi^{-1} \circ \phi=P_{h_\Gamma}(f\circ \psi), \end{align}\] where the pull-back operator \(P_h\) is defined by \(P_hu:=u \circ h\) for an orientation-preserving homeomorphism \(h: \mathbb{R}\to \mathbb{R}\). Thus, by Lemma 1 we know \(h_\Gamma\) is quasisymmetric and hence \(\Gamma\) is a quasicircle.
Conversely, assume \(\Gamma\) is a quasicircle. By Lemma 1 again, \(P_{h_\Gamma}\) is an isomorphism on \(B_p(\mathbb{R})\). This yields \(B_p^{\phi}(\Gamma) \simeq B_p^{\psi}(\Gamma)\), from which we conclude \(I^1_p(\Omega^+) \simeq I^1_p(\Omega^-)\). In fact, the bounded isomorphism between \(I^1_p(\Omega^+)\) and \(I^1_p(\Omega^-)\) is determined by \[F \xrightarrow{\circ \phi} \widetilde{F} \xrightarrow{\text{Trace}} \widetilde{f} \xrightarrow{\circ h_\Gamma^{-1}} \widetilde{g} \xrightarrow{\text{Poisson}} \widetilde{G} \xrightarrow{\circ \psi^{-1}} G.\] ◻
We begin by stating a basic geometric estimate (Lemma 3) from [21], whose proof mainly relies on the distortion theorem for quasiconformal mappings.
Let \(\phi\) be a conformal mapping from \(\mathbb{H}\) onto a quasidisk \(\Omega\) with boundary \(\Gamma\). For any \(w \in \Omega\), there exists a unique preimage \(z = \phi^{-1}(w) = x + iy \in \mathbb{H}\). Define the paths \(L(w)\) and \(\widetilde{L}(w)\) as \[L(w) =\phi (\{x +i(y+t): t\ge0\}); \quad \widetilde{L}(w)=\phi (\{x +it: t\ge0\}).\] If \(w_0 = \phi(x + iy_0)\) and \(0\le y_0 < y\), then the path \(L({w_0, w})\) is defined as \[L({w_0, w}) = \phi\{(x + it): y_0\leq t\leq y\}.\] This type of definition ensures that \(L(w)\) is a conformal vertical ray that behaves predictably.
Notation as above, we have the following lemma.
Lemma 3. For \(\varepsilon>0\) and \(w=\phi(x+iy)\) belonging to quasidisk \(\Omega\), we have \[\int_{L(w)} \delta(\xi)^{-1-\varepsilon} |d\xi| \asymp \delta(w)^{-\varepsilon}.\] Furthermore, if \(w_0 \in \Gamma\) and \(\varepsilon<2\), then \[\int_{L(w_0,w)} \delta(\xi)^{1-\varepsilon} |d\xi| \asymp \delta(w)^{2-\varepsilon}.\]
Then we have the following result.
Theorem 1. Let \(\phi\) conformally map \(\mathbb{H}\) onto a quasidisk \(\Omega\) with boundary \(\Gamma\) such that \(\phi(\infty) = \infty\). Let \(u\) be a harmonic function on \(\Omega\) satisfying the vanishing condition stated in section 1. Then for any \(n \ge 2\) and \(1<p<\infty\) we have the following norm equivalences and the higher-order conformal quasi-invariance: \[I^1_p(u, \Omega) \asymp I^n_p(u, \Omega) \asymp I^n_p(u \circ \phi, \mathbb{H}).\]
Proof. Let us prove that \(I^1_p(u, \Omega) \asymp I^n_p(u, \Omega)\) for \(n \ge 2\). Since \(u\) can be decomposed as \(u_1 + \overline{u_2}\) where \(u_1\) and \(u_2\) are holomorphic, we may assume \(u\) is holomorphic.
