--- title: Weighted Korenblum Spaces url: https://www.emergentmind.com/topics/weighted-korenblum-spaces type: topic --- # Weighted Korenblum Spaces Weighted Korenblum spaces are spaces of holomorphic functions in which admissible boundary growth is controlled by a weight or majorant. In the literature represented here, the term refers not to a single canonical object but to several closely related scales on the unit disk and the upper half-plane: weighted Banach spaces \(H_v^\infty\), classical Korenblum growth spaces \(A^{-\gamma}\), Fréchet and \((LB)\) limits built from those Banach steps, inductive-limit spaces \(A_\Lambda^{-\infty}\) defined by exponential majorants, and upper half-plane spaces \(H_{\Omega,t}^\infty(\mathbb C_+)\). Across these models, the common theme is that a vanishing weight near the boundary permits controlled blow-up of analytic functions, and this growth structure interacts sharply with operator theory, cyclicity, basis questions, and boundary-measure decompositions [1805.07804] [1210.0947] [1712.00280] [2503.20054] [2508.07294]. ## 1. Definitions and principal models On the unit disk \(\mathbb D\), one basic weighted model is the weighted Banach space \[ H_v^\infty=\left\{f\in H(\mathbb D): \|f\|_{H_v^\infty}:=\sup_{z\in\mathbb D}|f(z)|\,v(z)<\infty\right\}, \] where \(v\) is a continuous radial weight, normalized by \(\sup_{z\in\mathbb D}v(z)=1\) in one formulation, or more generally assumed radial, continuous, non-increasing, and such that \(v(r)\to0\) as \(r\to1^-\). Its vanishing subspace is \[ H^\infty_{v,0}=\left\{f\in H^\infty_v:\ \lim_{|z|\to 1}|f(z)|v(z)=0\right\}, \] or, equivalently in parallel notation, \[ H_v^0:=\left\{ f\in H(\mathbb D): \lim_{|z|\to 1^-} v(z)\,|f(z)|=0\right\}. \] For the standard power weights \[ v_\gamma(r)=(1-r)^\gamma,\qquad \gamma>0, \] these become the classical Korenblum-type growth Banach spaces. A closely related and standard normalization is \[ H^\infty_\alpha=\left\{f\in H(\mathbb D): \|f\|_{H^\infty_\alpha}:=\sup_{z\in\mathbb D}(1-|z|^2)^\alpha |f(z)|<\infty\right\},\qquad 0<\alpha<1. \] In this formulation, the parameter \(\alpha\) measures the allowed boundary growth of \(f\): larger \(\alpha\) means more vanishing weight near the boundary and hence permits faster growth of \(f\) as \(|z|\to1\) [1805.07804] [2407.17646]. A second, more flexible model uses a majorant \(A=\Lambda\) satisfying positivity, monotonicity, convexity, doubling-type control \[ A(t^2)\le C\,A(t), \] and auxiliary regularity conditions such as \(A(0)=+\infty\), \(tA(t)\to0\) as \(t\to0^+\), and the existence of \(a\in(0,1)\) such that \(t^aA(t)\) is non-decreasing. The corresponding weighted Korenblum-type space is \[ A^{-\infty}=A_\Lambda^{-\infty}:=\bigcup_{C>0} A_C, \] where \[ A_C=\left\{ f\in \mathrm{Hol}(\mathbb D): |f(z)|\le \exp\!