March 14, 2026
We study Hardy–Sobolev spaces \(H_n^p(\mathbb{C}^+)\) on the upper half-plane for \(1\le p\le\infty\) and \(n\in\mathbb{N}\), from both function-theoretic and operator-theoretic viewpoints. We establish an isometric boundary characterization of \(H_n^p(\mathbb{C}^+)\) via nontangential limits, together with a Sobolev-type embedding theorem, a Cauchy integral representation, a direct-sum decomposition of \(W_n^p(\mathbb{R})\) for \(1<p<\infty\), and a generalized Banach algebra structure under pointwise multiplication. We also obtain a finer Fourier-analytic description in the Hilbert case \(p=2\) by proving a Paley–Wiener theorem and deriving the reproducing kernel of \(H_n^2(\mathbb{C}^+)\). On the operator-theoretic side, we prove the spectral formula for multiplication operators and establish two verifiable sufficient conditions for the boundedness of weighted composition operators. These results provide a systematic theory of Hardy–Sobolev spaces on the upper half-plane beyond the Hilbert setting.
Hardy-Sobolev spaces ,upper half-plane ,boundary values ,Paley-Wiener theorem ,reproducing kernels ,spectrum of multiplication operators ,weighted composition operators
Hardy–Sobolev spaces are natural analytic function spaces that combine the boundary control of Hardy spaces with the additional regularity encoded by Sobolev norms. They occupy an important position in complex analysis, harmonic analysis, and operator theory. On bounded domains such as the unit ball \(\mathbb{B}^n\) \((n\ge 1)\), the theory of Hardy–Sobolev spaces is by now well developed. For the spaces \(H_s^p(\mathbb{B}^n)\), Carleson measures were characterized in [1], [2], and a variety of operator-theoretic problems have been studied, including Toeplitz and Hankel operators, Fredholmness, essential spectra, multipliers, composition operators, and related integral operators; see, for instance, [3]–[11] and the references therein.
By contrast, the Hardy–Sobolev theory on unbounded domains remains much less developed, even in one complex variable. A notable feature of the available literature is that many results are concentrated in the Hilbert case \(p=2\), where reproducing kernel Hilbert space techniques and Fourier–Laplace methods provide powerful structural tools. Outside the Hilbert setting, however, reproducing kernels are no longer available as a general mechanism, and it is far from clear whether the existing theory can be extended to the full Banach range \(1\le p\le\infty\). This leads to the central problem of the present paper: how to develop a systematic Hardy–Sobolev theory on the upper half-plane beyond the Hilbert framework, and what can replace reproducing kernels in that setting.
In this paper we study Hardy–Sobolev spaces on the upper half-plane \[\mathbb{C}^+=\{z\in\mathbb{C}: Im(z)>0\}.\] Let \(H(\mathbb{C}^+)\) denote the space of holomorphic functions on \(\mathbb{C}^+\), and let \(H^p(\mathbb{C}^+)\) be the classical Hardy space on \(\mathbb{C}^+\) for \(1\le p\le\infty\). For \(n\in\mathbb{N}\) and \(1\le p\le\infty\), we define \[H_n^p(\mathbb{C}^+)=\Bigl\{F\in H(\mathbb{C}^+):F^{(k)}\in H^p(\mathbb{C}^+),\;k=0,1,\dots,n\Bigr\},\] equipped with the norm \[\|F\|_{H_n^p}= \begin{cases} \Bigl(\sum_{k=0}^n\|F^{(k)}\|_{H^p}^p\Bigr)^{1/p}, & 1\le p<\infty,\\[6pt] \sum_{k=0}^n\|F^{(k)}\|_{H^\infty}, & p=\infty. \end{cases}\] These spaces provide a natural half-plane analogue of classical Hardy–Sobolev spaces on bounded domains, while also reflecting the special interaction among holomorphic boundary values on \(\mathbb{R}\), derivative regularity, and Fourier analysis in the upper half-plane.
Several earlier works already indicate the richness of the half-plane Hardy–Sobolev setting. Early appearances of related Sobolev–Hardy structures may be found in the study of rational wavelets in [12]. Dang–Qian–You [13] and Dang–Qian–Yang [14] studied Hardy–Sobolev derivatives and obtained, in particular, the direct-sum decomposition \[W_1^2(\mathbb{R})=H_1^2(\mathbb{C}^+)\oplus H_1^2(\mathbb{C}^-)\] in the sense of isometric isomorphism. In a more general Hilbert-space Laplace-transform framework, Kucik [15] introduced the spaces \(A_{(m)}^2\) and proved spectral inclusion results for multiplication operators. Another Hilbert Hardy–Sobolev space on the right half-plane, \[\mathscr{H}_n^2(\mathbb{C}_+)=\Bigl\{F\in H(\mathbb{C}_+):z^kF^{(k)}\in H^2(\mathbb{C}_+),\; k=0,1,\dots,n\Bigr\},\] was investigated in [16], [17], where Paley–Wiener type theorems, reproducing kernels, and boundedness criteria for weighted composition operators were established. These works show that half-plane Hardy–Sobolev spaces form a natural and fruitful framework, but they also indicate that much of the existing theory is still essentially Hilbertian.
The main purpose of this paper is to develop a systematic theory of \(H_n^p(\mathbb{C}^+)\) in the full Banach range \(1\le p\le\infty\), from both function-theoretic and operator-theoretic viewpoints. Our basic principle is that, beyond the Hilbert case, the structural role played by reproducing kernels should be replaced by boundary theory. More precisely, we show that nontangential boundary values provide an isometric identification of \(H_n^p(\mathbb{C}^+)\) with a natural Hardy–Sobolev space on \(\mathbb{R}\), denoted by \(H_n^p(\mathbb{R})\). This boundary model is the starting point of the paper and serves as the main mechanism from which the subsequent function-theoretic and operator-theoretic results are derived.
Our first main contribution is the Banach-range structure theory of \(H_n^p(\mathbb{C}^+)\). In Section 3 we prove an isometric boundary characterization of \(H_n^p(\mathbb{C}^+)\) via nontangential limits, together with a Sobolev-type embedding estimate and a Cauchy integral representation. For \(1<p<\infty\), we further obtain a direct-sum decomposition of \(W_n^p(\mathbb{R})\) into upper and lower half-plane Hardy–Sobolev parts, thereby extending the known Hilbert-space decomposition to the full reflexive Banach range. We also show that, for \(n\ge1\), the space \(H_n^p(\mathbb{C}^+)\) admits a generalized Banach algebra structure under pointwise multiplication. This phenomenon has no counterpart in the classical Hardy space case \(n=0\) and plays a crucial bridging role in the operator-theoretic part of the paper.
Our second main contribution concerns the special Hilbert case \(p=2\). In Section 4 we prove a Paley–Wiener theorem for \(H_n^2(\mathbb{C}^+)\) and derive an explicit reproducing kernel. These results provide a finer Fourier-analytic description of the Hilbertian Hardy–Sobolev space and complement the boundary-based Banach theory developed earlier.
Our third main contribution lies in operator theory. In Section 5 we study multiplication operators and weighted composition operators on \(H_n^p(\mathbb{C}^+)\). For multipliers \(\psi\in\mathscr{M}_{n,p}\), we prove the sharp spectral formula \[\sigma(T_\psi)=\overline{\psi(\mathbb{C}^+)},\] which extends the known spectral inclusion in the Hilbert setting to an exact spectral description in our Banach framework. As an immediate consequence, the only compact multiplication operator on \(H_n^p(\mathbb{C}^+)\) is the zero operator. We also establish two verifiable sufficient conditions for the boundedness of weighted composition operators on \(H_n^p(\mathbb{C}^+)\).
We emphasize that the novelty of the paper lies not only in the statements of these results, but also in the method. In the non-Hilbert setting, our arguments do not rely on reproducing kernel techniques; instead, they are based on boundary behavior, weak derivative structure, Sobolev embedding, and Cauchy-type representations. This provides a unified framework for treating Hardy–Sobolev spaces on the upper half-plane throughout the full Banach range.
The principal results of the paper are Theorems 5, 11, and 12 in Section 3, Theorems 13 and 15 in Section 4, and Theorems 23, 25, and 26 in Section 5.
The paper is organized as follows. Section 2 collects preliminaries on Hardy spaces, Sobolev spaces, and weighted Lebesgue spaces. Section 3 develops the Banach-range theory of \(H_n^p(\mathbb{C}^+)\), including the boundary characterization, Sobolev-type embedding, Cauchy integral representation, direct-sum decomposition, and generalized Banach algebra structure. Section 4 is devoted to the Hilbert case \(p=2\), where we establish the Paley–Wiener theorem and derive the reproducing kernel of \(H_n^2(\mathbb{C}^+)\). Finally, Section 5 treats multiplication operators and weighted composition operators on \(H_n^p(\mathbb{C}^+)\).
In this section we collect the notation and standard facts that will be used throughout the paper. We recall basic properties of Hardy spaces on the upper half-plane, Sobolev spaces on the real line, and certain weighted Lebesgue spaces on \(\mathbb{R}^+\) that arise naturally in the Fourier-analytic description of \(H_n^2(\mathbb{C}^+)\).