First, we show \(I^1_p(u,\Omega) \lesssim I^2_p(u,\Omega)\). For any \(z \in \Omega\), we by \(u'(\infty)=0\) have \[|u'(z)| \le \int_{L(z)} |u''(w)| |dw|.\] We introduce a small parameter \(\varepsilon > 0\) and apply Hölder’s inequality to get \[|u'(z)|^p \le \bigg( \int_{L(z)} \delta(w)^{-1-\frac{\varepsilon}{p-1}} |dw| \bigg)^{p-1} \int_{L(z)} |u''(w)|^p \delta(w)^{p-1+\varepsilon} |dw|.\] By Lemma 3, \[|u'(z)|^p \lesssim \delta(z)^{-\varepsilon} \int_{L(z)} |u''(w)|^p \delta(w)^{p-1+\varepsilon} |dw|.\] Integrating \(|u'(z)|^p\) over \(\Omega\) with respect to the weight \(\delta(z)^{p-2}\) yields that \[I^1_p(u,\Omega)\lesssim \int_{\Omega} \delta(z)^{p-2-\varepsilon} \bigg( \int_{L(z)} |u''(w)|^p \delta(w)^{p-1+\varepsilon} |dw| \bigg) dm(z).\] Note that \(\{z:L(z)\ni w\}=L(w_0, w)\) for a fixed \(w\). Then by Fubini, \[I^1_p(u,\Omega) \lesssim \int_{\Omega} |u''(w)|^p \delta(w)^{p-1+\varepsilon} \bigg( \int_{L(w_0, w)} \delta(z)^{p-2-\varepsilon} |dz| \bigg) {dm(w)}.\] By Lemma 3 again, yielding that \(I^1_p(u,\Omega) \lesssim I^2_p(u,\Omega)\).
We use the subharmonicity of \(|u'|^p\) to show \(I^2_p(u,\Omega)\lesssim I^1_p(u,\Omega)\). Let \(0<c<\frac{sec:1}{8}\). Since \(u'\) is holomorphic on \(B(z,c\delta(z))\), we by Cauchy integral formula see \[\begin{align} |u''(z)|^p\lesssim{\delta(z)^{-p}}\sup \{|u'(\zeta)|^p: \zeta\in B(z,c\delta(z)). \} \end{align}\] Note the subharmonicity of \(|u'|^p\) implies \[|u'(\zeta)|^p \lesssim {\delta(z)^{-2}}\int_{B(\zeta, c\delta(z))} |u'(w)|^p dm(w),\] Thus, \[\begin{align} |u''(z)|^p\lesssim \frac{sec:1}{\delta(z)^{p+2}}\int_{B(z,\frac{sec:1}{4}\delta(z))} |u'(w)|^p dm(w). \end{align}\] By integrating over \(\Omega\) with weight \(\delta(z)^{2p-2}\), it follows that \[I^2_p(u,\Omega) \lesssim\int_{\Omega} \delta(z)^{p-4} \int_{B(z,\frac{sec:1}{4}\delta(z))} |u'(w)|^p dm(w) dm(z)\] Let \(\widetilde{B}(z)\) denote \(\{z: B(z,\frac{sec:1}{4}\delta(z))\ni w\}\). By Fubini, \[\begin{align} I^2_p(u,\Omega) &\lesssim\int_{\Omega} \int_{\widetilde{B}(z)} \delta(z)^{p-4}|u'(w)|^p dm(z) dm(w)\\ &\asymp \int_{\Omega} \delta(w)^{p-2} |u'(w)|^p dm(w). \end{align}\] It is similar to get \(I^{n-1}_p(u,\Omega) \asymp I^n_p(u,\Omega)\) using the above arguments.
It now remains to prove the conformal quasi-invariance. Notice \(I^1_p(u,\Omega) \asymp I^n_p(u,\Omega)\), the quasi-invariance will follow \[I^n_p(u, \Omega) \asymp I^1_p(u, \Omega) = I^1_p(u \circ \phi, \mathbb{H}) \asymp I^n_p(u \circ \phi, \mathbb{H}).\] But we need to rigorously prove that \(F := u \circ \phi\) satisfies \(F^{(k)}(z) \to 0\) as \(y \to \infty\) in \(\mathbb{H}\). This critical step justifies \(I_p^1(F, \mathbb{H}) \asymp I_p^n(F, \mathbb{H})\). We break into four steps.
