Fock–Carleson Measure in Fock Spaces

• Fock–Carleson measure is a concept that defines when a positive measure controls the norm of entire functions in Gaussian-weighted Fock spaces.
• It employs analytic and geometric criteria, such as reproducing kernels and ball conditions, to ensure boundedness of associated operators.
• Its framework extends to weighted, vector-valued, Sobolev, and quaternionic settings, influencing Toeplitz operator theory and spectral analysis.

A Fock–Carleson measure is a precise measure-theoretic concept central to operator theory and function space geometry in the Fock space, the Hilbert or Banach space of entire functions endowed with a Gaussian-weighted $L^p$ norm. Fock–Carleson measures dictate boundedness, compactness, and invertibility properties of integral and Toeplitz operators acting on Fock spaces and their generalizations, and are deeply connected to sampling, interpolation, and geometric function theory.

1. Definition and Core Characterization

Let $d\lambda(z) = \frac{1}{2} e^{-|z|^2/2}dA(z)$ denote Gaussian measure on $\mathbb{C}$, and let $\mathcal{F}^2$ be the Hilbert Fock space of entire $f$ with finite norm $$\|f\|^2 = \int_\mathbb{C} |f(z)|^2 e^{-|z|^2/2} dA(z) < \infty$$. A positive Borel measure $\mu$ on $\mathbb{C}$ is a Fock–Carleson measure if there exists $C>0$ such that, for all $f\in\mathcal{F}^2$,

$$\int_\mathbb{C} |f(z)|^2 e^{-|z|^2/2} d\mu(z) \leq C \int_\mathbb{C} |f(z)|^2 e^{-|z|^2/2} dA(z).$$

This extends to Fock spaces $\mathcal{F}^p$, $0 < p \leq \infty$, and weighted Fock spaces $F^p_\alpha$ and vector-valued or Sobolev-type variants (Chen et al., 2024, Cho et al., 2012, Cascante et al., 2016, Esmeral et al., 2019, Huang et al., 14 Feb 2025, Arroussi et al., 21 Apr 2025, Mengestie, 2015).

A measure is a reverse Fock–Carleson measure if the same inequality holds in reverse with $c>0$ (i.e., $\int |f(z)|^2 e^{-|z|^2/2} d\mu(z) \geq c \|f\|^2$). These conditions regulate the boundedness and invertibility of associated operators.

2. Equivalent Formulations and Geometric Criteria

The Fock–Carleson property admits several equivalent analytic and geometric characterizations:

• Via reproducing kernels $K_z(w) = e^{z\overline{w}/2}$,

$$\sup_{z\in\mathbb{C}} \int_{\mathbb{C}} |K_z(w)|^2 e^{-|w|^2/2} d\mu(w) < \infty.$$

• Through ball conditions: For every $r > 0$, there exists $C_r$ with

$$\mu(B(z, r)) \leq C_r \quad \forall z\in\mathbb{C}.$$

For reverse Carleson,

$$C^{-1} \leq \mu(B(z, r)) \leq C \quad \forall z\in\mathbb{C}.$$

Weighted and Sobolev Fock spaces introduce modifications in the ball conditions, incorporating growth rates of $(1+|z|)^{mp}$ for Fock–Sobolev spaces $F^{p,m}$ (Cho et al., 2012).

In multi-dimensional and vector-valued settings, similar criteria use balls in $\mathbb{C}^n$ or generalized Dall'Ara weights and require local averages against the underlying measure (Arroussi et al., 21 Apr 2025).

3. Berezin Transform, Operator Theory, and Examples

The Berezin transform of a measure $\mu$ is

$$B_\mu(z) = \int_\mathbb{C} |k_z(w)|^2 d\mu(w) = \frac{1}{2} \int_\mathbb{C} e^{-|z-w|^2/2} d\mu(w),$$

where $k_z$ is the normalized kernel (Chen et al., 2024, Huang et al., 14 Feb 2025, Arroussi et al., 21 Apr 2025, Mengestie, 2015).

For Toeplitz operators $T_\mu$ with symbol $\mu$, boundedness on $\mathcal{F}^2$ is equivalent to $\mu$ being Carleson, and invertibility to $\mu$ being reverse Carleson (Chen et al., 2024). While boundedness can often be detected via uniform control of $B_\mu(z)$, invertibility is stricter; pointwise lower bounds for $B_\mu(z)$ do not suffice (see the lattice counterexample in (Chen et al., 2024)). This exemplifies the subtlety: bounded Berezin transforms do not guarantee invertibility—local mass lower bounds (reverse Carleson) are essential.

