Logarithmic Bloch Spaces

• Logarithmic Bloch spaces are Banach spaces of holomorphic functions on the unit disk defined via a logarithmically weighted derivative norm, bridging classical Bloch spaces and H∞.
• They provide a precise framework for studying operator theory, embedding, and duality with clear norm structures and inclusion relationships between analytic function spaces.
• Their applications range from analyzing Hilbert, composition, and integral operators to extending results in multidimensional settings with Carleson measure criteria.

Logarithmic Bloch spaces are Banach spaces of holomorphic functions on the unit disk (and their multidimensional analogues) defined by augmenting the classical Bloch semi-norm with a logarithmic weight. These spaces interpolate between the classical Bloch space, which controls the size of the derivative with $(1-|z|^2)$ near the boundary, and $H^\infty$, while admitting a precise operator-theoretic, function-theoretic, and duality-based framework with connections to integral operators, Carleson measures, and multiplier theory. Logarithmic Bloch spaces play a critical role in endpoint operator theory, in embedding and interpolation problems, and as models for analytic function spaces with the weakest decay on the derivative necessary for nontrivial bounded operator theory.

1. Formal Definitions and Norm Structure

Let $D = \{z \in \mathbb{C} : |z| < 1\}$ and $H(D)$ be the space of analytic functions on $D$. The logarithmically weighted Bloch space $\mathcal{B}_{\log}$ is defined by

$$\mathcal{B}_{\log} = \left\{ f \in H(D) : \|f\|_{\mathcal{B}_{\log}} := |f(0)| + \sup_{z \in D} (1 - |z|^2) \log\left( \frac{e}{1 - |z|^2} \right) |f'(z)| < \infty \right\}.$$

This norm is Banach and equips $\mathcal{B}_{\log}$ with a complete structure (Ye et al., 27 Oct 2025).

A generalization, the logarithmic Bloch-type space $\mathcal{B}_{\log}^\gamma$ with parameters $\beta > 0$, $\gamma \ge 0$, consists of those $f \in H(D)$ such that

$$\|f\|_{\mathcal{B}_{\log}^\gamma} = |f(0)| + \sup_{z \in D} (1 - |z|^2)^\beta |f'(z)| \left( \log \frac{e}{1 - |z|^2} \right)^\gamma < \infty,$$

covering the standard logarithmic Bloch space at $\beta=1$, $\gamma=1$ (Yang et al., 14 Sep 2025).

Further families include

$$B_{\log^\alpha} = \left\{ f \in H(D) : \|f\|_{B_{\log^\alpha}} = |f(0)| + \sup_{z \in D} (1 - |z|^2) \left( \log \frac{1}{1 - |z|} \right)^{\alpha} |f'(z)| < \infty \right\}$$

for real $\alpha$, with associated little Bloch, predual, and $L^1$-Bloch norm structures (Pavlović, 2011). Typical alternate weights (e.g., $(1 - |z|)\log(4/(1-|z|))$) yield equivalent norms (Malavé-RamiÍrez et al., 2014).

2. Inclusion Relations and Function-Theoretic Properties

Logarithmic Bloch spaces are situated strictly between the classical Bloch space $\mathcal{B}$ and $H^\infty$: $$\mathcal{B} \subsetneq \mathcal{B}_{\log} \subsetneq H^\infty.$$ The classical Bloch norm imposes

$$\|f\|_{\mathcal{B}} = |f(0)| + \sup_{z \in D} (1 - |z|^2) |f'(z)|$$

while $\mathcal{B}_{\log}$ relaxes the requirement for decay in $|f'|$ as $|z| \to 1$, admitting functions with

$$|f'(z)| = O\left( \frac{1}{1 - |z|^2} \cdot \left[ \log \left(\frac{e}{1 - |z|^2}\right) \right]^{-1} \right),$$

and hence a strictly larger class with milder behavior near the boundary (Ye et al., 27 Oct 2025, Malavé-RamiÍrez et al., 2014).

For each $\alpha > 0$, the $\alpha$-Bloch space $\mathcal{B}^\alpha$ with $(1 - |z|^2)^\alpha$ in the norm satisfies $$\mathcal{B}^\alpha \subsetneq \mathcal{B}^\alpha_{\log}$$ for every $\alpha$ (Ye et al., 27 Oct 2025). Polynomial functions and certain slowly growing outer functions are always in $\mathcal{B}_{\log}$, while most inner functions are excluded (Malavé-RamiÍrez et al., 2014). The monomial $g_n(z) = z^n$ has norm $\|g_n\|_{\mathcal{B}_{\log}} \asymp n/\log n$ as $n \gg 1$ (Malavé-RamiÍrez et al., 2014).

