Bergman Dual: Spaces, Metrics and Kernels

• Bergman Dual is a framework that unifies various dual constructions, including integral pairings, transform realizations, and geometric dualizations using Bergman kernels and metrics.
• It connects Banach-space duality and transform methods, such as the Fantappié and Laplace transforms, to produce concrete holomorphic realizations across different domains.
• Research shows that duality in Bergman spaces is highly domain-dependent, with projection regularity, boundary behavior, and curvature conditions dictating its validity and scope.

“Bergman dual” is not a single universally fixed object. In current usage, it denotes several dual constructions attached to Bergman spaces, Bergman kernels, and Bergman metrics. In Banach-space theory it refers to the continuous dual, predual, or annihilator of a Bergman space under an integral pairing; in transform theory it refers to concrete realizations of that dual by holomorphic functions on a different domain or on $\mathbb C^n$ with an exponential weight; in Kähler geometry it denotes the metric built from the dualized kernel $K_D(z,-\bar z)$; and in bundle-theoretic settings it refers to Bergman kernels defined on dual line bundles or from inner products on $O(E)^*$ (Gori et al., 2019, Chatterjee, 3 Jun 2025, Loi et al., 7 Oct 2025, Ebenfelt et al., 25 Oct 2025, Lempert, 2021).

1. Terminological scope and basic models

The expression occurs in several technically distinct settings.

Context	Construction	Resulting object
Bergman-space duality	Integral pairing on $A^p(D)$ or weighted variants	Continuous dual, predual, or annihilator
Transform realization	Fantappié or Laplace transform of analytic functionals	Holomorphic space on $D^*$ or weighted space on $\mathbb C^n$
Geometric dualization	$K_D^*(z,\bar z)=K_D(z,-\bar z)$	Bergman dual $(D^*,g_D^*)$
Bundle/CR setting	Bergman kernel of disk bundle in $L^*$	“Dual” Bergman kernel form on $D\subset L^*$
Dual section space	Inner products on $K_D(z,-\bar z)$0	Bergman kernel $K_D(z,-\bar z)$1 and Bergman section

For a bounded domain $K_D(z,-\bar z)$2, the classical Bergman framework begins with holomorphic $K_D(z,-\bar z)$3-spaces and the Bergman projection. In the most standard reflexive regime, one expects duality through the pairing

$K_D(z,-\bar z)$4

However, the literature shows that this expectation is highly domain-dependent, changes at nonreflexive endpoints, and generalizes in different directions to Orlicz, meromorphic, quaternionic, and geometric settings (Bhat, 2024, Chakrabarti et al., 2018, Sehba et al., 2015, Ghiloufi et al., 2023, Vicente-Benítez, 2024).

The unifying feature is that duality is controlled by reproducing structures: either the Bergman kernel itself, an adapted projection operator, or a dualized kernel/potential. This suggests that “Bergman dual” is best understood as a family of kernel-mediated dual objects rather than a single invariant.

2. Classical Banach-space duality and its failure on irregular domains

For a bounded domain $K_D(z,-\bar z)$5, Bhat formulates the $K_D(z,-\bar z)$6-Bergman space

$K_D(z,-\bar z)$7

and proves that if the Bergman projection $K_D(z,-\bar z)$8 extends to a bounded operator on both $K_D(z,-\bar z)$9 and $O(E)^*$0, then for $O(E)^*$1 with $O(E)^*$2 there is a canonical isometric isomorphism

$O(E)^*$3

with each $O(E)^*$4 inducing

$O(E)^*$5

The same work states that the properties “$O(E)^*$6 extends boundedly on $O(E)^*$7 and on $O(E)^*$8,” “$O(E)^*$9 via the pairing,” and “$A^p(D)$0 is of weak type $A^p(D)$1 and $A^p(D)$2” are equivalent, and it further identifies the interpolation identity

$A^p(D)$3

as part of the same analytic package (Bhat, 2024).