Illustrative example: A “thin” lattice of point masses $\nu = \sum_k \delta_{a_k}$, spaced widely, gives $\nu(B(z, r)) \simeq 1$ so $B_\nu$ is bounded below and above, yet $\nu$ may fail the reverse Carleson inequality globally, making $T_\nu$ non-invertible (Chen et al., 2024).

4. Fock–Carleson Measures in Weighted, Vector, and Sobolev Settings

For weighted Fock spaces $F^p_\alpha$ or vector-valued analogs $F^2_\phi(H)$ with subharmonic weights, the Carleson property is similarly formulated with respect to the weighted norm and reproducing kernel. The scalar-valued and operator-valued Berezin transforms and box averages (e.g. $\mu(D(z,r))$ normalized by local scale) provide equivalent criteria for boundedness and compactness of Toeplitz operators (Arroussi et al., 21 Apr 2025).

Carleson measures for Fock–Sobolev spaces $F^{p,m}$ are characterized by polynomial growth controls:

$$\sup_{a\in\mathbb{C}} \frac{\mu(D(a, r))}{(1+|a|)^{m p}} < \infty,$$

and weighted Fock–Sobolev variants with $A_\infty$-type weights require analogous arithmetic on the covering balls and associated averages (Cho et al., 2012, Cascante et al., 2016, Mengestie, 2015).

Averaging functions and sequences derived from such local ball conditions or Berezin transforms describe both the embedding and operator-theoretic aspects (Mengestie, 2015).

5. Advanced Extensions: Higher Derivatives, Quaternionic, and Lagrangian-Invariant Cases

Generalizations include Fock–Carleson measures for $k$-th derivatives (horizontal Fock–Carleson type measures, $k$-hFC), defined by

$$\sum_{|\alpha|=k} \int_{\mathbb{C}^n} |\partial^\alpha f(z)|^2 e^{-\|z\|^2/2} d\mu(z) \leq C_k \|f\|^2_{Fock},$$

and codifferentials define an isomorphism to a commutative $C^*$-algebra via the Bargmann transform (Esmeral et al., 2019).

Quaternionic Fock spaces on $\mathbb{H}$ extend the notion: $\mu$ is a right-linear Fock–Carleson measure if, for slice-regular $f$,

$$\int_\mathbb{H} |f(w) e^{-\alpha |w|^2/2}|^p d\mu(w) \leq C \|f\|^p_{p,\alpha},$$

with equivalent slice and global formulations. The slice-averaging over symmetric Carleson boxes introduces new geometric complexity, but the embedding and compactness criteria parallel the complex case (Lin et al., 15 Jan 2026).

Measures invariant under Lagrangian translations in phase space $\mathbb{C}^n$ are mapped (via unitary transformation) to horizontal Fock–Carleson measures, and the associated Toeplitz operators are again commutative and diagonalizable (Esmeral et al., 2019).

6. Functional Analytic Applications and Connections

Fock–Carleson measures are foundational for the study of Toeplitz operator spectra, Schatten class criteria, interpolation and sampling in entire function spaces, boundedness/compactness of weighted composition operators, and extension to $p$-$q$ Carleson measure theory for $F^p_\alpha \rightarrow L^q(\mu)$ embedding (Mengestie, 2015).

In operator theory, they determine exactly when integral or kernel-induced operators, Berezin transforms, and composition-type mappings are bounded, compact, or invertible. The interplay of local mass conditions, Berezin transforms, lattice averaging, and geometric measure properties underpins a unified theory spanning classical, weighted, Sobolev, vector-valued, and quaternionic Fock spaces (Chen et al., 2024, Cho et al., 2012, Cascante et al., 2016, Esmeral et al., 2019, Huang et al., 14 Feb 2025, Lin et al., 15 Jan 2026, Arroussi et al., 21 Apr 2025, Mengestie, 2015).

7. Summary Table: Key Criteria

Property	Analytic Criterion	Geometric Criterion	Operator-Theoretic Outcome
Fock–Carleson	$\sup_z \int |K_z(w)|^2 d\mu(w) < \infty$	$\forall z, \mu(B(z,r)) \leq C_r$	$T_\mu$ bounded on Fock space
Reverse Fock–Carleson	$\int |f(z)|^2 d\mu(z) \geq c \|f\|^2$	$C^{-1} \leq \mu(B(z,r)) \leq C$	$T_\mu$ invertible on Fock space
Berezin-based (boundedness)	$B_\mu(z)$ bounded above	-	Bounded embedding/Toeplitz
Berezin-based (invertible)	$B_\mu(z)$ bounded below insufficient	-	Full reverse Carleson needed

The table summarizes the essential analytic, geometric, and operator relations: boundedness is guaranteed by Carleson-type local upper bounds; invertibility requires two-sided control (reverse Carleson), and Berezin transform criteria—though necessary for boundedness—do not suffice for invertibility.