A "little logarithmic Bloch space" $\mathcal{B}_{\log,0}^\gamma$ consists of those $f$ in $\mathcal{B}_{\log}^\gamma$ with

$$\lim_{|z|\to 1^-} (1 - |z|^2)^\beta |f'(z)| \left( \log \frac{e}{1 - |z|^2} \right)^\gamma = 0$$

(Yang et al., 14 Sep 2025).

3. Operator Theory: Hilbert, Composition, and Integral Operators

The norm and mapping properties of classical operators (Hilbert, Cesàro, Libera, composition) on logarithmic Bloch spaces have been characterized with sharp constants and extremals.

Hilbert Matrix Operator: The operator $\mathcal{H}$ defined via

$$\mathcal{H} f(z) = \sum_{n=0}^\infty \left( \sum_{k=0}^\infty \frac{a_k}{n+k+1} \right) z^n = \int_0^1 \frac{f(t)}{1- t z} dt$$

maps $\mathcal{B} \to \mathcal{B}_{\log}$ with exact operator norm

$$\|\mathcal{H}\|_{\mathcal{B} \to \mathcal{B}_{\log}} = \frac{3}{2}$$

and from $H^\infty \to H^\infty_{\log}$ with norm $1$. For the generalized setting $$\mathcal{H} : \mathcal{B}^\alpha \to \mathcal{B}^\alpha_{\log}$$ with $1<\alpha<2$, explicit lower and upper bounds are given using Beta and Gamma integrals, and the operator is unbounded outside this range (Ye et al., 27 Oct 2025).

Weighted Composition Operators: For $W_{u,\varphi}(f) = u(z)f(\varphi(z))$, essential norm estimates on $\mathcal{B}_{\log}$ are derived in terms of discrete norms of monomial sequences, integrals of iterates, and test function techniques. In particular,

$$\|W_{u,\varphi}\|_{e}^{\mathcal{B}_{\log} \to \mathcal{B}_{\log}} \asymp \max \left\{ \limsup_{n\to\infty} \frac{\|(n+1)J_u(\varphi^n)\|_{\mathcal{B}_{\log}}}{\|g_{n+1}\|_{\mathcal{B}_{\log}}},\ \limsup_{n\to\infty} \frac{\|I_u(\varphi^n)\|_{\mathcal{B}_{\log}}}{\|g_n\|_{\mathcal{B}_{\log}}} \right\}$$

Compactness corresponds to vanishing of both limsups (Malavé-RamiÍrez et al., 2014).

Integral-type Hilbert Operators: For measures $\mu$ on $[0,1)$ and $\alpha > -1$, the operator (Tang et al., 2022)

$$\mathcal{I}_{\mu_{\alpha+1}}(f)(z) = \int_0^1 \frac{f(t)}{(1 - t z)^{\alpha+1}} d\mu(t)$$

acts boundedly on $\mathcal{B}_{\log}$ iff the measure $\mu$ is logarithmic Carleson of order $\alpha+1$, i.e.

$$\sup_{0 \leq t < 1} \frac{ \mu([t, 1)) \left[ \log \left( \frac{e}{1-t} \right) \right]}{(1-t)^{\alpha+1} } < \infty$$

and compacts iff this quantity vanishes as $t \to 1^-$.

Cesàro and Libera Transforms: The Cesàro operator $C$ maps $B_{\log^\alpha} \to B_{\log^{\alpha+1}}$ boundedly for $\alpha>-1$, and the Libera transform $L$ shifts indices in the negative direction for $\alpha>0$, with explicit coefficient criteria for boundedness (Pavlović, 2011).

4. Duality, Preduals, and Functional Decomposition

Functional duality and explicit decompositions characterize the internal structure of logarithmic Bloch spaces.

• The predual $\mathfrak{B}^1_{\log^\alpha}$ comprises those $f \in H(D)$ with

$$\|f\|_{\mathfrak{B}^1_{\log^\alpha}} = |f(0)| + \int_D |f'(z)| (\log \tfrac{1}{1 - |z|})^\alpha dA(z) < \infty,$$

while the "little" $b_{\log^\alpha}$ space is the closure of polynomials in $B_{\log^\alpha}$ norm or those $f$ with vanishing weighted derivative at the boundary (Pavlović, 2011).