Step 1: Local distortion of \(\phi'\). Fix \(z \in \mathbb{H}\) with \(y = \text{Im} z\). We normalize \(\phi\) by defining \[g(w) = \frac{\phi(z + y w) - \phi(z)}{y \phi'(z)}, \quad w \in \mathbb{D}.\]
Since \(\phi\) is univalent, \(g \in \mathcal{S}\) (the class of normalized univalent functions on \(\mathbb{D}\) with \(g(0)=0\) and \(g'(0)=1\)). By the classical Koebe distortion theorem for class \(\mathcal{S}\), we have \[|g'(w)| \le \frac{1+|w|}{(1-|w|)^3}.\]
For any \(\zeta \in B_z:=B(z,y/2)\), we can write \(\zeta = z + yw\) for some \(w \in \mathbb{D}\) with \(|w| \le 1/2\). Then \[|g'(w)| \le \frac{1 + 1/2}{(1 - 1/2)^3} = 12.\] Since \(g'(w) = \phi'(\zeta)/\phi'(z)\), we obtain \[\label{eq:local95distortion} |\phi'(\zeta)| \le 12 |\phi'(z)|, \quad \zeta \in B(z, y/2).\tag{3}\]
Step 2: Higher-order Koebe distortion estimates. For any integer \(j \ge 1\), we have \[\phi^{(j)}(z) = \frac{(j-1)!}{2\pi i} \int_{|\zeta - z| = y/2} \frac{\phi'(\zeta)}{(\zeta - z)^j} d\zeta.\] Then by 3 , \[|\phi^{(j)}(z)| \lesssim \frac{sec:1}{y^{j-1}} \max_{|\zeta - z| = y/2} |\phi'(\zeta)| \lesssim \frac{sec:1}{y^{j-1}} |\phi'(z)|.\] Noting that \(y|\phi'(z)| \asymp \delta(\phi(z))\), we obtain the higher-order bound: \[\label{eq:higher95koebe} |\phi^{(j)}(z)| \lesssim\frac{\delta\circ\phi(z)}{y^j}.\tag{4}\]
Step 3: Higher-order Bloch embedding. By the subharmonicity of \(|u^{(k)}|^p\), we obtain \[\begin{align} |u^{(k)}(w)|^p &\lesssim \frac{sec:1}{\delta(w)^2} \int_{B(w, \delta(w)/2)} |u^{(k)}(\zeta)|^p dm(\zeta) \\ &\lesssim \frac{sec:1}{\delta(w)^{kp}} \int_{B(w, \delta(w)/2)}|u^{(k)}(\zeta)|^p \delta(\zeta)^{kp-2} dm(\zeta)\\ &\lesssim \delta(w)^{-kp} \|u\|^p. \end{align}\] This yields the Besov space can be embedded into the \(k\)-order Bloch space \(\mathcal{B}^k(\Omega)\): \[\label{eq:bloch95embed} |u^{(k)}(w)| \lesssim \delta(w)^{-k}.\tag{5}\]
Step 4: Estimate \(F^{(k)}(z)\) as \(y\to\infty\). Recall that Faà di Bruno’s formula, \[F^{(k)}(z) = \sum \frac{k! u^{(m)}\circ\phi(z) }{m_1! 1!^{m_1} m_2! 2!^{m_2} \cdots m_k! k!^{m_k}} \prod_{j=1}^k \big(\phi^{(j)}(z)\big)^{m_j},\] where the sum is taken over all \(k\)-tuples of non-negative integers \((m_1, \dots, m_k)\) satisfying \(\sum_{j=1}^k j m_j = k\), and \(m = \sum_{j=1}^k m_j \ge 1\).
Then we by 4 have \[\begin{align} \prod_{j=1}^k \big|\phi^{(j)}(z)\big|^{m_j} &\lesssim \prod_{j=1}^k \left(\delta\circ\phi(z) y^{-j} \right)^{m_j}\\ & = \delta^{\sum_{j=1}^k m_j}\circ\phi(z) \cdot y^{-\sum_{j=1}^k j m_j}\\ &=\delta^{m}\circ\phi(z) \cdot y^{-k}. \end{align}\] And by 5 , \[|u^{(m)}\circ\phi(z)| \prod_{j=1}^k \big|\phi^{(j)}(z)\big|^{m_j} \lesssim\delta^{-m}\circ\phi(z) \cdot \delta^{m}\circ\phi(z) \cdot y^{-k} = y^{-k}.\] We conclude that \(|F^{(k)}(z)| \le C y^{-k}\) holds uniformly. Finally, we obtain \(\lim\limits_{y \to \infty}|F^{(k)}(z)| = 0\), which confirms the vanishing condition on \(\mathbb{H}\). We have finished the proof. ◻
Now let us return to the characterization of chord-arc curves by the isomorphism among these spaces (Theorem 2).
To prove it, we need to introduce a crucial characterization of Ahlfors-David regular curves. This result is originally due to Yves Meyer, whose proof first appeared in David’s [22] studying the \(L^2\) boundedness of the Cauchy integral on the regular curves (see also [12]). Later, Bruna and González provided an alternative proof in [23].