For \(0<p\le\infty\), the Hardy space \(H^p(\mathbb{C}^+)\) consists of all holomorphic functions \(F\) on the upper half-plane such that \[\|F\|_{H^p} = \begin{cases} \displaystyle \sup_{y>0}\left(\int_{-\infty}^{\infty}|F(x+iy)|^p\,dx\right)^{1/p}<\infty, & 0<p<\infty,\\[12pt] \displaystyle \sup_{z\in\mathbb{C}^+}|F(z)|<\infty, & p=\infty. \end{cases}\]
For \(1\le p<\infty\), every function \(F\in H^p(\mathbb{C}^+)\) admits nontangential boundary values \(F_l\) almost everywhere on \(\mathbb{R}\). The boundary function belongs to the Hardy boundary space \[\begin{align} H^p(\mathbb{R}) :&= \{\,f\in L^p(\mathbb{R}): f \text{ is the nontangential boundary value of } F\in H^p(\mathbb{C}^+)\,\} \\ &= \left\{\,f\in L^p(\mathbb{R}): \int_{-\infty}^{\infty}\frac{f(x)}{x-\overline{z}}\,dx=0,\;\forall z\in\mathbb{C}^+ \right\}. \end{align}\] If \(p=\infty\), we write \[H^\infty(\mathbb{R}) = \left\{\,f\in L^\infty(\mathbb{R}): \int_{-\infty}^{\infty}\frac{f(x)}{(x-\overline{z})(x+i)}\,dx=0,\; \forall z\in\mathbb{C}^+ \right\},\] and \(F\) is uniquely determined by \(F_l\). Moreover, \[\|F\|_{H^p}=\|F_l\|_{L^p}.\]
For \(1\le p<\infty\), each \(F\in H^p(\mathbb{C}^+)\) admits the Cauchy integral representation \[\label{e8} F(z)=\frac{1}{2\pi i}\int_{-\infty}^{\infty}\frac{F_l(x)}{x-z}\,dx, \qquad z\in\mathbb{C}^+.\tag{1}\] Consequently, \[F(x+iy) = \frac{1}{\pi}\int_{-\infty}^{\infty} F_l(t)\,\frac{y}{(x-t)^2+y^2}\,dt.\]
For \(f\in L^p(\mathbb{R})\) with \(1<p<\infty\), define \[F_+(z)=\frac{1}{2\pi i}\int_{-\infty}^{\infty}\frac{f(x)}{x-z}\,dx, \qquad z\in\mathbb{C}^+,\] and \[F_-(z)=-\frac{1}{2\pi i}\int_{-\infty}^{\infty}\frac{f(x)}{x-z}\,dx, \qquad z\in\mathbb{C}^-.\] Their nontangential boundary values are \[\frac{1}{2} f+\frac{i}{2}Hf \qquad\text{and}\qquad \frac{1}{2} f-\frac{i}{2}Hf,\] respectively, where \(H\) denotes the Hilbert transform \[Hf(x)=\frac{1}{\pi}\lim_{\varepsilon\to0} \int_{|x-t|>\varepsilon}\frac{f(t)}{x-t}\,dt.\] These formulas are consequences of the Plemelj theorem; see, for instance, [18] or [19]. When \(p=1\), the same boundary relations remain valid, although \(F_\pm\) need not belong to \(H^1(\mathbb{C}^\pm)\).
We write \(\prescript{+}{}{H^p}(\mathbb{R})\) and \(\prescript{-}{}{H^p}(\mathbb{R})\) for the boundary Hardy spaces corresponding to the upper and lower half-planes, respectively. The preceding boundary formulas imply the classical decomposition \[\label{e4} L^p(\mathbb{R})=\prescript{+}{}{H^p}(\mathbb{R})\oplus\prescript{-}{}{H^p}(\mathbb{R}), \qquad 1<p<\infty,\tag{2}\] where the sum is direct because \[\prescript{+}{}{H^p}(\mathbb{R})\cap \prescript{-}{}{H^p}(\mathbb{R})=\{0\}.\] Moreover, \(\prescript{+}{}{H^2}(\mathbb{R})\) and \(\prescript{-}{}{H^2}(\mathbb{R})\) are orthogonal in \(L^2(\mathbb{R})\).
In the Hilbert case \(p=2\), the classical Paley–Wiener theorem states that the holomorphic Fourier transform \[\mathscr{F}(f)(z)=\int_0^\infty f(x)e^{izx}\,dx, \qquad z\in\mathbb{C}^+,\] defines an isometric isomorphism from \(L^2(\mathbb{R}^+,2\pi\,dx)\) onto \(H^2(\mathbb{C}^+)\); see [20]. We refer to [18], [19], [21] for further background on Hardy spaces.
Let \(L^1_{\mathrm{loc}}(\mathbb{R})\) denote the space of locally integrable functions on \(\mathbb{R}\), and let \(C_c^\infty(\mathbb{R})\) denote the space of test functions. A function \(F\in L^1_{\mathrm{loc}}(\mathbb{R})\) is said to be weakly differentiable if there exists a function \(G\in L^1_{\mathrm{loc}}(\mathbb{R})\) such that \[\int_{-\infty}^{\infty} F(x)\varphi'(x)\,dx = -\int_{-\infty}^{\infty} G(x)\varphi(x)\,dx\] for all \(\varphi\in C_c^\infty(\mathbb{R})\). In this case we write \(D(F)=G\) and call \(G\) the weak derivative of \(F\).
Define inductively \[W_1^0(\mathbb{R}) = \{\,F:\mathbb{R}\to\mathbb{C}: F \text{ is weakly differentiable}\,\},\] and, for \(n\ge2\), \[W_n^0(\mathbb{R}) = \{\,F\in W_{n-1}^0(\mathbb{R}): D^{(n-1)}F\in W_1^0(\mathbb{R})\,\}.\]
The following standard characterization will be used repeatedly; see, for example, [22].
Proposition 1. Let \(n\in\mathbb{N}^+\). A function \(F\) belongs to \(W_n^0(\mathbb{R})\) if and only if there exists a finite sequence \(\{F_k\}_{k=0}^n\subset L^1_{\mathrm{loc}}(\mathbb{R})\) such that \(F=F_0\) almost everywhere and \[F_k(b)-F_k(a)=\int_a^b F_{k+1}(x)\,dx\] for all \(-\infty<a<b<\infty\) and \(k=0,\dots,n-1\).
For \(1\le p\le\infty\) and \(n\in\mathbb{N}^+\), the Sobolev space \(W_n^p(\mathbb{R})\) is defined by \[W_n^p(\mathbb{R}) = \{\,F\in L^p(\mathbb{R})\cap W_n^0(\mathbb{R}): D^{(k)}F\in L^p(\mathbb{R}),\;k=1,\dots,n\,\},\] equipped with the Sobolev norm \[\|F\|_{W_n^p} = \begin{cases} \displaystyle \left(\sum_{k=0}^n \|D^{(k)}F\|_{L^p}^p\right)^{1/p}, & 1\le p<\infty,\\[12pt] \displaystyle \sum_{k=0}^n \|D^{(k)}F\|_{L^\infty}, & p=\infty. \end{cases}\]
Proposition 1 immediately yields the following criterion.
Corollary 2. Let \(n\in\mathbb{N}^+\) and \(1\le p\le\infty\). A function \(F:\mathbb{R}\to\mathbb{C}\) belongs to \(W_n^p(\mathbb{R})\) if and only if there exists a finite sequence \(\{F_k\}_{k=0}^n\subset L^p(\mathbb{R})\) such that \(F=F_0\) almost everywhere and \[F_k(b)-F_k(a)=\int_a^b F_{k+1}(x)\,dx\] for all \(-\infty<a<b<\infty\) and \(k=0,\dots,n-1\).
We refer to [22]–[24] for background on Sobolev spaces and to [15], [25] for related weighted Fourier-side constructions.
For \(1\le p<\infty\) and \(n\in\mathbb{N}\), define the measure \[d\mu_{n,p}=2\pi x^{np}\,dx\] on \(\mathbb{R}^+=(0,\infty)\). We write \(L^p(\mathbb{R}^+,d\mu_{n,p})\) for the corresponding weighted Lebesgue space, with norm \[\|F\|_{\mu_{n,p}} = \left(\int_0^\infty |F(x)|^p\,2\pi x^{np}\,dx\right)^{1/p}.\]
Set \[L_n^p(\mathbb{R}^+) = \bigcap_{k=0}^n L^p(\mathbb{R}^+,d\mu_{k,p}),\] equipped with the norm \[\|F\|_{L_n^p} = \left(\sum_{k=0}^n \|F\|_{\mu_{k,p}}^p\right)^{1/p}.\] Equivalently, \(L_n^p(\mathbb{R}^+)=L^p(\mathbb{R}^+,dv_{n,p})\) with equivalent norm, where \[dv_{n,p}=2\pi(1+x^p)^n\,dx.\]
The following elementary equivalence will be used in Section 4.
Proposition 3. For \(1\le p<\infty\) and \(n\in\mathbb{N}\), \[L_n^p(\mathbb{R}^+)=L^p(\mathbb{R}^+,d\mu_{0,p})\cap L^p(\mathbb{R}^+,d\mu_{n,p}).\]
By Proposition 3, the norm \[\|F\|_{n,p} = \left(\|F\|_{\mu_{0,p}}^p+\|F\|_{\mu_{n,p}}^p\right)^{1/p}\] is equivalent to \(\|\cdot\|_{L_n^p}\).
The next embedding will also be used in the Hilbert case.
Proposition 4. If \(n\ge1\), then \[L_n^p(\mathbb{R}^+)\hookrightarrow L_1^p(\mathbb{R}^+)\hookrightarrow L^1(\mathbb{R}^+).\]
When \(p=2\), the space \(L_n^2(\mathbb{R}^+)\) is a Hilbert space under the inner product \[\langle f,g\rangle_{L_n^2} = \sum_{k=0}^n \int_0^\infty f(x)\overline{g(x)}\,2\pi x^{2k}\,dx.\] This Hilbert-space structure will be used in Section 4.
This section contains the principal new function-theoretic results of the paper in the full Banach range \(1\le p\le\infty\). Our starting point is an isometric boundary characterization of \(H_n^p(\mathbb{C}^+)\) via nontangential limits. Its significance is not merely that it extends the classical Hardy boundary theory to the Sobolev setting, but that it provides a boundary-based structural model for \(H_n^p(\mathbb{C}^+)\) beyond the Hilbert framework. From this model we derive several fundamental consequences, including Sobolev-type embedding estimates, a Cauchy integral representation, a direct-sum decomposition of \(W_n^p(\mathbb{R})\) for \(1<p<\infty\), and, for \(n\ge1\), a generalized Banach algebra structure under pointwise multiplication. The latter will also serve as a key bridge to the operator-theoretic part of the paper.