• Coefficient block decompositions (using frequency polynomials $V_n$) yield norm equivalences, succinctly relating analytic and sequence-space norms:
  • $f \in B_{\log^\alpha}$ iff $\sup_n (n+1)^{-\alpha}\|V_n*f\|_\infty < \infty$,
  • $f \in \mathfrak{B}^1_{\log^\alpha}$ iff $$\sum_{n=0}^\infty (n+1)^{-\alpha} \|V_n * f\|_\infty < \infty$$.
• Duality is realized via a canonical Bloch pairing on Fourier coefficients: $$\langle f, g \rangle = \lim_{r \to 1^-} \sum_{k=0}^\infty a_k b_k 2^{-k} r^k,$$ yielding $(b_{\log^\alpha})^* \cong B_{\log^\alpha}$, $$(B_{\log^\alpha})^* \cong \mathfrak{B}^1_{\log^\alpha}$$, and $$(\mathfrak{B}^1_{\log^\alpha})^*\cong b_{\log^\alpha}$$ (Pavlović, 2011).

5. Carleson Measures, Tent Spaces, and Closures

Precise descriptions of closures, multipliers, and subspace properties are given via Carleson-type measure criteria and derivative tent spaces.

• Tent Spaces: The closure of derivative tent spaces $DT_p^q(\alpha)$ in $\mathcal{B}_{\log}^\gamma$ is characterized depending on parameter regimes. For $\beta = 1$, $\gamma = 1$ (the standard logarithmic Bloch), the closure is described by the finiteness of certain level-set integrals over non-tangential approach regions ("tents") (Yang et al., 14 Sep 2025). In a critical regime, this closure equals exactly the logarithmic Bloch space for values outside a threshold, while in the critical case, membership is determined by "Carleson tent integrals" of the derivative weighted with the logarithmic factor.
• Special Cases: This encompasses byproducts such as the closure of Dirichlet-type spaces in the Bloch or logarithmic Bloch norm, with explicit necessity/sufficiency via Carleson-type tent integrals involving weighted derivatives.
• Further Structure: Embedding theorems show $B_{\log^\alpha} \subset H^1$ for $\alpha \ge 0$ and into the disk algebra for $\alpha < 0$ with explicit modulus of continuity.

6. Generalizations to High Dimensions and Multipliers

In the unit polydisc $D^n$, two main logarithmic Bloch spaces are used:

• The product logarithmic Bloch space $\mathcal{B}_{\log}(D^n)$, with norm

$$\|f\|_{\mathcal{B}_{\log}} = |f(0)| + \sup_{z \in D^n} \sum_{j=1}^n (1 - |z_j|^2) \log\left( \frac{2}{1 - |z_j|} \right) |D f(z)|$$

• The pointwise variant $\mathbb{B}_{\log}(D^n)$, with

$$\|f\|_{\mathbb{B}_{\log}} = |f(0)| + \sup_{z \in D^n} \sum_{j=1}^n (1 - |z_j|^2) \log\left( \frac{2}{1 - |z_j|^2} \right) |D_j f(z)|$$

(Sehba, 2013).

The symbols for endpoint bounded Hankel operators on the Bergman space $A^1(D^n)$ are exactly the product logarithmic Bloch space, while pointwise logarithmic Bloch spaces capture the symbols for boundedness from the product to the pointwise Bloch space.

For multipliers:

• $\mathcal{B}(D^n) \to \mathbb{B}(D^n)$ multipliers are those analytic $\varphi$ in $H^\infty(D^n) \cap \mathbb{B}_{\log}(D^n)$,
• $\mathbb{B}(D^n) \to \mathbb{B}(D^n)$ multipliers are functions in $$H^\infty \cap \mathcal{B}\mathcal{L}\mathcal{L}(D^n)$$, where the last denotes a space with double-logarithmic weighted mixed derivatives.

These results establish the logarithmic Bloch spaces as "endpoint" symbols for many multidimensional operator theories.

7. Open Problems and Directions

Several unresolved questions and directions persist:

• Determination of essential norms for more general integral or Hankel-type operators in the logarithmic Bloch context.
• Extension of Carleson measure and duality theory for logarithmic Bloch spaces, especially for non-radial or highly singular weighting functions (Malavé-RamiÍrez et al., 2014, Tang et al., 2022).
• Detailed interpolation and sampling theory, stability of composition operators, and explicit structure of preduals, especially in several variables.
• Investigation of mapping properties and shift index formulas for classical transforms (Cesàro, Libera) in the full parameter range, including the "double-logarithmic" regime (Pavlović, 2011).

Logarithmic Bloch spaces thus serve as a central object in the study of analytic operator theory with precise endpoint regularity, sharp embedding properties, and a spectrum of open theoretical questions.