\bigl(CA(1-|z|)\bigr)\right\}, \qquad \|f\|_{A_C}=\sup_{z\in\mathbb D}|f(z)|\,e^{-CA(1-|z|)}<\infty. \] Typical examples are \[ A(t)=\log\log(1/t),\qquad A(t)=\bigl(\log(1/t)\bigr)^p,\quad p>0. \] Here the boundary growth is governed by the majorant \(A(1-|z|)\), and larger \(A\) means a larger space [1210.0947]. A third variant is the upper half-plane model. For a positive weight \(\Omega\) and \(t\ge0\), \[ L_{\Omega,t}^\infty(\mathbb C_+) := \left\{ f:\mathbb C_+\to\mathbb C \text{ measurable}:\ \|f\|_{L_{\Omega,t}^\infty}:=\sup_{z\in\mathbb C_+}(\Im z)^t\Omega(z)|f(z)|<\infty\right\}, \] and the weighted Korenblum space is \[ H_{\Omega,t}^\infty(\mathbb C_+) := \mathcal H(\mathbb C_+)\cap L_{\Omega,t}^\infty(\mathbb C_+). \] In the main applications there, \(\Omega=\omega^k\), where \(\omega\) is the standard logarithmic function used in the upper half-plane setting [2508.07294]. | Model | Definition | Typical notation | |---|---|---| | Weighted sup-norm space on \(\mathbb D\) | \(\sup |f|\,v<\infty\) | \(H_v^\infty\), \(H_v^0\) | | Standard Korenblum weight | \(\sup (1-|z|^2)^\alpha |f(z)|<\infty\) or \(\sup (1-|z|)^\gamma |f(z)|<\infty\) | \(H^\infty_\alpha\), \(A^{-\gamma}\) | | Majorant-defined inductive limit | \(|f(z)|\le \exp(CA(1-|z|))\) | \(A_\Lambda^{-\infty}\) | | Upper half-plane weighted model | \(\sup (\Im z)^t\Omega(z)|f(z)|<\infty\) | \(H_{\Omega,t}^\infty(\mathbb C_+)\) | A recurrent source of confusion is notational rather than mathematical. The cited papers use \(H^\infty_\alpha\), \(H^\infty_{w_\alpha}\), \(A^{-\gamma}\), \(A^{-\gamma\pm}\), \(A_\Lambda^{-\infty}\), and \(H_{\Omega,t}^\infty(\mathbb C_+)\) for related but non-identical constructions. The common structure is weighted boundary growth control, but the topology and operator theory depend strongly on which scale is chosen. ## 2. Scale structure, topology, and basis theory For \(p>0\), the weighted Banach norm \[ \|f\|_{p}=\sup_{0\gamma} A^{-p}, \qquad A_-^{-\gamma}:=\bigcup_{p<\gamma} A^{-p}. \] The special Korenblum space is \[ A^{-}:=A_-^{-\infty}=\bigcup_{p>0} A^{-p}. \] In another notation, the Fréchet Korenblum-type space is \[ A^\gamma:=\bigcap_{u>\gamma} A^u, \] while the \((LB)\)-version is \[ A^{-\gamma}:=\bigcup_{00} A^{-(\gamma+\varepsilon)}, \qquad A^{-\gamma-} :=\bigcup_{\varepsilon>0} A^{-(\gamma-\varepsilon)}, \] and \[ A^{-\infty} := \operatorname{ind}\, H_{(1-|z|)^n}^\infty. \] These formulations are all built from weighted Banach steps, but they encode different locally convex topologies [1712.00280] [2003.13573] [2401.09406]. The topological structure is concrete. For the majorant model, \[ f_n\to f \text{ in } A^{-\infty} \iff \exists N>0\ \text{such that } f_n,f\in A_N \text{ and } \|f_n-f\|_{A_N}\to 0. \] For projective constructions such as \(A_+^{-\gamma}\), convergence is determined by all norms above the threshold, while for inductive constructions such as \(A_-^{-\gamma}\), convergence is controlled within one Banach step. The monomial system \[ e_n(z)=z^n,\qquad \Lambda=(e_n)_{n=0}^\infty, \] has markedly different behavior at the Banach and limit-space levels. For every \(\gamma>0\), \(\Lambda\) is a Schauder basis of \(A_+^{-\gamma}\) and of \(A_-^{-\gamma}\). Thus every \(f\) in either space has a unique expansion \[ f(z)=\sum_{n=0}^\infty a_n z^n \] with convergence in the corresponding locally convex topology. By contrast, Lusky had shown that the monomials are not a Schauder basis for the closure of the polynomials in weighted Banach spaces of \(H^\infty\)-type. The positive basis theorem is therefore specific to the Fréchet and \((LB)\) Korenblum-type limit spaces, not to the underlying weighted Banach steps [1712.00280]. The same work gives an explicit sequence-space model. For each \(\gamma>0\), \(A_+^{-\gamma}\) is isomorphic to a Köthe echelon space \(\Lambda_0(A_\gamma)\), and \(A_-^{-\gamma}\) is represented as a Köthe co-echelon space \(k_0(V_\gamma)\). This places weighted Korenblum scales within the standard framework of locally convex sequence spaces and explains why dyadic block decompositions and discrete-continuous norm comparisons are effective in their analysis. ## 3. Cyclicity, premeasures, and weighted Korenblum–Roberts theory For weighted Korenblum-type spaces \(A_\Lambda^{-\infty}\), cyclicity is studied for the multiplication operator \[ (M_z f)(z)=z f(z). \] If \(f\in X\), where \(X=A^{-\infty}\) or one Banach stage \(A_C\), the cyclic subspace generated by \(f\) is \[ [f]_X=\overline{\operatorname{span}\{p(z)f(z): p \text{ polynomial}\}}, \] and \(f\) is cyclic if \([f]_X=X\). For zero-free normalized functions \(f(0)=1\), the key boundary datum is a premeasure \(\mu_f\) in the representation \[ f(z)=\exp\!\left(\int_{\mathbb T}\frac{\zeta+z}{\zeta-z}\,d\mu_f(\zeta)\right). \] The main theorem states that cyclicity is characterized by the vanishing of the \(A\)-singular part of this premeasure. Under the growth hypotheses labeled (C1) and (C2), the paper proves \[ f_\mu \text{ is cyclic in } A^{-\infty} \iff (\mu)_s=0. \] This gives a positive answer to Deninger’s conjecture in the weighted Korenblum setting [1210.0947]. The same paper develops the relevant boundary apparatus. A premeasure \(\mu\) is \(A\)-bounded if \[ |\mu(I)|\le C\,A(|I|) \] for all arcs \(I\). For a closed \(F\subset\mathbb T\), the \(A\)-entropy is \[ \operatorname{Entr}_A(F)=\sum_n |I_n|\,A(|I_n|), \] where \(\{I_n\}\) are the complementary arcs of \(F\). The structural theorem \[ \mu \text{ is \(A\)-absolutely continuous} \iff \mu_s=0 \] is the bridge between boundary decomposition and cyclicity. In this framework, the singular part of the boundary data is exactly the obstruction to density of polynomial multiples. A more recent weighted extension is the weighted Korenblum–Roberts theory for spaces \[ P^t(\mu)=\overline{P}^{\,L^t(\mu)}, \qquad d\mu = dA_\alpha + w\,dm, \qquad \alpha>-1, \] and also for more general radial disk weights \[ d\mu = G_{ah}\,dA + w\,dm, \qquad G_{ah}(z)=\exp\!\Big(-a\frac{h(1-|z|)}{1-|z|}\Big), \] with the specific gauge \[ h(x)=x\log(e/x). \] Here the boundary weight \(w\) determines the family of associated Beurling–Carleson sets \[ Assoc(w)=\Big\{E\in BC:\int_E \log w\,dm>- \infty\Big\}. \] The main cyclicity theorem is \[ S_\nu \text{ is cyclic in } P^t(\mu) \quad\Longleftrightarrow\quad \nu(E)=0\ \text{for all }E\in Assoc(w), \] assuming \(P^t(\mu)\) is analytic. In the unweighted Bergman case, the relevant Beurling–Carleson sets have Lebesgue measure zero; in the weighted theory, associated sets can have either zero or positive Lebesgue measure. That distinction is one of the key structural differences from the classical Korenblum–Roberts