For the classical Hardy space \(H^p(\mathbb{C}^+)\), nontangential boundary values provide an isometric identification with the Hardy boundary space \(\prescript{+}{}{H^p}(\mathbb{R}):=H^p(\mathbb{R})\). The first main result of this section shows that an analogous statement remains valid at the Hardy–Sobolev level. The point is that, for \(n\ge1\), one must not only recover the boundary values of the function itself, but also identify the Sobolev structure carried by its derivatives. This leads naturally to the boundary Hardy–Sobolev space \[H_n^p(\mathbb{R}):=\prescript{+}{}{H^p}(\mathbb{R})\cap W_n^p(\mathbb{R}),\] equipped with the Sobolev norm.
Theorem 5. Let \(1\le p\le\infty\) and \(n\in\mathbb{N}\). Then the nontangential boundary value map defines an isometric isomorphism from \(H_n^p(\mathbb{C}^+)\) onto \(H_n^p(\mathbb{R})\).
The proof combines the classical Hardy boundary theory with the weak derivative characterization of Sobolev spaces and repeated integration by parts. In this way, one can recover the full boundary Sobolev structure of \(H_n^p(\mathbb{C}^+)\) without relying on reproducing-kernel or Hilbert-space arguments.
Proof. The case \(n=0\) is exactly the classical boundary theory of Hardy spaces on the upper half-plane. Assume now that \(n\ge1\) and let \(F\in H_n^p(\mathbb{C}^+) \subset H^p(\mathbb{C}^+)\). Denote by \(F_l\) the nontangential boundary value of \(F\). Since \(F\in H^p(\mathbb{C}^+)\), we have \[\int_{-\infty}^{\infty}\frac{F_l(x)}{x-\overline{z}}\,dx=0, \qquad z\in\mathbb{C}^+.\] Moreover, for every \(k=0,1,\dots,n-1\), \[\label{e1-new} F^{(k)}(b+iy)-F^{(k)}(a+iy) = \int_a^b F^{(k+1)}(x+iy)\,dx\tag{3}\] for all \(-\infty<a<b<\infty\).
Let \[F_l^{(k)}(x)=\lim_{y\to0^+}F^{(k)}(x+iy),\] which exists for almost every \(x\in\mathbb{R}\) because \(F^{(k)}\in H^p(\mathbb{C}^+)\) for \(k=0,1,\dots,n\). For fixed \(a<b\), Hölder’s inequality gives \[\left| \int_a^b F^{(k)}(x+iy)\,dx-\int_a^b F_l^{(k)}(x)\,dx \right| \le (b-a)^{1/q} \|F^{(k)}(\cdot+iy)-F_l^{(k)}\|_{L^p(\mathbb{R})},\] with the usual interpretation when \(p=1\). Since \(F^{(k)}(\cdot+iy)\to F_l^{(k)}\) in \(L^p(\mathbb{R})\) as \(y\to0^+\), it follows that \[\int_a^b F^{(k)}(x+iy)\,dx \to \int_a^b F_l^{(k)}(x)\,dx.\] Passing to the limit in 3 , we obtain \[\label{e7} F_l^{(k)}(b)-F_l^{(k)}(a)=\int_a^b F_l^{(k+1)}(x)\,dx\tag{4}\] for almost every \(a<b\) and every \(k=0,1,\dots,n-1\). By Corollary 2, this shows that \(F_l\in W_n^p(\mathbb{R})\). Together with the Hardy boundary condition above, we conclude that \(F_l\in H_n^p(\mathbb{R})\). Moreover, \[\label{e2-new} F_l^{(k)}=D^{(k)}F_l \qquad \text{a.e. on }\mathbb{R},\quad k=0,1,\dots,n.\tag{5}\]
Conversely, let \(G\in H_n^p(\mathbb{R})\). For \(1\le p<\infty\), define \[F(z)=\frac{1}{2\pi i}\int_{-\infty}^{\infty}\frac{G(x)}{x-z}\,dx, \qquad z\in\mathbb{C}^+.\] Then \(F\in H^p(\mathbb{C}^+)\) and its nontangential boundary value equals \(G\) almost everywhere. For \(k=1,\dots,n\), differentiating under the integral sign yields \[F^{(k)}(z)=\frac{k!}{2\pi i}\int_{-\infty}^{\infty} \frac{G(x)}{(x-z)^{k+1}}\,dx.\] Using integration by parts, equivalently the weak derivative structure of \(G\), we obtain \[F^{(k)}(z)=\frac{(-1)^{k-1}}{2\pi i}\int_{-\infty}^{\infty} \frac{D^{(k-1)}G(x)}{(x-z)^2}\,dx.\] Since \(G\in H_n^p(\mathbb{R})\), we also have \[0 = k!\int_{-\infty}^{\infty}\frac{G(x)}{(x-\overline{z})^{k+1}}\,dx = (-1)^{k-1}\int_{-\infty}^{\infty} \frac{D^{(k-1)}G(x)}{(x-\overline{z})^2}\,dx.\] Subtracting the two expressions gives \[F^{(k)}(z) = \frac{(-1)^k}{2\pi i} \int_{-\infty}^{\infty} D^{(k-1)}G(t) \left[ \frac{1}{(t-\overline{z})^2}-\frac{1}{(t-z)^2} \right]dt.\] Writing \(z=x+iy\), we note that \[\frac{\partial}{\partial t}\frac{2iy}{(x-t)^2+y^2} = \frac{1}{(t-\overline{z})^2}-\frac{1}{(t-z)^2}.\] Hence, integrating by parts once more, \[F^{(k)}(x+iy) = \frac{(-1)^{k+1}}{\pi} \int_{-\infty}^{\infty} D^{(k)}G(t)\,\frac{y}{(x-t)^2+y^2}\,dt.\] The right-hand side is the Poisson integral of \((-1)^{k+1}D^{(k)}G\), so \(F^{(k)}\in H^p(\mathbb{C}^+)\) for all \(k=0,1,\dots,n\), and therefore \(F\in H_n^p(\mathbb{C}^+)\). Moreover, by the classical boundary isometry for Hardy spaces and 5 , \[\|F^{(k)}\|_{H^p} = \|F_l^{(k)}\|_{L^p} = \|D^{(k)}G\|_{L^p}, \qquad k=0,1,\dots,n.\] Thus the boundary-value map is an isometry.
For \(p=\infty\), one uses the normalized Cauchy integral \[F(z)=\frac{1}{2\pi i}\int_{-\infty}^{\infty} G(x)\left(\frac{1}{x-z}-\frac{1}{x+i}\right)\,dx, \qquad z\in\mathbb{C}^+,\] and argues similarly. The subtraction of \((x+i)^{-1}\) ensures absolute convergence, and the same boundary and derivative arguments show that \(F\in H_n^\infty(\mathbb{C}^+)\) with boundary value \(G\). ◻
As immediate consequences of Theorem 5, we record several basic structural properties of \(H_n^p(\mathbb{C}^+)\).
Corollary 6. Let \(1\le p\le\infty\) and \(0\le m\le n\). Then \[H_n^p(\mathbb{C}^+) \hookrightarrow H_m^p(\mathbb{C}^+).\]
Corollary 7. For every \(1\le p\le\infty\) and \(n\in\mathbb{N}\), the space \(H_n^p(\mathbb{C}^+)\) is a Banach space.
Proof. For \(1\le p<\infty\), by Theorem 5 it suffices to prove that \(H_n^p(\mathbb{R})\) is complete. Let \(\{F_j\}\subset H_n^p(\mathbb{R})\) be Cauchy. Since \(W_n^p(\mathbb{R})\) is complete, there exists \(G\in W_n^p(\mathbb{R})\) such that \[\lim_{j\to\infty}\|F_j-G\|_{W_n^p}=0.\] It remains to show that \(G\in H_n^p(\mathbb{R})\). Fix \(z\in\mathbb{C}^+\). By Hölder’s inequality, \[\left|\int_{-\infty}^{\infty}\frac{G(x)}{x-\overline{z}}\,dx\right| = \left|\int_{-\infty}^{\infty}\frac{G(x)-F_j(x)}{x-\overline{z}}\,dx\right| \le \|G-F_j\|_{L^p} \left\|\frac{1}{\,\cdot-\overline{z}\,}\right\|_{L^q(\mathbb{R})},\] with the usual interpretation when \(p=1\). Letting \(j\to\infty\) gives \[\int_{-\infty}^{\infty}\frac{G(x)}{x-\overline{z}}\,dx=0,\] so \(G\in H_n^p(\mathbb{R})\).
For \(p=\infty\), let \(\{F_j\}\) be Cauchy in \(H_n^\infty(\mathbb{C}^+)\). Then for each \(k=0,1,\dots,n\), the sequence \(\{F_j^{(k)}\}\) converges uniformly on \(\mathbb{C}^+\) to some \(F_k\in H^\infty(\mathbb{C}^+)\). Standard arguments show that \(F_k=F_0^{(k)}\) for all \(k\), so \(F_0\in H_n^\infty(\mathbb{C}^+)\). Hence \(H_n^\infty(\mathbb{C}^+)\) is complete. ◻
Corollary 8. Let \(F\in H_n^p(\mathbb{C}^+)\) with \(n\ge1\). Then, for each integer \(0\le k\le n-1\), the derivative \(F^{(k)}\) extends continuously to \(\overline{\mathbb{C}^+}\).