Lemma 4 ([24]). Let \(\Gamma\) be a locally rectifiable curve. Then \(\Gamma\) is a Ahlfors-David regular curve iff there exists a constant \(C > 0\) such that for any \(w \notin \Gamma\), \[\label{Eq:32Meyer-David-Integral} \int_{\Gamma} \frac{|dz|}{|z - w|^2} \le \frac{C}{\delta(w)}.\tag{6}\]
Theorem 2. Let \(\Gamma\) be a locally rectifiable Jordan curve passing through \(\infty\), bounding domains \(\Omega^+\) and \(\Omega^-\). For any \(p \in (1, \infty)\), \(\Gamma\) is a chord-arc curve if and only if \[\label{Eq:32CAcondition} B_p(\Gamma) \asymp I^1_p(\Omega^+) \asymp I^1_p( \Omega^-).\qquad{(1)}\]
Proof. \(\Longrightarrow\) Note that when \(\Gamma\) is a quasicircle, it follows readily from Lemma 2 that \[B_p^{\phi}(\Gamma) \simeq B_p^{\psi}(\Gamma)\simeq I^1_p(\Omega^+) \asymp I^1_p( \Omega^-).\] We denote the space by \(\mathcal{B}_p(\Gamma)\) and assign a norm \(\|\cdot\|_{\mathcal{B}_p(\Gamma)}\) to be \(\|\cdot\|_{B_p^{\phi}(\Gamma)}\) or \(\|\cdot\|_{B_p^{\psi}(\Gamma)}\). Furthermore, the isomorphism \(B_p(\Gamma) \simeq \mathcal{B}_p(\Gamma)\) in the case of chord-arc curves has been shown in our previous paper [13].
\(\Longleftarrow\) Now suppose ?? is valid, we show \(\Gamma\) is a chord-arc curve. Note that \(I^1_p(\Omega^+) \asymp I^1_p(\Omega^-)\) implies \(\Gamma\) is a quasicircle by Lemma 2.
Next, we aim to show that \(\Gamma\) is Ahlfors-David regular. It should be pointed out that the case \(p=2\) has been proved in [12]. Let us briefly review their idea. They used a test function \(F(z) := (w-z)^{-1}\) to check the regularity condition by Meyer’s Lemma 4. This function first appeared in the work of Bruna and González [23] on Hardy spaces defined on chord-arc domains. Noting that for \(p=2\) \[\|F\|^2_{B_2(\Gamma)} = \int_\Gamma \int_\Gamma \frac{sec:1}{|w-\zeta|^2 |w-\eta|^2} |d\zeta| |d\eta| = \left( \int_\Gamma \frac{|d\zeta|}{|\zeta - w|^2} \right)^2.\] Meanwhile, \[I^1_2(F, \Omega^{\pm}) =\int_{\Omega^{\pm}} \frac{sec:1}{|z-w|^4} dm(z) \lesssim \frac{sec:1}{\delta(w)^2}.\] Then \(\Gamma\) is Ahlfors regular by Lemma 4.
Now, we extend this to the general case \(1 < p < \infty\). For \(z \in \Omega^+ \cup \Gamma\), we still use \(F(z) := (w-z)^{-1}\), where \(w\in\Omega^{-}\). Since \(F\) is analytic on \(\Omega^+ \cup \Gamma\), we estimate its \(p\)-norm by Theorem 1. Write (\(n\ge2\)) \[\label{norm32of32F} \begin{align} I^1_p(F,\Omega^+)\asymp I^n_p(F,\Omega^+) &= \int_{\Omega^+} |F^{(n)}(z)|^{p} \delta(z)^{np-2}\,dx\,dy \\ &\leq \int_{\Omega^+} \frac{(n!)^p}{|z-w|^{n(p+1)}}\, |z-w|^{np-2}\,dx\,dy\\ &= \int_{\Omega^+} \frac{(n!)^p}{|z-w|^{p+2}}\,dx\,dy \\ &\le \int_{|z-w|\ge \delta(w)} \frac{(n!)^p}{|z-w|^{p+2}}\,dx\,dy \\ &=2\pi (n!)^p \int_{\delta(w)}^{+\infty} \frac{sec:1}{r^{p+1}} dr\asymp {\delta(w)^{-p}}. \end{align}\tag{7}\]
Now we estimate the boundary norm. For any \(w \in \Omega^-\), choose \(w^* \in \Gamma\) such that \(|w - w^*| = \delta(w)\), and define \(\Gamma_{kw} := \Gamma \cap B(w, k\delta(w))\) for \(k\ge2\) and \(\Gamma^*_{kw}:=\Gamma\setminus\Gamma_{kw}\). It is easy to see \(\ell(\Gamma_{2w}) \ge 2\delta(w)\) by \(\Gamma\cap B(w^*,\delta(w))\subset\Gamma_{2w}\). To get 6 , we divide into two cases.