theorem [2503.20054]. The same paper computes the Thomson decomposition \[ P^t(\mu) = P^t(dA_\alpha+w_c\,dm)\oplus \mathcal{L}^t(w_r\,dm), \] with \(w_c=w|_{core(w)}\) and \(w_r=w|_{res(w)}\). This isolates the maximal analytic part of the space and shows that analyticity itself is a weighted boundary phenomenon. ## 4. Hilbert matrix and generalized Hilbert operators The Hilbert matrix operator is defined on Taylor coefficients by \[ f(z)=\sum_{k=0}^\infty a_k z^k \quad\Longrightarrow\quad H(f)(z)=\sum_{n=0}^\infty \left(\sum_{k=0}^\infty \frac{a_k}{n+k+1}\right) z^n. \] A crucial device is its representation as an integral of weighted composition operators, \[ \mathcal H f(z)=\int_0^1 T_t f(z)\,dt, \qquad T_t f(z)=w_t(z)\,f(\varphi_t(z)), \] with \[ w_t(z)=\frac1{1-(1-t)z}, \qquad \varphi_t(z)=\frac{t}{1-(1-t)z}. \] For a weighted composition operator \(uC_\varphi f=u\cdot(f\circ\varphi)\) on \(H^\infty_\alpha\), the norm formula used in the analysis is \[ \|uC_\varphi\|_{H^\infty_\alpha\to H^\infty_\alpha} = \sup_{z\in\mathbb D} \frac{(1-|z|^2)^\alpha\,|u(z)|}{(1-|\varphi(z)|^2)^\alpha}. \] This formula reduces operator norm estimation to a supremum over the disk and is the technical heart of the Hilbert-matrix analysis on weighted Korenblum spaces [1805.07804] [2410.16598]. For the Korenblum spaces \[ H^\infty_\alpha=\left\{f\in H(\mathbb D): \sup_{z\in\mathbb D}(1-|z|^2)^\alpha |f(z)|<\infty\right\}, \qquad 0<\alpha<1, \] the 2018 norm computation gives a sharp formula in the range \[ 0<\alpha\le \frac23: \qquad \|\mathcal H\|_{H^\infty_\alpha\to H^\infty_\alpha} = \frac{\pi}{\sin(\pi\alpha)}. \] The lower bound is obtained by testing on \[ f_\alpha(z)=(1-z)^{-\alpha}, \qquad \|f_\alpha\|_{H^\infty_\alpha}=2^\alpha, \] and evaluating a Beta integral. For \[ \frac23<\alpha<1, \] the same paper proves boundedness and an explicit upper estimate, but does not claim sharpness. The structure of Lemma 4.2 suggests a change in norm behavior at the threshold \(\alpha=2/3\) [1805.07804]. A later result computes the exact norm and essential norm on the equivalent Korenblum weights \[ w_\alpha(z)=(1-|z|)^\alpha,\qquad 0<\alpha<1, \] showing that \[ \|H\|_{H^\infty_{w_\alpha}\to H^\infty_{w_\alpha}} = \|H\|_{e,\,H^\infty_{w_\alpha}\to H^\infty_{w_\alpha}} = \frac{\pi}{\sin(\pi\alpha)}. \] The same value holds for the equivalent standard weight \((1-|z|^2)^\alpha\). Thus, in this normalization, the exact constant is valid for all \(0<\alpha<1\), and the essential norm already attains the full operator norm. More generally, for generalized Hilbert matrix operators on weighted Banach spaces, \[ \|I_K\|_{e,\,H^\infty_u\to H^\infty_u} = \sup_{t\in(0,1)} \limsup_{z\to 1} \frac{|T_t(z)|u(z)}{u(\phi_t(z))}. \] A common misconception is that the exact constant \(\pi/\sin(\pi\alpha)\) was known uniformly across all Korenblum normalizations already in 2018. What the cited results show is more specific: exactness up to \(\alpha=2/3\) in one treatment, and exactness for all \(0<\alpha<1\) for the equivalent weight \(w_\alpha(z)=(1-|z|)^\alpha\), with explicit identification of norm and essential norm [2201.09591]. The generalized Hilbert