Proof. By 4 we have \[F_l^{(k)}(x)=F_l^{(k)}(a)+\int_a^x F_l^{(k+1)}(t)\,dt,\] so \(F_l^{(k)}\) is continuous for every \(0\le k\le n-1\). Therefore \(F^{(k)}\) is the Poisson integral of a continuous boundary function and extends continuously to \(\overline{\mathbb{C}^+}\). ◻
Analogously, one may define Hardy–Sobolev spaces on the lower half-plane, denoted by \(H_n^p(\mathbb{C}^-)\). The corresponding boundary value space is denoted by \(\prescript{-}{}{H_n^p}(\mathbb{R}):=\prescript{-}{}{H^p}(\mathbb{R})\cap W_n^p(\mathbb{R})\). All of the above statements have natural counterparts in this setting, and we omit the details.
The next estimate is a decisive consequence of the boundary model established in Theorem 5. In particular, it shows that once one passes to the Sobolev level \(n\ge1\), the space \(H_n^p(\mathbb{C}^+)\) acquires an \(H^\infty\)-type control that is absent in the classical Hardy case. This fact will be crucial in the proof of the generalized Banach algebra property below.
Proposition 9. Let \(1\le p\le\infty\) and \(n\ge1\). Then \[H_n^p(\mathbb{C}^+) \hookrightarrow H^\infty(\mathbb{C}^+).\] More precisely, there exists a constant \(0<C\le e^{1/e}\) such that \[\label{e3} \|F\|_{H^\infty}\le C\|F\|_{H_n^p}, \qquad F\in H_n^p(\mathbb{C}^+).\qquad{(1)}\]
Proof. By Theorem 5, it suffices to prove the corresponding estimate for boundary values in \(H_n^p(\mathbb{R})\subset W_n^p(\mathbb{R})\). Since \(n\ge1\), the classical one-dimensional Sobolev embedding theorem yields \[\|f\|_{L^\infty(\mathbb{R})}\le C\|f\|_{W_n^p(\mathbb{R})}\] for all \(f\in W_n^p(\mathbb{R})\), where \(0<C\le e^{1/e}\); see the footnote in [22]. Applying this to the boundary value \(f=F_l\in H_n^p(\mathbb{R})\) and using the isometric identification in Theorem 5, we obtain \[\|F\|_{H^\infty} = \|F_l\|_{L^\infty} \le C\|F_l\|_{W_n^p} = C\|F\|_{H_n^p}.\] This proves ?? . ◻
We next turn to a Cauchy integral representation of functions in \(H_n^p(\mathbb{C}^\pm)\). This representation is the natural analytic counterpart of the boundary characterization in Theorem 5, and it serves as the main tool in the proof of the direct-sum decomposition in the next subsection.
Proposition 10 (Integral representation). Let \(1\le p<\infty\) and \(n\in\mathbb{N}\).
A function \(F\) belongs to \(H_n^p(\mathbb{C}^\pm)\) if and only if there exists a unique \(f\in \prescript{\pm}{}{H_n^p}(\mathbb{R})\) such that \[F_\pm(z)=\frac{\pm1}{2\pi i}\int_{-\infty}^{\infty} \frac{f(x)}{x-z}\,dx, \qquad z\in\mathbb{C}^\pm,\] and then \(\lim_{y\to0^\pm}F_\pm(x+iy)=f(x)\) almost everywhere.
If \(f\in W_n^p(\mathbb{R})\) and \(p\neq1\), then the functions \(F_\pm\) defined above belong to \(H_n^p(\mathbb{C}^\pm)\) and satisfy \[\label{e9} \lim_{y\to0^\pm}F_\pm(x+iy) =\frac{1}{2} f(x)\pm\frac{i}{2}(Hf)(x) \qquad \text{a.e.}\qquad{(2)}\] For \(p=1\), the limit relation ?? still holds, but \(F_\pm\) need not belong to \(H_n^1(\mathbb{C}^\pm)\).
Proof. Part (a) is an immediate consequence of the converse direction of Theorem 5 applied on \(\mathbb{C}^\pm\).
For part (b), let \(f\in W_n^p(\mathbb{R})\) and define \[F_\pm(z)=\frac{\pm1}{2\pi i}\int_{-\infty}^{\infty} \frac{f(x)}{x-z}\,dx, \qquad z\in\mathbb{C}^\pm.\] These functions are holomorphic in the corresponding half-planes. For \(k=0\), this is exactly the Cauchy integral itself. For \(1\le k\le n\), differentiating under the integral sign and using integration by parts, or equivalently the weak derivative structure of \(f\), we obtain \[F_\pm^{(k)}(z) = \pm\frac{k!}{2\pi i}\int_{-\infty}^{\infty}\frac{f(x)}{(x-z)^{k+1}}\,dx = \pm\frac{(-1)^{k-1}}{2\pi i} \int_{-\infty}^{\infty}\frac{D^{(k)}f(x)}{x-z}\,dx.\] Since \(D^{(k)}f\in L^p(\mathbb{R})\) for \(k=0,1,\dots,n\), the classical Hardy-space theory implies that \(F_\pm^{(k)}\in H^p(\mathbb{C}^\pm)\) whenever \(p>1\). Therefore \(F_\pm\in H_n^p(\mathbb{C}^\pm)\) for \(p>1\).
The boundary relation ?? follows from the Plemelj theorem. When \(p=1\), the same boundary formula remains valid, but \(F_\pm\) need not belong to \(H_n^1(\mathbb{C}^\pm)\). Indeed, if this were true for every \(f\in W_n^1(\mathbb{R})\), then the Hilbert transform would map every \(W_n^1(\mathbb{R})\)-function into \(L^1(\mathbb{R})\), hence in particular every compactly supported smooth function into \(L^1(\mathbb{R})\), which is false. ◻
For the classical Hardy spaces, every function in \(L^p(\mathbb{R})\) with \(1<p<\infty\) admits a decomposition into the sum of an upper half-plane and a lower half-plane boundary Hardy function. The next result lifts this classical decomposition to the Sobolev level. In the Hilbert case it recovers the known orthogonal decomposition, while for general \(1<p<\infty\) it yields the corresponding Banach-space decomposition throughout the full reflexive range.
From Proposition 10(b) we obtain \[W_n^p(\mathbb{R})=\prescript{+}{}{H_n^p}(\mathbb{R})+\prescript{-}{}{H_n^p}(\mathbb{R}).\] Since \(\prescript{+}{}{H_n^p}(\mathbb{R})\) and \(\prescript{-}{}{H_n^p}(\mathbb{R})\) are subspaces of \(\prescript{+}{}{H^p}(\mathbb{R})\) and \(\prescript{-}{}{H^p}(\mathbb{R})\), respectively, and since \[\prescript{+}{}{H^p}(\mathbb{R})\cap\prescript{-}{}{H^p}(\mathbb{R})=\{0\}\] for \(1<p<\infty\), the above sum is direct.
Theorem 11. For \(1<p<\infty\) and \(n\in\mathbb{N}\), \[W_n^p(\mathbb{R})=\prescript{+}{}{H_n^p}(\mathbb{R})\oplus\prescript{-}{}{H_n^p}(\mathbb{R}).\] Moreover, when \(p=2\), this decomposition is orthogonal with respect to the \(W_n^2\) inner product.
Proof. Only the orthogonality for \(p=2\) requires comment. If \(F\in\prescript{+}{}{H_n^2}(\mathbb{R})\) and \(G\in\prescript{-}{}{H_n^2}(\mathbb{R})\), then for each \(k=0,1,\dots,n\) we have \(D^{(k)}F\in\prescript{+}{}{H^2}(\mathbb{R})\) and \(D^{(k)}G\in\prescript{-}{}{H^2}(\mathbb{R})\). Since \(\prescript{+}{}{H^2}(\mathbb{R})\) and \(\prescript{-}{}{H^2}(\mathbb{R})\) are mutually orthogonal in \(L^2(\mathbb{R})\), it follows that \[\langle F,G\rangle_{W_n^2} =\sum_{k=0}^n \langle D^{(k)}F,D^{(k)}G\rangle_{L^2}=0.\] ◻
Remark. The decomposition in Theorem 11 does not extend to the endpoint cases \(p=1\) and \(p=\infty\). For \(p=\infty\), nonzero constant functions belong to both \(\prescript{+}{}{H_n^\infty}(\mathbb{R})\) and \(\prescript{-}{}{H_n^\infty}(\mathbb{R})\), so the intersection is nontrivial. For \(p=1\), if such a decomposition were valid, then the Cauchy integral of every function in \(W_n^1(\mathbb{R})\) would belong to \(H_n^1(\mathbb{C}^+)\), contradicting Proposition 10(b). \(\qedsymbol\)
This decomposition has several useful consequences. First, it is closely related to adaptive Fourier decomposition (AFD); see [19]. Second, it is noteworthy from the Banach-space point of view, since not every closed subspace of a Banach space admits a complement. Finally, even when a subspace is known to be complemented, an explicit description of a complementary subspace is often highly nontrivial.
We now arrive at one of the main structural consequences of the previous results. Once \(n\ge1\), the Sobolev-type embedding shows that \(H_n^p(\mathbb{C}^+)\) is stable under pointwise multiplication and, in fact, carries a generalized Banach algebra structure. This is a genuinely new phenomenon at the Hardy–Sobolev level: it fails in the classical Hardy space case \(n=0\), and it will play a central role in the operator-theoretic developments of Section 5.
Recall that a Banach space \((X,\|\cdot\|)\) is called a Banach algebra if it is equipped with a multiplication making it an algebra and satisfying \[\|xy\|\le \|x\|\,\|y\|\] for all \(x,y\in X\). More generally, if there exists a constant \(C>0\) such that \[\|xy\|\le C\|x\|\,\|y\|\] for all \(x,y\in X\), then \((X,\|\cdot\|)\) is called a generalized Banach algebra.