This picture fails on general pseudoconvex domains. On the classical Hartogs triangle

$A^p(D)$4

Edholm and McNeal show that for $A^p(D)$5 and $A^p(D)$6, the monomial $A^p(D)$7 lies in $A^p(D)$8 but not in $A^p(D)$9. The coefficient functional $D^*$0 is bounded on $D^*$1 but cannot be represented by $D^*$2 with $D^*$3, and $D^*$4 is not dense in $D^*$5. Thus the naive pattern

$D^*$6

breaks down even in low dimension (Chakrabarti et al., 2018).

The same paper isolates a positive mechanism. If an integral operator $D^*$7 satisfies the three mapping properties (H1), (H2), and (H3), then the pairing map $D^*$8 is onto. On generalized Hartogs triangles $D^*$9, a “sub-Bergman” projection $\mathbb C^n$0 satisfies the required properties for each $\mathbb C^n$1, recovering a concrete dual description in terms of $\mathbb C^n$2-allowable Laurent monomials (Chakrabarti et al., 2018).

A common misconception is that Riesz-type Bergman duality is automatic for bounded domains. The available results show the opposite: duality depends on projection regularity, and projection regularity depends sharply on domain geometry.

3. Nonreflexive endpoints and generalized dual objects

At the nonreflexive endpoint $\mathbb C^n$3, the dual need not be another Bergman space. For the upper half-plane $\mathbb C^n$4 with weighted measure

$\mathbb C^n$5

Galanopoulos and Girela identify the predual of the nonreflexive weighted Bergman space $\mathbb C^n$6 as the little Bloch space vanishing at $\mathbb C^n$7,

$\mathbb C^n$8

with norm $\mathbb C^n$9. They prove

$K_D^*(z,\bar z)=K_D(z,-\bar z)$0

under the pairing

$K_D^*(z,\bar z)=K_D(z,-\bar z)$1

On the identified predual, the adjoint scaling and translation groups act by

$K_D^*(z,\bar z)=K_D(z,-\bar z)$2

and form groups of surjective isometries; for the scaling generator $K_D^*(z,\bar z)=K_D(z,-\bar z)$3, the point spectrum is empty, the full spectrum is $K_D^*(z,\bar z)=K_D(z,-\bar z)$4, and the resolvent norm equals $K_D^*(z,\bar z)=K_D(z,-\bar z)$5 (Gori et al., 2019).

Other generalized Bergman duals preserve the same principle but change the target category. For large Bergman–Orlicz spaces on the unit ball, Sehba and Tchoundja show two distinct duality regimes: when $K_D^*(z,\bar z)=K_D(z,-\bar z)$6 is convex and satisfies the $K_D^*(z,\bar z)=K_D(z,-\bar z)$7-condition together with its complementary function $K_D^*(z,\bar z)=K_D(z,-\bar z)$8, one has

$K_D^*(z,\bar z)=K_D(z,-\bar z)$9

whereas for concave $(D^*,g_D^*)$0 with $(D^*,g_D^*)$1,

$(D^*,g_D^*)$2

under the integral pairing $(D^*,g_D^*)$3 (Sehba et al., 2015).

For the meromorphic Bergman spaces on the pointed disc $(D^*,g_D^*)$4, the dual remains of the same meromorphic type: for $(D^*,g_D^*)$5 and Hölder conjugate $(D^*,g_D^*)$6,

$(D^*,g_D^*)$7

isometrically under

$(D^*,g_D^*)$8

despite the admissibility of controlled poles at $(D^*,g_D^*)$9 (Ghiloufi et al., 2023).

In the biquaternionic Vekua setting on bounded Liapunov domains $L^*$0, the Bergman space is $L^*$1, and its annihilator in $L^*$2 is identified by

$L^*$3

with the closure removed under the smallness condition $L^*$4 (Vicente-Benítez, 2024).