operator associated with a positive finite Borel measure \(\mu\) on \([0,1)\) is governed by the moments \[ \mu_n:=\int_{[0,1)} t^n\,d\mu(t), \] and the integral operator \[ I_\mu(f)(z):=\int_{[0,1)} \frac{f(t)}{1-tz}\,d\mu(t). \] On spaces where both are defined and polynomials are dense, \(I_\mu=H_\mu\). For weighted Korenblum-type spaces \(H_v^\infty\) and \(H_v^0\), the main control quantity is \[ C(v,w):=\sup_{01\), the well-definedness criterion becomes \[ \int_{[0,1)} \frac{d\mu(t)}{(1-t)^\gamma}<\infty, \qquad\text{equivalently}\qquad \sum_{n=0}^\infty \mu_n\, n^{\gamma-1}<\infty. \] The same paper emphasizes that this behavior differs in essential ways from the Cesàro-type setting [2407.17646]. The Hilbert matrix has also been studied on the logarithmically weighted Korenblum space \[ H^\infty_{\alpha,\log} = \left\{f\in H(\mathbb D): \sup_{z\in\mathbb D}(1-|z|^2)^\alpha \log\!\frac{2e^\alpha}{1-|z|^2}\,|f(z)|<\infty\right\}, \] with \(H^\infty_{\alpha,\log}\subset H^\infty_\alpha\). The exact norm of \(\mathcal H:H^\infty_{\alpha,\log}\to H^\infty_\alpha\) and of \(\mathcal H\) on \(H^\infty_{\alpha,\log}\) is expressed by an explicit supremum of integral kernels, and the lower estimate \[ \frac{\pi}{\sin(\pi\alpha)} \] again arises from the Beta identity. This extends the Hilbert-matrix norm problem from standard Korenblum weights to logarithmic variants [2410.16598]. ## 5. Cesàro, Volterra, and integration operators Given a positive finite Borel measure \(\mu\) on \([0,1)\), the Cesàro-type operator is \[ C_\mu(f)(z) = \int_0^1 \frac{f(tz)}{1-tz}\,d\mu(t), \] and if \[ f(z)=\sum_{n=0}^\infty a_n z^n, \] then \[ C_\mu(f)(z) = \sum_{n=0}^\infty \mu_n\Big(\sum_{k=0}^n a_k\Big) z^n. \] For general weights \(v,w\), a sufficient continuity criterion is \[ C(v,w):=\sup_{00\) and \(\delta\in(-\gamma,1)\), then \[ C_\mu:H_{v_\gamma}^\infty\to H_{v_{\gamma+\delta}}^\infty \text{ is continuous} \iff \mu_n = O(n^{-(1-\delta)}) \] \[ \iff \mu([t,1))=O\big((1-t)^{1-\delta}\big), \] and compactness is characterized by \[ \mu_n=o(n^{-(1-\delta)}) \] or, equivalently, the vanishing \((1-\delta)\)-Carleson condition. These Banach-step results transfer to the Korenblum-type scales: \[ \mu \text{ is an }s\text{-Carleson measure for every }00. \] Moreover, \[ C_\mu:A^{-\infty}\to A^{-\infty} \] is continuous for every positive finite Borel measure \(\mu\). Under a Carleson assumption, \[ \sigma_p(C_\mu, A^{-\infty})=\{\mu_n:n\in\mathbb N_0\}, \qquad \sigma(C_\mu,A^{-\infty})=\{\mu_n:n\in\mathbb N_0\}\cup\{0\}. \] Thus the moment sequence governs not only continuity and compactness but also the spectrum on the classical Korenblum space [2401.09406]. For generalized Volterra operators, \[ (V_g f)(z):=\int_0^z f(\zeta)g'(\zeta)\,d\zeta, \] the relevant Banach spaces are \[ A^{-\gamma} :=\left\{f\in H(\mathbb D): \sup_{z\in\mathbb D}(1-|z|)^\gamma |f(z)|<\infty\right\}, \] and \[ A_0^{-\gamma} :=\left\{f\in H(\mathbb D): \lim_{|z|\to1^-}(1-|z|)^\gamma |f(z)|=0\right\}. \] Boundedness and compactness are completely controlled by the Bloch and little Bloch classes: \[ V_g:A^{-\gamma}\to A^{-\gamma}\text{ is continuous} \iff V_g:A_0^{-\gamma}\to