Theorem 12. Let \(n\in\mathbb{N}^+\) and \(1\le p\le\infty\). Then \(H_n^p(\mathbb{C}^+)\), equipped with pointwise multiplication, is a generalized Banach algebra. More precisely, \[\label{e10} \|FG\|_{H_n^p}\le \begin{cases} \displaystyle C\left(\frac{2^{p(n+1)}-1}{2^p-1}\right)^{1/p} \|F\|_{H_n^p}\|G\|_{H_n^p}, & 1\le p<\infty,\\[12pt] \displaystyle (2^{n+1}-1)\|F\|_{H_n^\infty}\|G\|_{H_n^\infty}, & p=\infty, \end{cases}\qquad{(3)}\] where \(C\) is the constant in the Sobolev embedding inequality ?? .
Proof. Assume first that \(1\le p<\infty\). For \(F,G\in H_n^p(\mathbb{C}^+)\) and \(0\le k\le n\), the Leibniz rule gives \[(FG)^{(k)}=\sum_{j=0}^k \binom{k}{j}F^{(k-j)}G^{(j)}.\] Hence \[\|(FG)^{(k)}\|_{H^p} \le \sum_{j=0}^k \binom{k}{j}\|F^{(k-j)}G^{(j)}\|_{H^p}.\] Using the embedding inequality ?? , we obtain \[\|F^{(k-j)}G^{(j)}\|_{H^p} \le \|F^{(k-j)}\|_{H^\infty}\|G^{(j)}\|_{H^p} \le C\|F\|_{H_n^p}\|G\|_{H_n^p},\] and similarly when the roles of \(F\) and \(G\) are interchanged. Therefore, \[\|(FG)^{(k)}\|_{H^p} \le C\,2^k\,\|F\|_{H_n^p}\|G\|_{H_n^p}.\] Raising to the \(p\)th power and summing over \(k=0,\dots,n\) yields ?? for \(1\le p<\infty\).
For \(p=\infty\), Leibniz’ rule gives \[\|(FG)^{(k)}\|_{H^\infty} \le \sum_{j=0}^k \binom{k}{j} \|F^{(k-j)}\|_{H^\infty}\|G^{(j)}\|_{H^\infty} \le 2^k \|F\|_{H_n^\infty}\|G\|_{H_n^\infty}.\] Summing over \(k\) gives \[\|FG\|_{H_n^\infty} \le (2^{n+1}-1)\|F\|_{H_n^\infty}\|G\|_{H_n^\infty},\] which proves ?? . ◻
Remark. Theorem 12 fails completely in the classical case \(n=0\) when \(1\le p<\infty\). Indeed, if \(H^p(\mathbb{C}^+)\) were a generalized Banach algebra under pointwise multiplication, then \(F^2\in H^p(\mathbb{C}^+)\) would hold for every \(F\in H^p(\mathbb{C}^+)\), which would imply \(H^p(\mathbb{C}^+)\subseteq H^{2p}(\mathbb{C}^+)\). This is false. For example, \[F(z)=\left(\frac{1}{\sqrt{z}\,(z+i)}\right)^{1/p}, \qquad z\in\mathbb{C}^+,\] belongs to \(H^p(\mathbb{C}^+)\) but not to \(H^{2p}(\mathbb{C}^+)\). \(\qedsymbol\)
The boundary-based theory developed in Section 3 applies throughout the full Banach range \(1\le p\le\infty\). In the special case \(p=2\), however, \(H_n^2(\mathbb{C}^+)\) carries additional Hilbert-space structure, and this allows a finer Fourier-analytic description than is available in the general Banach setting. The purpose of this section is to complement the boundary model of Section 3 by a frequency-side Hilbert model. The two main results are a Paley–Wiener theorem identifying \(H_n^2(\mathbb{C}^+)\) with a weighted Fourier-side space on \(\mathbb{R}^+\), and an explicit formula for the reproducing kernel of \(H_n^2(\mathbb{C}^+)\).
We begin with the first main refinement specific to the Hilbert case. While Section 3 identifies \(H_n^2(\mathbb{C}^+)\) through boundary values, in the Hilbert setting one can go further and realize this space isometrically on the Fourier side. The natural model is the weighted space \(L_n^2(\mathbb{R}^+)\), and the next theorem shows that the holomorphic Fourier transform provides the precise Paley–Wiener correspondence between these two spaces.
Theorem 13 (Paley–Wiener). For every \(n\in\mathbb{N}\), the holomorphic Fourier transform \[\mathscr{F}(f)(z)=\int_0^\infty f(x)e^{izx}\,dx, \qquad z\in\mathbb{C}^+,\] defines an isometric isomorphism from \(L_n^2(\mathbb{R}^+)\) onto \(H_n^2(\mathbb{C}^+)\).
Proof. Let \(F\in H_n^2(\mathbb{C}^+)\). Since \(F\in H^2(\mathbb{C}^+)\), the classical Paley–Wiener theorem yields a unique function \(f\in L^2(\mathbb{R}^+,2\pi\,dx)\) such that \[F(z)=\int_0^\infty f(x)e^{izx}\,dx, \qquad z\in\mathbb{C}^+.\] Differentiating under the integral sign, we obtain \[F^{(k)}(z)=\int_0^\infty (ix)^k f(x)e^{izx}\,dx, \qquad k=0,1,\dots,n.\] Since \(F^{(k)}\in H^2(\mathbb{C}^+)\) for each \(k\), the classical Paley–Wiener theorem again implies that \(x^k f\in L^2(\mathbb{R}^+,2\pi\,dx)\) for \(k=0,1,\dots,n\). Hence \(f\in L_n^2(\mathbb{R}^+)\).
Conversely, let \(f\in L_n^2(\mathbb{R}^+)\) and define \[F(z)=\mathscr{F}(f)(z)=\int_0^\infty f(x)e^{izx}\,dx, \qquad z\in\mathbb{C}^+.\] Then \[F^{(k)}(z)=\int_0^\infty (ix)^k f(x)e^{izx}\,dx, \qquad k=0,1,\dots,n.\] Since \(x^k f\in L^2(\mathbb{R}^+,2\pi\,dx)\), the classical Paley–Wiener theorem shows that \(F^{(k)}\in H^2(\mathbb{C}^+)\) for each \(k\), so \(F\in H_n^2(\mathbb{C}^+)\).
Finally, for each \(k=0,1,\dots,n\), the classical Paley–Wiener identity gives \[\|F^{(k)}\|_{H^2}^2 =\int_0^\infty |f(x)|^2\,2\pi x^{2k}\,dx.\] Summing over \(k\) yields \[\|\mathscr{F}(f)\|_{H_n^2}^2=\|f\|_{L_n^2}^2,\] which proves that \(\mathscr{F}\) is an isometric isomorphism. ◻
The Paley–Wiener theorem gives a convenient frequency-side model for \(H_n^2(\mathbb{C}^+)\). In particular, combined with Proposition 3, it yields the following simple characterization.
Corollary 14. For every \(n\in\mathbb{N}\), \[H_n^2(\mathbb{C}^+) = \bigl\{\,F\in H(\mathbb{C}^+): F\in H^2(\mathbb{C}^+)\;\text{and}\;F^{(n)}\in H^2(\mathbb{C}^+)\,\bigr\}.\]
Proof. By Theorem 13, a function \(F\) belongs to \(H_n^2(\mathbb{C}^+)\) if and only if \(F=\mathscr{F}(f)\) for some \(f\in L_n^2(\mathbb{R}^+)\). By Proposition 3, this is equivalent to requiring that both \(f\in L^2(\mathbb{R}^+,2\pi\,dx)\) and \(x^n f\in L^2(\mathbb{R}^+,2\pi\,dx)\). By the classical Paley–Wiener theorem, these conditions are equivalent to \(F\in H^2(\mathbb{C}^+)\) and \(F^{(n)}\in H^2(\mathbb{C}^+)\), respectively. ◻
Remark. Corollary 14 shows that, in the Hilbert case, the full Hardy–Sobolev norm can be recovered from the two extreme conditions \(F\in H^2(\mathbb{C}^+)\) and \(F^{(n)}\in H^2(\mathbb{C}^+)\). This is a particularly simple consequence of the Fourier-side model provided by Theorem 13. \(\qedsymbol\)
The Paley–Wiener theorem above provides an explicit Hilbert-space model for \(H_n^2(\mathbb{C}^+)\). Our second main result in this section is that this model also leads to an explicit reproducing kernel formula. This gives a concrete realization of the Hilbertian structure of \(H_n^2(\mathbb{C}^+)\) that has no analogue in the general Banach setting.
Recall that a Hilbert space of functions on a set \(E\) is called a reproducing kernel Hilbert space if point evaluations are bounded. In that case, for each \(z\in E\), there exists a unique kernel function \(K_z\) such that \[F(z)=\langle F,K_z\rangle\] for all \(F\) in the space. Since \(H_n^2(\mathbb{C}^+)\hookrightarrow H^\infty(\mathbb{C}^+)\) by the Sobolev-type embedding established in Section 3, point evaluations are bounded on \(H_n^2(\mathbb{C}^+)\), and hence \(H_n^2(\mathbb{C}^+)\) is a reproducing kernel Hilbert space.
Theorem 15. The reproducing kernel of \(H_n^2(\mathbb{C}^+)\) is given by \[K_n(z,w)=\frac{1}{2\pi}\int_0^\infty \frac{1-x^2}{1-x^{2n+2}}\,e^{ix(w-\overline{z})}\,dx, \qquad z,w\in\mathbb{C}^+.\]
Proof. Fix \(z\in\mathbb{C}^+\) and define \[K_{n,z}(w) = \frac{1}{2\pi}\int_0^\infty \frac{1-x^2}{1-x^{2n+2}}\,e^{ix(w-\overline{z})}\,dx.\] We may write \[K_{n,z}(w)=\mathscr{F}(g)(w),\] where \[g(x)=\frac{1}{2\pi}\,\frac{(1-x^2)e^{-ix\overline{z}}}{1-x^{2n+2}}.\]
The function \[x\mapsto \frac{1-x^2}{1-x^{2n+2}} = \frac{1}{1+x^2+\cdots+x^{2n}}\] extends continuously to \((0,\infty)\) and is bounded there. Since \(e^{-ix\overline{z}}\) decays exponentially like \(e^{-x \, Im(z)}\), there exists a constant \(M>0\) such that \[|g(x)|\le M e^{-x\, Im(z)}, \qquad |x^n g(x)|\le M e^{-x\,Im(z)}, \qquad x>0.\] Hence \(g\in L_n^2(\mathbb{R}^+)\) by Proposition 3, and therefore \(K_{n,z}\in H_n^2(\mathbb{C}^+)\) by Theorem 13.