4. Transform realizations via Fantappié and Laplace theory

A different meaning of Bergman dual arises when the continuous dual of $L^*$5 is realized as a holomorphic function space on another domain. Chatterjee studies this for bounded domains $L^*$6 by embedding $L^*$7 into the analytic functionals $L^*$8 through

$L^*$9

The Fantappié transform is then defined by

$D\subset L^*$0

where

$D\subset L^*$1

For $D\subset L^*$2,

$D\subset L^*$3

If $D\subset L^*$4 is bounded, $D\subset L^*$5-smooth, strongly lineally-convex, contains $D\subset L^*$6, and $D\subset L^*$7 is also strongly convex, then

$D\subset L^*$8

is a topological isomorphism for every $D\subset L^*$9; in particular,

$K_D(z,-\bar z)$00

via the Fantappié transform (Chatterjee, 3 Jun 2025).

The same paper gives a Laplace realization. On $K_D(z,-\bar z)$01,

$K_D(z,-\bar z)$02

and on $K_D(z,-\bar z)$03,

$K_D(z,-\bar z)$04

With support function

$K_D(z,-\bar z)$05

and weight

$K_D(z,-\bar z)$06

the map

$K_D(z,-\bar z)$07

is a topological isomorphism, and the inverse is expressed through the $K_D(z,-\bar z)$08th-degree Borel transform

$K_D(z,-\bar z)$09

This extends the planar results of Napalkov Jr–Yulumukhametov from simply connected or convex domains to higher-dimensional strongly convex domains (Chatterjee, 3 Jun 2025).

The higher-dimensional theory is not a routine generalization. For the $K_D(z,-\bar z)$10-ball

$K_D(z,-\bar z)$11

$K_D(z,-\bar z)$12 and $K_D(z,-\bar z)$13 remain bounded and injective, but their ranges are proper subspaces of the corresponding target Bergman spaces. Mere convexity is therefore insufficient once $K_D(z,-\bar z)$14 (Chatterjee, 3 Jun 2025).

5. Geometric Bergman duals from the kernel $K_D(z,-\bar z)$15

In Kähler geometry, “Bergman dual” denotes a dual metric constructed directly from the Bergman kernel. For a bounded domain $K_D(z,-\bar z)$16 containing the origin, with Bergman kernel $K_D(z,-\bar z)$17, the diagonal kernel $K_D(z,-\bar z)$18 yields the Bergman metric $K_D(z,-\bar z)$19 through

$K_D(z,-\bar z)$20

The dualized kernel is defined by

$K_D(z,-\bar z)$21

Since $K_D(z,-\bar z)$22 is real-analytic near the origin, there is a maximal connected open neighborhood $K_D(z,-\bar z)$23 of $K_D(z,-\bar z)$24 on which $K_D(z,-\bar z)$25. The Bergman dual of $K_D(z,-\bar z)$26 is then $K_D(z,-\bar z)$27, with Kähler form

$K_D(z,-\bar z)$28

and local coefficients

$K_D(z,-\bar z)$29

When $K_D(z,-\bar z)$30 is a bounded symmetric domain, Harish–Chandra theory implies that $K_D(z,-\bar z)$31 never vanishes on $K_D(z,-\bar z)$32, so $K_D(z,-\bar z)$33; in that case $K_D(z,-\bar z)$34 is the compact dual restricted to its big cell (Loi et al., 7 Oct 2025).

For Cartan–Hartogs domains

$K_D(z,-\bar z)$35

the Bergman dual enters a rigidity theorem. With $K_D(z,-\bar z)$36 and Bergman metric $K_D(z,-\bar z)$37, the following are equivalent: $K_D(z,-\bar z)$38 is biholomorphic to the unit ball $K_D(z,-\bar z)$39; $K_D(z,-\bar z)$40 is Kähler–Einstein; $K_D(z,-\bar z)$41 is a Kähler–Ricci soliton with $K_D(z,-\bar z)$42; and, up to a positive constant $K_D(z,-\bar z)$43, the Bergman dual metric $K_D(z,-\bar z)$44 is finitely projectively induced. In the ball case, the dual Bergman metric on $K_D(z,-\bar z)$45 is $K_D(z,-\bar z)$46 times the Fubini–Study metric restricted to the affine chart, and the rescaling constant is $K_D(z,-\bar z)$47 (Loi et al., 7 Oct 2025).