Case 1 (\(1 < p < 2\)): For all \(\zeta, \eta \in \Gamma_{2w}\), we have \[|\zeta-\eta| \le |\zeta-w| + |\eta-w| \le 4\delta(w).\] Then \(p<2\) implies \[\begin{align} \|F\|_{B_p(\Gamma)}^p &\ge \int_{\Gamma_{2w}}\int_{\Gamma_{2w}} \frac{|\zeta-\eta|^{p-2}}{|\zeta-w|^p |\eta-w|^p} |d\zeta| |d\eta| \\ &\ge \frac{sec:1}{(2\delta(w))^{2p}} \int_{\Gamma_{2w}}\int_{\Gamma_{2w}} |\zeta-\eta|^{p-2} |d\zeta| |d\eta| \\ &\ge \frac{(4\delta(w))^{p-2}}{(2\delta(w))^{2p}} \int_{\Gamma_{2w}}\int_{\Gamma_{2w}} |d\zeta| |d\eta| \\ &\asymp\delta(w)^{-p-2} \ell(\Gamma_{2w})^2. \end{align}\] Then we by 7 have \[{\ell(\Gamma_{2w}) ^2}{\delta(w)^{-p-2}}\lesssim \|F\|_{B_p(\Gamma)}^p \lesssim I^n_p(F,\Omega^+)\lesssim{\delta(w)^{-p}}\Longrightarrow \ell(\Gamma_{2w}) \lesssim \delta(w), \; \text{and}\] \[\label{eq:near951} \int_{\Gamma_{2w}} \frac{|d\zeta|}{|\zeta - w|^2} \le \frac{\ell(\Gamma_{2w})}{\delta(w)^2} \lesssim \delta(w)^{-1}.\tag{8}\]
For \(\zeta\in \Gamma^*_{2w}\) and \(\eta\in \Gamma_{2w}\), we have \[|\zeta-\eta|\leq|\zeta-w|+|\eta-w|\leq 2|\zeta-w|.\] Thus for \(1<p<2\), \[\begin{align} \|F\|_{B_p(\Gamma)}^p &\ge \int_{\Gamma^*_{2w}} \int_{\Gamma_{2w}} \frac{|\zeta - \eta|^{p-2}}{|\zeta - w|^p |\eta - w|^p} |d\zeta| |d\eta|\\ &\ge 2^{p-2} \int_{\Gamma^*_{2w}}\int_{\Gamma_{2w}} \frac{|d\eta||d\zeta| }{|\zeta - w|^2|\eta - w|^p}\\ &\gtrsim \frac{\ell(\Gamma_{2w})}{(2\delta(w))^p}\int_{\Gamma^*_{2w}} \frac{|d\zeta|}{|\zeta - w|^2}\\ &\gtrsim \delta(w)^{1-p} \int_{\Gamma^*_{2w}} \frac{|d\zeta|}{|\zeta - w|^2}. \end{align}\] Noting that \(\|F\|_{B_p(\Gamma)}^p \lesssim \delta(w)^{-p}\), we obtain \[\label{eq:far951} \int_{\Gamma^*_{2w}} \frac{|d\zeta|}{|\zeta - w|^2} \lesssim \delta(w)^{-1}.\tag{9}\] Combining 9 with 8 implies for \(w\in\Omega^-\) \[\int_\Gamma \frac{|d\zeta|}{|\zeta - w|^2} = \int_{\Gamma^*_{2w}}\frac{|d\zeta|}{|\zeta - w|^2} + \int_{\Gamma_{2w}} \frac{|d\zeta|}{|\zeta - w|^2} \lesssim \delta(w)^{-1}.\]
Case 2 (\(p \ge 2\)): Let \(\Gamma^o_{4w} := \Gamma \cap \{z: 5\delta(w) \le |z - w| \le 6\delta(w)\}\) ensure \(\ell(\Gamma^o_{4w}) \ge 2\delta(w)\). For any \(\eta \in \Gamma_{4w}\) and \(\zeta \in \Gamma^o_{4w}\), we have \(|\zeta - \eta| \ge \delta(w)\) while \(|w - \eta| \le 4\delta(w)\) and \(|w - \zeta| \le 6\delta(w)\). This gives \[\begin{align} \|F\|_{B_p(\Gamma)}^p &> \int_{\Gamma^o_{4w}} \int_{\Gamma_{4w}} \frac{|\zeta - \eta|^{p-2}}{|\zeta - w|^p |\eta - w|^p} |d\eta| |d\zeta| \\ &\ge \int_{\Gamma^o_{4w}} \int_{\Gamma_{4w}} \frac{\delta(w)^{p-2}}{4^p6^{p}\delta(w)^{2p} } |d\eta| |d\zeta|\\ &\asymp \delta(w)^{-p-2} \ell(\Gamma^o_{4w}) \ell(\Gamma_{4w})\\ &\gtrsim \delta(w)^{-p-1} \ell(\Gamma_{4w}). \end{align}\] Comparing this with 7 forces \(\ell(\Gamma_{4w}) \lesssim \delta(w)\). As in 8 , we get \[\label{near952} \int_{\Gamma_{4w}} \frac{|d\zeta|}{|\zeta - w|^2} \lesssim \delta(w)^{-1}.\tag{10}\]