Now let \(F\in H_n^2(\mathbb{C}^+)\), and let \(f\in L_n^2(\mathbb{R}^+)\) be the unique function such that \[F(w)=\int_0^\infty f(x)e^{iwx}\,dx, \qquad w\in\mathbb{C}^+.\] Then \[\begin{align} \langle F,K_{n,z}\rangle_{H_n^2} &=\langle \mathscr F(f),\mathscr F(g)\rangle_{H_n^2} =\langle f,g\rangle_{L_n^2} \\ &=\sum_{k=0}^n \int_0^\infty f(x)\overline{g(x)}\,2\pi x^{2k}\,dx \\ &=\int_0^\infty f(x)\, \frac{1-x^2}{1-x^{2n+2}}\,e^{ixz} \left(\sum_{k=0}^n x^{2k}\right)\,dx. \end{align}\] Since \[\sum_{k=0}^n x^{2k}=\frac{1-x^{2n+2}}{1-x^2},\] it follows that \[\langle F,K_{n,z}\rangle_{H_n^2} =\int_0^\infty f(x)e^{ixz}\,dx =F(z).\] Thus \(K_n\) is the reproducing kernel of \(H_n^2(\mathbb{C}^+)\). ◻
As useful consequences of the explicit kernel formula, we record a uniform bound for the kernel norms and a corresponding product estimate.
Corollary 16. Let \(n\in\mathbb{N}^+\) and \(z\in\mathbb{C}^+\). Then:
\[\sup_{z\in\mathbb{C}^+}\|K_{n,z}\|_{H_n^2}^2\le \frac{1}{4};\]
if \(F\in H_n^2(\mathbb{C}^+)\) and \(G\in H^2(\mathbb{C}^+)\), then \[\|FG\|_{H^2} \le \frac{1}{2}\,\|F\|_{H_n^2}\,\|G\|_{H^2} \le \frac{1}{2}\,\|F\|_{H_n^2}\,\|G\|_{H_n^2}.\]
Proof. For part (a), by the reproducing property we have \[\|K_{n,z}\|_{H_n^2}^2=K_n(z,z) = \frac{1}{2\pi}\int_0^\infty \frac{1-x^2}{1-x^{2n+2}}\,e^{-2x \, Im(z)}\,dx.\] Hence \[\|K_{n,z}\|_{H_n^2}^2 \le \frac{1}{2\pi}\int_0^\infty \frac{1-x^2}{1-x^{2n+2}}\,dx.\] Since \[\frac{1-x^2}{1-x^{2n+2}} \le \frac{1}{1+x^2} \qquad (x\ge0),\] it follows that \[\|K_{n,z}\|_{H_n^2}^2 \le \frac{1}{2\pi}\int_0^\infty \frac{dx}{1+x^2} =\frac{1}{4}.\]
For part (b), we estimate \[\begin{align} \|FG\|_{H^2}^2 &= \sup_{y>0}\int_{-\infty}^\infty |F(x+iy)G(x+iy)|^2\,dx \\ &\le \sup_{z\in\mathbb{C}^+}|F(z)|^2\,\|G\|_{H^2}^2 \\ &= \sup_{z\in\mathbb{C}^+}|\langle F,K_{n,z}\rangle_{H_n^2}|^2\,\|G\|_{H^2}^2 \\ &\le \sup_{z\in\mathbb{C}^+}\|K_{n,z}\|_{H_n^2}^2\,\|F\|_{H_n^2}^2\,\|G\|_{H^2}^2 \\ &\le \frac{1}{4}\,\|F\|_{H_n^2}^2\,\|G\|_{H^2}^2. \end{align}\] Taking square roots gives the desired inequality. ◻
Remark. The second inequality in Corollary 16(b) is immediate from the inclusion \(H_n^2(\mathbb{C}^+)\subset H^2(\mathbb{C}^+)\). More generally, if \(G^{(n)}\in H^2(\mathbb{C}^+)\), the same argument gives \[\|F\,G^{(n)}\|_{H^2} \le \frac{1}{2}\,\|F\|_{H_n^2}\,\|G^{(n)}\|_{H^2} \le \frac{1}{2}\,\|F\|_{H_n^2}\,\|G\|_{H_n^2},\] which will be useful in the estimate below. \(\qedsymbol\)
The kernel estimate also yields a sharper multiplicative bound in the Hilbert case.
Corollary 17. For every \(n\in\mathbb{N}^+\) and all \(F,G\in H_n^2(\mathbb{C}^+)\), \[\|FG\|_{H_n^2}^2 \le \frac{1}{3}\,(4^n-1)\,\|F\|_{H_n^2}^2\,\|G\|_{H_n^2}^2.\]
Proof. The case \(n=1\) is exactly the result of Kucik [15]. Assume the statement holds for some \(n=k\ge1\). Then for \(n=k+1\), the Leibniz rule gives \[\begin{align} \|FG\|_{H_{k+1}^2}^2 &= \|FG\|_{H_k^2}^2+\|(FG)^{(k+1)}\|_{H^2}^2 \\ &\le \frac{1}{3}(4^k-1)\|F\|_{H_k^2}^2\|G\|_{H_k^2}^2 + \left[ \sum_{m=0}^{k+1}\binom{k+1}{m} \|F^{(m)}G^{(k+1-m)}\|_{H^2} \right]^2. \end{align}\] For each \(0\le m\le k+1\), we apply Corollary 16(b) to \(F^{(m)}\in H_{k+1-m}^2(\mathbb{C}^+)\) and \(G^{(k+1-m)}\in H^2(\mathbb{C}^+)\). Since \[\|F^{(m)}\|_{H_{k+1-m}^2}\le \|F\|_{H_{k+1}^2}, \qquad \|G^{(k+1-m)}\|_{H^2}\le \|G\|_{H_{k+1}^2},\] it follows that \[\|F^{(m)}G^{(k+1-m)}\|_{H^2} \le \frac{1}{2}\,\|F\|_{H_{k+1}^2}\,\|G\|_{H_{k+1}^2}\] for all \(0\le m\le k+1\). Hence \[\begin{align} \|FG\|_{H_{k+1}^2}^2 &\le \frac{1}{3}(4^k-1)\|F\|_{H_{k+1}^2}^2\|G\|_{H_{k+1}^2}^2 + \left[ \frac{1}{2}\sum_{m=0}^{k+1}\binom{k+1}{m} \right]^2 \|F\|_{H_{k+1}^2}^2\|G\|_{H_{k+1}^2}^2 \\ &= \frac{1}{3}(4^k-1)\|F\|_{H_{k+1}^2}^2\|G\|_{H_{k+1}^2}^2 + 4^k\|F\|_{H_{k+1}^2}^2\|G\|_{H_{k+1}^2}^2 \\ &= \frac{1}{3}(4^{k+1}-1)\|F\|_{H_{k+1}^2}^2\|G\|_{H_{k+1}^2}^2. \end{align}\] This completes the induction. ◻
For comparison, we briefly relate \(H_n^2(\mathbb{C}^+)\) to the Hilbert-type space \(\mathscr{H}_n^2(\mathbb{C}^+)\) mentioned in the Introduction. Although both spaces admit Fourier-side descriptions, the corresponding models are substantially different, and the following examples show that neither space contains the other.
We now compare \(H_n^2(\mathbb{C}^+)\) with the Hilbert-type space \(\mathscr{H}_n^2(\mathbb{C}^+)\) introduced in the Introduction. The Paley–Wiener description above shows that \(H_n^2(\mathbb{C}^+)\) is modeled by the weighted space \(L_n^2(\mathbb{R}^+)\), whereas the corresponding Fourier-side model for \(\mathscr{H}_n^2(\mathbb{C}^+)\) is the space \(\mathscr{T}_n^2\) from [25].
Theorem 1 ([25]). Let \(n\in\mathbb{N}^+\). The holomorphic Fourier transform is an isometric isomorphism from \(\mathscr{T}_n^2\) onto \(\mathscr{H}_n^2(\mathbb{C}^+)\), where \(\mathscr{T}_n^2\) consists of all functions \(f:\mathbb{R}^+\to\mathbb{C}\) such that \(f^{(k)}\) exists for \(k=0,1,\dots,n-1\), the derivative \(f^{(n-1)}\) is absolutely continuous, and each function \(x\mapsto x^k f^{(k)}(x)\) belongs to \(L^2(\mathbb{R}^+)\) for \(k=0,1,\dots,n\).
The first example is the function \[f(x)= \begin{cases} W(x), & 0<x<1,\\ 0, & x\ge1, \end{cases}\] where \(W\) denotes the everywhere continuous but nowhere differentiable Weierstrass function [26]. Since \(x^k f(x)\) is bounded on \((0,1)\) and vanishes on \([1,\infty)\) for every \(k=0,1,\dots,n\), it follows that \(f\in L_n^2(\mathbb{R}^+)\). However, \(f\) is nowhere differentiable on \((0,1)\), so \(f\notin\mathscr{T}_n^2\).