This geometric duality comes with limitations. Completeness of $K_D(z,-\bar z)$48 does not imply completeness of $K_D(z,-\bar z)$49. The paper also poses open problems: if $K_D(z,-\bar z)$50, must the original $K_D(z,-\bar z)$51 be the ball? In a rank $K_D(z,-\bar z)$52 example with $K_D(z,-\bar z)$53, $K_D(z,-\bar z)$54 vanishes along a real hypersurface in $K_D(z,-\bar z)$55, so $K_D(z,-\bar z)$56, and the induced metric is incomplete with unbounded negative curvature near the boundary (Loi et al., 7 Oct 2025).

6. Dual Bergman kernels on bundles and spaces of sections

A further line of development shifts the dualization from domains in $K_D(z,-\bar z)$57 to line bundles and their disk bundles. Let $K_D(z,-\bar z)$58 be a polarized Kähler manifold, and let $K_D(z,-\bar z)$59 be the disk bundle of the dual line bundle. Ebenfelt, Xiao, and Xu define the Bergman space

$K_D(z,-\bar z)$60

and its Bergman kernel form

$K_D(z,-\bar z)$61

for an orthonormal basis $K_D(z,-\bar z)$62. If $K_D(z,-\bar z)$63, then $K_D(z,-\bar z)$64 admits a Fefferman-type boundary expansion, and the log-term is controlled by the Tian–Yau–Zelditch–Catlin coefficients $K_D(z,-\bar z)$65 of the twisted Bergman kernels

$K_D(z,-\bar z)$66

They prove that on an open set $K_D(z,-\bar z)$67, the vanishing of all coefficients $K_D(z,-\bar z)$68 for $K_D(z,-\bar z)$69 is equivalent to Bergman-logarithmic-flatness of the corresponding piece of the circle bundle $K_D(z,-\bar z)$70. As an application, if $K_D(z,-\bar z)$71 is compact and $K_D(z,-\bar z)$72 is locally homogeneous, then the circle bundle is Bergman logarithmically flat (Ebenfelt et al., 25 Oct 2025).

Lempert studies a related dual construction for a very ample line bundle $K_D(z,-\bar z)$73, but now from the finite-dimensional space $K_D(z,-\bar z)$74. For an arbitrary real-bilinear, Hermitian-symmetric inner product $K_D(z,-\bar z)$75 on $K_D(z,-\bar z)$76, there is a unique holomorphic section

$K_D(z,-\bar z)$77

satisfying the reproducing identity

$K_D(z,-\bar z)$78

Its diagonal restriction

$K_D(z,-\bar z)$79

is the Bergman section. This construction extends the Fubini–Study map by setting

$K_D(z,-\bar z)$80

The classical $K_D(z,-\bar z)$81 is an injective immersion, but its image is in general not closed in $K_D(z,-\bar z)$82. To obtain a closed range, one enlarges the domain to certain semidefinite forms on $K_D(z,-\bar z)$83: the map $K_D(z,-\bar z)$84 on the set $K_D(z,-\bar z)$85 of nonnegative forms with rather ample underlying subspace is injective and proper, and the closure of $K_D(z,-\bar z)$86 is exactly $K_D(z,-\bar z)$87 (Lempert, 2021).

These bundle-theoretic constructions show that Bergman duality is not restricted to Banach duals of holomorphic $K_D(z,-\bar z)$88-spaces. It also appears as a dualization of kernels, potentials, and section spaces, with boundary singularities, projective inducedness, and closure phenomena determined by curvature and ampleness data rather than by $K_D(z,-\bar z)$89-mapping alone.