A_0^{-\gamma}\text{ is continuous} \iff g\in\mathcal B, \] and \[ V_g:A^{-\gamma}\to A^{-\gamma}\text{ is compact} \iff V_g:A_0^{-\gamma}\to A_0^{-\gamma}\text{ is compact} \iff g\in\mathcal B_0. \] When \(g\in\mathcal B\) is nonconstant, the optimal domains are Banach spaces and admit the precise descriptions \[ [V_g,A^{-\gamma}] = \left\{f\in H(\mathbb D): f\,g'\in A^{-(\gamma+1)}\right\}, \] \[ [V_g,A_0^{-\gamma}] = \left\{f\in H(\mathbb D): f\,g'\in A_0^{-(\gamma+1)}\right\}. \] Their multiplier spaces are \[ M([V_g,A^{-\gamma}])=M([V_g,A_0^{-\gamma}])=H^\infty. \] The classical Cesàro operator appears as the special case \(g_0(z)=-\log(1-z)\), and for every \(\gamma>0\), \[ A^{-\gamma}\subsetneq [C,A^{-\gamma}], \qquad A_0^{-\gamma}\subsetneq [C,A_0^{-\gamma}]. \] This shows that optimal domains are often strictly larger than the original weighted Korenblum space [2502.00755]. A different operator-theoretic rigidity appears for the integration operator \[ (Jf)(z)=\int_0^z f(s)\,ds. \] On weighted Fréchet and \((LB)\) spaces of holomorphic functions, including the Korenblum-type examples built from the weights \[ v_u(r)=(1-r)^u,\qquad u>0, \] the proper closed invariant subspaces of \(J\) are exactly the jet-vanishing spaces \[ A_K=\{f\in E:\ f^{(j)}(0)=0,\ 0\le j1\) by \(p=1\), with \(k\le0\) and a loss \(\varepsilon>0\) [2508.07294]. This upper half-plane formulation complements the disk theory rather than duplicating it. On the disk, weighted Korenblum spaces are frequently defined directly by radial growth; on \(\mathbb C_+\), the same growth class is recovered through operator-symbol duality. A plausible implication is that the weighted Korenblum condition is robust under substantial changes of ambient geometry, provided the boundary growth is encoded in the correct weighted analytic pairing. The basis and sequence-space results also reinforce a broader structural theme. Weighted Korenblum spaces can behave poorly at the single Banach-step level—for example, the monomials need not form a Schauder basis there—yet acquire strong approximation and decomposition properties at the projective or inductive limit level. Conversely, operator-theoretic phenomena can sharpen when passing to a single weighted Banach step, as in the exact Hilbert-matrix norm formulas or the Bloch characterization of Volterra boundedness. This suggests that the topology of the scale is not a secondary detail but a central part of the theory [1712.00280]. Taken together, the cited results show that weighted Korenblum spaces form a family of analytically natural growth spaces whose structure is simultaneously function-theoretic, boundary-measure-theoretic, and operator-theoretic. Exact norm formulas for Hilbert operators, cyclicity criteria via premeasures and associated Beurling–Carleson sets, optimal-domain descriptions for Volterra operators, and Hankel-symbol characterizations in the upper half-plane all point to the same organizing principle: weighted boundary growth is the mechanism by which these spaces encode both analytic regularity and singular boundary behavior. Source: https://www.emergentmind.com/topics/weighted-korenblum-spaces