Conversely, consider \[g(x)= \begin{cases} \displaystyle \sum_{m=0}^n (1-x)^m, & 0<x<1,\\[4pt] \displaystyle \frac{1}{x}, & x\ge1. \end{cases}\] Since \(g(x)=x^{-1}\) for \(x\ge1\), we have \(g\notin L^1(\mathbb{R}^+)\), and hence \(g\notin L_n^2(\mathbb{R}^+)\) by Proposition 4. On the other hand, \(g^{(n-1)}\) is absolutely continuous on \((0,\infty)\), and for each \(k=0,1,\dots,n\) the function \(x^k g^{(k)}(x)\) is square-integrable on \(\mathbb{R}^+\). Indeed, on \((0,1)\) this is immediate since \(g\) is a polynomial, while on \([1,\infty)\) one has \(g^{(k)}(x)=(-1)^k k! \, x^{-k-1}\), so \[x^k g^{(k)}(x)=(-1)^k k!\,x^{-1}\in L^2(1,\infty).\] Thus \(g\in\mathscr{T}_n^2\).
Hence the spaces \(\mathscr{H}_n^2(\mathbb{C}^+)\) and \(H_n^2(\mathbb{C}^+)\) intersect, but neither contains the other.
Finally, we note one further distinction between these two spaces. While \(H_n^2(\mathbb{C}^+)\) is a generalized Banach algebra by Theorem 12, \(\mathscr{H}_n^2(\mathbb{C}^+)\) is not.
Proposition 18. The space \(\mathscr{H}_n^2(\mathbb{C}^+)\) cannot be made into a generalized Banach algebra under pointwise multiplication by any equivalent norm.
Proof. Assume, for contradiction, that \(\mathscr{H}_n^2(\mathbb{C}^+)\) were a generalized Banach algebra. By [25], it is a reproducing kernel Hilbert space whose reproducing kernel \(K_n\) satisfies the two-sided estimate \[\frac{1}{(n-1)!\sqrt{2n-1}}\frac{1}{\sqrt{|z|}} \le \|K_{n,z}\|_{\mathscr{H}_n^2} \le \frac{\sqrt{\pi}}{(n-1)!\sqrt{n}}\frac{1}{\sqrt{|z|}}, \qquad z\in\mathbb{C}^+.\] On the other hand, Kucik proved in [15] that if a Hilbert function space over a domain is a generalized Banach algebra under pointwise multiplication, then its reproducing kernels must satisfy \[\sup_{z\in\Omega}\|K_z\|_{\mathscr H}<\infty.\] But the lower bound above shows that \[\|K_{n,z}\|_{\mathscr{H}_n^2}\to\infty \qquad\text{as } |z|\to0^+,\] which is a contradiction. Therefore \(\mathscr{H}_n^2(\mathbb{C}^+)\) is not a generalized Banach algebra. ◻
The Hilbertian description developed in this section complements the boundary-based function theory established in Section 3. We now turn to operator-theoretic applications on \(H_n^p(\mathbb{C}^+)\).
In this section we turn to operator-theoretic consequences of the function theory developed in Section 3. In particular, the point-evaluation structure and the generalized Banach algebra property obtained there provide the basic tools for studying multiplication operators and weighted composition operators on \(H_n^p(\mathbb{C}^+)\). Our main result in the first part of the section is a precise spectral description of multiplication operators induced by multipliers. In the second part, we establish two useful sufficient conditions for the boundedness of weighted composition operators.
Let \[\mathscr{M}_{n,p} = \left\{ \psi\in H(\mathbb{C}^+) : \psi F\in H_n^p(\mathbb{C}^+)\;\text{for all } F\in H_n^p(\mathbb{C}^+) \right\}\] denote the multiplier space of \(H_n^p(\mathbb{C}^+)\). For each \(\psi\in\mathscr{M}_{n,p}\), the associated multiplication operator is defined by \[T_\psi F=\psi F, \qquad F\in H_n^p(\mathbb{C}^+).\]
We begin with the basic boundedness of multiplication operators and several elementary structural properties of the multiplier space.
Proposition 19. Let \(\psi\in\mathscr{M}_{n,p}\). Then the multiplication operator \(T_\psi:H_n^p(\mathbb{C}^+)\to H_n^p(\mathbb{C}^+)\) is bounded.
Proof. Since \(H_n^p(\mathbb{C}^+)\) is a Banach space, it suffices by the closed graph theorem to show that the graph of \(T_\psi\) is closed. Let \(\{F_m\}\) be a sequence in \(H_n^p(\mathbb{C}^+)\) such that \[F_m\to0 \quad\text{and}\quad T_\psi(F_m)\to G\] in \(H_n^p(\mathbb{C}^+)\) for some \(G\in H_n^p(\mathbb{C}^+)\), and prove that \(G=0\).
For \(n\ge1\), boundedness of point evaluations follows from the Sobolev-type embedding established in Section 3; for \(n=0\), it is part of the classical Hardy-space theory. Thus, for each fixed \(z\in\mathbb{C}^+\), there exists a constant \(C_z>0\) such that \[|F(z)|\le C_z\|F\|_{H_n^p} \qquad\text{for all } F\in H_n^p(\mathbb{C}^+).\] Hence \[\begin{align} |G(z)| &\le |G(z)-\psi(z)F_m(z)|+|\psi(z)F_m(z)| \\ &\le C_z\|G-T_\psi(F_m)\|_{H_n^p} +|\psi(z)|\,C_z\|F_m\|_{H_n^p}\to0 \end{align}\] as \(m\to\infty\). Since \(z\in\mathbb{C}^+\) was arbitrary, we conclude that \(G\equiv0\). Therefore the graph of \(T_\psi\) is closed, and hence \(T_\psi\) is bounded. ◻
The space \(\mathscr{M}_{n,p}\) is naturally equipped with the operator norm \[\|\psi\|_{\mathscr M}:=\|T_\psi\|_{\mathscr B(H_n^p)}.\] Thus the boundedness problem for multiplication operators is equivalent to describing the multiplier space itself.
Recall that for each \(z\in\mathbb{C}^+\), the evaluation functional \[M_z:H_n^p(\mathbb{C}^+)\to\mathbb{C},\qquad F\mapsto F(z),\] is bounded. The next proposition shows that these point evaluations play the role of eigenvectors for the adjoint multiplication operator.
Proposition 20. If \(\psi\in\mathscr{M}_{n,p}\), then \[T_\psi^*(M_z)=\psi(z)M_z \qquad\text{for every } z\in\mathbb{C}^+.\]
Proof. For any \(F\in H_n^p(\mathbb{C}^+)\), \[(T_\psi^*M_z)(F)=M_z(T_\psi F)=M_z(\psi F)=\psi(z)F(z)=\psi(z)M_z(F),\] which proves the claim. ◻
Remark. In a reproducing kernel Hilbert space, bounded multiplication operators are often detected through the fact that kernel functions are eigenvectors of the adjoint. Proposition 20 is the corresponding point-evaluation version of this observation in the present setting. \(\qedsymbol\)
We next record several basic inclusion relations for the multiplier space.
Proposition 21. Let \(1\le p\le\infty\) and \(n\in\mathbb{N}\).
\(H_n^\infty(\mathbb{C}^+)\hookrightarrow \mathscr{M}_{n,p}\);
\(H_n^p(\mathbb{C}^+)\hookrightarrow \mathscr{M}_{n,p}\) whenever \(n\ge1\);
\(\mathscr{M}_{n,p}\hookrightarrow H^\infty(\mathbb{C}^+)\).
Proof. For (a), if \(n=0\), then for every \(F\in H^p(\mathbb{C}^+)\), \[\|\psi F\|_{H^p}\le \|\psi\|_{H^\infty}\|F\|_{H^p},\] so \(\psi\in\mathscr M_{0,p}\). If \(n\ge1\), the same conclusion follows from Leibniz’ rule together with the estimate used in the proof of Theorem 12, replacing one factor by \(\psi\) and controlling all derivatives of \(\psi\) by the \(H_n^\infty\) norm.
Part (b) is an immediate consequence of Theorem 12, since \(H_n^p(\mathbb{C}^+)\) is a generalized Banach algebra for \(n\ge1\).
For (c), Proposition 20 implies that each value \(\psi(z)\) belongs to the point spectrum of \(T_\psi^*\), hence to the spectrum of \(T_\psi\). Thus \[|\psi(z)|\le r(T_\psi)\le \|T_\psi\|=\|\psi\|_{\mathscr M}, \qquad z\in\mathbb{C}^+,\] and therefore \(\psi\in H^\infty(\mathbb{C}^+)\). ◻
Remark. In general, none of the inclusions in Proposition 21 is reversible. For instance, when \(p=2\) and \(n=1\):
\(H_1^2(\mathbb{C}^+)\not\subset H_1^\infty(\mathbb{C}^+)\);
\(\mathscr M_{1,2}\not\subset H_1^2(\mathbb{C}^+)\), since nonzero constants belong to \(\mathscr M_{1,2}\) but not to \(H_1^2(\mathbb{C}^+)\);
\(H^\infty(\mathbb{C}^+)\not\subset \mathscr M_{1,2}\).
A necessary and sufficient condition for membership in \(\mathscr M_{n,2}\) is given in [15]. \(\qedsymbol\)
To obtain the spectral description of multiplication operators, we need the following invertibility lemma.
Lemma 22. If \(\psi\in\mathscr{M}_{n,p}\) and \[\inf_{z\in\mathbb{C}^+}|\psi(z)|>0,\] then \(1/\psi\in\mathscr{M}_{n,p}\).
Proof. Clearly \(\psi\in H^\infty(\mathbb{C}^+)\) by Proposition 21(c), and \(1/\psi\in H^\infty(\mathbb{C}^+)\) because \(\inf_{z\in\mathbb{C}^+}|\psi(z)|>0\).