For \(\zeta\in \Gamma^*_{4w}\) and \(\eta\in \Gamma_{2w}\), we have \[|\zeta - \eta| \ge |\zeta - w| - |\eta - w| \ge \frac{sec:1}{2}|\zeta - w|.\] Thus for \(p\ge2\), \[\begin{align} \|F\|_{B_p(\Gamma)}^p &\ge \int_{\Gamma^*_{4w}} \int_{\Gamma_{2w}} \frac{|\zeta - \eta|^{p-2}}{|\zeta - w|^p|\eta - w|^p} |d\eta| |d\zeta| \\ &\ge 2^{2-p} \int_{\Gamma^*_{4w}} \frac{sec:1}{|\zeta - w|^2} \int_{\Gamma_{2w}} \frac{|d\eta|}{|\eta - w|^p} |d\zeta| \\ &\gtrsim \delta(w)^{1-p} \int_{\Gamma^*_{4w}} \frac{|d\zeta|}{|\zeta - w|^2}. \end{align}\] Again, \[\label{eq:far952} \int_{\Gamma^*_{4w}} \frac{|d\zeta|}{|\zeta - w|^2} \lesssim \delta(w)^{-1}.\tag{11}\] Combining 11 with 10 for \(w\in\Omega^-\) ensures 6 holds.
In all cases, we can get 6 for \(w\in\Omega^+\) by a similar argument. This completes the proof of Theorem 2. ◻
Finally, we end the paper with some remarks and open problems.
Remark 3. For \(p > 2\), the estimate \(I^1_p(F, \Omega^+) \lesssim \delta(w)^{-p}\) does not rely on \(n\ge2\) as can be seen by checking 7 . By a completely different approach, a recent preprint [25] obtained \[\|F\|^p_{B_p(\Gamma)}\gtrsim\delta(w)^{1-p}\int_{\Gamma}\frac{|d\zeta|}{|\zeta-w|^{2}}.\]
Theorem 1 says that \(I^1_p(\Omega) \simeq I^n_p(\Omega)\) for quasidisk \(\Omega\), but the following converse problem remains open.
Problem 4. Let \(\Omega\) be a simply connected domain with boundary \(\Gamma\). What class of curves \(\Gamma\) is characterized by the isomorphism \(I^1_p(\Omega) \simeq I^n_p(\Omega)\) for some \(n\)?
This work was supported by the National Natural Science Foundation of China under Grants Nos. 12401095 (T. Liu), 12571083 (Y. Shen), and 12526204 (Y. Yang).
Conflict of interest: On behalf of all authors, the corresponding author states that there is no conflict of interest.
School of Mathematics and Physics, Jiangsu University of Technology, Changzhou 213001, China Email: ltlmath@jsut.edu.cn↩︎
Department of Mathematics, Soochow University, Suzhou 215006, China. Email: ylshen@suda.edu.cn↩︎
Beijing International Center for Mathematical Research (BICMR), Beijing 100871, China Email: yaosongyang@bicmr.pku.edu.cn↩︎