Let \(F\in H_n^p(\mathbb{C}^+)\). By the higher-order quotient rule [7], \[D^k\!\left(\frac{F}{\psi}\right) = \frac{(-1)^k}{\psi^{k+1}} \sum_{j=0}^k (-1)^j\binom{k+1}{j}\, \psi^j D^k\!\left(\psi^{\,k-j}F\right), \qquad k=0,1,\dots,n.\] Since \(\psi\in\mathscr M_{n,p}\), the multiplication operator \(T_\psi\) is bounded on \(H_n^p(\mathbb{C}^+)\). Hence, by iteration, multiplication by \(\psi^m\) is bounded on \(H_n^p(\mathbb{C}^+)\) for every integer \(m\ge1\). It follows that \(\psi^{k-j}F\in H_n^p(\mathbb{C}^+)\), and therefore \[D^k(\psi^{k-j}F)\in H^p(\mathbb{C}^+).\] Since both \(\psi\) and \(1/\psi\) are bounded, each term on the right-hand side belongs to \(H^p(\mathbb{C}^+)\), and hence \(D^k(F/\psi)\in H^p(\mathbb{C}^+)\) for all \(k=0,1,\dots,n\). Therefore \((1/\psi)F\in H_n^p(\mathbb{C}^+)\), proving that \(1/\psi\in\mathscr M_{n,p}\). ◻
The next theorem is the main result of this subsection. It gives a complete spectral description of multiplication operators on \(H_n^p(\mathbb{C}^+)\) and shows that the spectrum is determined exactly by the range of the multiplier symbol.
Theorem 23. Let \(\psi\in\mathscr{M}_{n,p}\). Then \[\sigma(T_\psi)=\overline{\psi(\mathbb{C}^+)}.\]
Proof. By Proposition 20, every value \(\psi(z)\) is an eigenvalue of \(T_\psi^*\), hence belongs to \(\sigma(T_\psi)\). Therefore \[\overline{\psi(\mathbb{C}^+)}\subseteq \sigma(T_\psi).\]
Conversely, let \(\lambda\notin \overline{\psi(\mathbb{C}^+)}\). Then there exists \(\delta>0\) such that \[|\lambda-\psi(z)|\ge \delta \qquad\text{for all } z\in\mathbb{C}^+.\] Applying Lemma 22 to \(\lambda-\psi\), we obtain \[\frac{1}{\lambda-\psi}\in\mathscr{M}_{n,p}.\] It is then immediate that \[(\lambda I-T_\psi)^{-1}=T_{1/(\lambda-\psi)}.\] Hence \(\lambda\notin \sigma(T_\psi)\), so \[\sigma(T_\psi)\subseteq \overline{\psi(\mathbb{C}^+)}.\] This proves the theorem. ◻
An immediate and useful consequence of Theorem 23 is that compact multiplication operators are completely rigid.
Corollary 24. Let \(\psi\in\mathscr{M}_{n,p}\). Then \(T_\psi\) is compact if and only if \(\psi\equiv0\).
Proof. If \(T_\psi\) is compact, then its spectrum is countable and has no accumulation point except possibly \(0\). By Theorem 23, \[\sigma(T_\psi)=\overline{\psi(\mathbb{C}^+)},\] so \(\overline{\psi(\mathbb{C}^+)}\) must be countable. Since a nonconstant holomorphic function maps open sets onto open sets, the open mapping theorem forces \(\psi\) to be constant. Since \(H_n^p(\mathbb{C}^+)\) is infinite-dimensional and \(0\in\sigma(T_\psi)\) for every compact operator on an infinite-dimensional Banach space, that constant must be \(0\). ◻
We now turn to weighted composition operators \[T_{\psi,\varphi}F=\psi\cdot(F\circ\varphi),\] where \(\psi\in H(\mathbb{C}^+)\) and \(\varphi\in\mathscr S(\mathbb{C}^+)\) is an analytic self-map of \(\mathbb{C}^+\). In contrast to multiplication operators, a complete description of bounded weighted composition operators on \(H_n^p(\mathbb{C}^+)\) seems to be substantially more delicate. We therefore restrict ourselves to two practical sufficient conditions for boundedness, both of which are readily verifiable in concrete situations.
Our first criterion is based on a lower bound involving the derivative of the symbol \(\varphi\).
Theorem 25. Let \(n\in\mathbb{N}^+\). Assume that \[\psi\in H_n^\infty(\mathbb{C}^+) \quad \text{or} \quad \psi\in H_n^p(\mathbb{C}^+),\] and that \(\varphi\in\mathscr S(\mathbb{C}^+)\) satisfies \(\varphi'\in H_{n-1}^\infty(\mathbb{C}^+)\) and the function \[A_\varphi(z,w)=\bigl|Re(\varphi'(z))+i\,Im(\varphi'(w))\bigr|, \qquad z,w\in\mathbb{C}^+,\] has a positive lower bound. Then \(T_{\psi,\varphi}\) is bounded on \(H_n^p(\mathbb{C}^+)\).
Proof. Using the Faà di Bruno formula for holomorphic composition, we may write \[(F\circ\varphi)^{(k)} = \sum_{a_1+2a_2+\cdots+ka_k=k} c_{(a_1,\dots,a_k)} (F^{(m_k)}\circ\varphi) (\varphi')^{a_1}\cdots(\varphi^{(k)})^{a_k},\] where \(m_k=a_1+\cdots+a_k\). Applying Leibniz’ rule to \(\psi(F\circ\varphi)\), we obtain \[\begin{align} \|(\psi(F\circ\varphi))^{(k)}\|_{H^p} \le \sum_{j=0}^k \sum_{a_1,\dots,a_j} \binom{k}{j}c_{(a_1,\dots,a_j)} \|F^{(m_j)}\circ\varphi\|_{H^p} \|\varphi'\|_{H^\infty}^{a_1}\cdots \|\varphi^{(j)}\|_{H^\infty}^{a_j} \|\psi^{(k-j)}\|_{H^\infty}. \end{align}\] Since \(\varphi'\in H_{n-1}^\infty(\mathbb{C}^+)\), all derivatives \(\varphi^{(j)}\) with \(1\le j\le n\) belong to \(H^\infty(\mathbb{C}^+)\). Moreover, the hypothesis on \(A_\varphi\) implies the boundedness of the composition operator \(C_\varphi:F\mapsto F\circ\varphi\) on \(H^p(\mathbb{C}^+)\) for \(1\le p<\infty\); see [27]. For \(p=\infty\) this is immediate. Finally, \(\psi^{(k-j)}\in H^\infty(\mathbb{C}^+)\) follows either from \(\psi\in H_n^\infty(\mathbb{C}^+)\) or from the Sobolev embedding when \(\psi\in H_n^p(\mathbb{C}^+)\).
Therefore each derivative of \(\psi(F\circ\varphi)\) up to order \(n\) belongs to \(H^p(\mathbb{C}^+)\), so \(T_{\psi,\varphi}\) maps \(H_n^p(\mathbb{C}^+)\) into itself. Since \(H_n^p(\mathbb{C}^+)\) is Banach, the closed graph theorem implies that \(T_{\psi,\varphi}\) is bounded. ◻
We also use the standard Julia–Carathéodory notion of angular derivative at infinity [28]. For \(\varphi\in\mathscr S(\mathbb{C}^+)\), we say that \(\varphi\) has a finite angular derivative at infinity if \[\sup_{z\in\mathbb{C}^+}\frac{Im(z)}{Im(\varphi(z))}<\infty.\] For \(0<p<\infty\), this is equivalent to the boundedness of the composition operator \(C_\varphi:F\mapsto F\circ\varphi\) on \(H^p(\mathbb{C}^+)\).
This yields a second useful sufficient condition.
Theorem 26. Let \(n\in\mathbb{N}^+\). Assume that \[\psi\in H_n^\infty(\mathbb{C}^+) \quad \text{or} \quad \psi\in H_n^p(\mathbb{C}^+),\] and that \(\varphi\in\mathscr S(\mathbb{C}^+)\) satisfies \(\varphi'\in H_{n-1}^\infty(\mathbb{C}^+)\) and has a finite angular derivative at infinity. Then \(T_{\psi,\varphi}\) is bounded on \(H_n^p(\mathbb{C}^+)\).
Proof. The proof is identical to that of Theorem 25. The only difference is that the boundedness of \(C_\varphi\) on \(H^p(\mathbb{C}^+)\) for \(1\le p<\infty\) now follows from the Elliott–Jury theorem [28], while the case \(p=\infty\) remains immediate. ◻
Remark. For \(1\le p<\infty\), a basic example of a symbol satisfying either of the above sufficient conditions is the affine map \[\varphi(z)=kz+b, \qquad k>0, \;Im(b)>0.\] It is also worth emphasizing that the assumption \(\psi\in H_n^p(\mathbb{C}^+)\), and not only \(\psi\in H_n^\infty(\mathbb{C}^+)\), is admissible because of the Sobolev embedding theorem established in Section 3. \(\qedsymbol\)
We conclude with a brief remark on the Hilbert case \(p=2\). In that setting, the boundedness of weighted composition operators on \(H_n^2(\mathbb{C}^+)\) also admits a characterization in terms of positivity of an associated kernel; see [17]. We record this criterion for completeness, but do not use it in what follows.
A kernel \(A:\mathbb{C}^+\times\mathbb{C}^+\to\mathbb{C}\) is called non-negative if \[\sum_{i,j=1}^{N} c_i\overline{c_j}\,A(z_i,z_j)\ge 0\] for every finite choice of points \(z_1,\dots,z_N\in\mathbb{C}^+\) and scalars \(c_1,\dots,c_N\in\mathbb{C}\). Thus the operator \(T_{\psi,\varphi}\) is bounded on \(H_n^2(\mathbb{C}^+)\) if and only if there exists a constant \(M\ge0\) such that the kernel \[A_n(z,w) = \frac{1}{2\pi}\int_0^\infty \Bigl[ M^2e^{ix(w-\overline{z})} - \overline{\psi(w)}\psi(z)\, e^{ix(\varphi(w)-\overline{\varphi(z)})} \Bigr] \frac{1-x^2}{1-x^{2n+2}}\,dx\] is non-negative on \(\mathbb{C}^+\times\mathbb{C}^+\).
This work was supported by the NSF of Guangdong Province, China (Grant No. 2025A1515011213) and the Science and Technology Development Fund of Macau SAR (No. 0020/2023/RIB1) .