Gevrey–Beurling Classes in Ultradifferentiable Analysis

• Gevrey–Beurling classes are ultradifferentiable spaces defined by universal estimates, contrasting with Roumieu classes by requiring conditions to hold for every scale parameter.
• They feature precise Fourier decay and spectral operator characterizations that convert ultradifferentiability into practical domain conditions for semigroup generation and PDE analysis.
• Extended Gevrey classes and noncommutative realizations illustrate the broad applicability of Beurling methods in ultraholomorphic extensions, wavelet theory, and microlocal analysis.

Gevrey–Beurling classes are Beurling-type ultradifferentiable classes in which Gevrey growth is imposed with universal, rather than existential, control parameters. In the classical setting they arise from the weight sequence $m_n=[n!]^\beta$, but the same “for every scale parameter” principle also appears in Carleman classes of vectors for operators, in Fourier descriptions on compact Lie groups and tori, in ultraholomorphic classes on sectors, and in generalized weight-matrix or operator-algebraic settings beyond the classical Gevrey hierarchy (Markin, 2016, Dasgupta et al., 2012, Teofanov et al., 2024, Chen et al., 3 Jul 2026).

1. Foundational definitions and the Beurling quantifier

A standard operator-theoretic formulation begins with a densely defined closed linear operator $A$ in a complex Banach space $X$ and a sequence of positive numbers $\{m_n\}_{n=0}^\infty$. The space of smooth vectors is

$C^\infty(A):=\bigcap_{n=0}^{\infty}D(A^n),$

and the associated Carleman classes are

$$C_{\{m_n\}}(A):=\left\{f\in C^\infty(A)\mid \exists \alpha>0\ \exists c>0:\ \|A^n f\|\le c\alpha^n m_n,\ n\in\mathbb Z_+\right\},$$

$$C_{(m_n)}(A):=\left\{f\in C^\infty(A)\mid \forall \alpha>0\ \exists c>0:\ \|A^n f\|\le c\alpha^n m_n,\ n\in\mathbb Z_+\right\}.$$

The decisive distinction is the quantifier pattern: Roumieu type requires one admissible $\alpha>0$, whereas Beurling type requires the estimate for every $\alpha>0$ (Markin, 2016).

For Gevrey regularity one takes

$m_n=[n!]^\beta,$

equivalently $A$0, and defines

$A$1

Thus Gevrey–Beurling ultradifferentiability of order $A$2 means membership in $A$3. In the same framework, $A$4 consists of analytic vector functions, $A$5 of entire vector functions, and $A$6 correspond to entire functions or vectors of exponential and minimal exponential type (Markin, 2016).

On compact Lie groups the same distinction is expressed directly by derivatives. If $A$7 is a compact Lie group and $A$8, then the Gevrey–Beurling class $A$9 consists of $X$0 such that for every $X$1 there exists $X$2 with

$X$3

for all multi-indices $X$4. The paper places these spaces in the projective-limit topology, in contrast with the inductive-limit topology on the Roumieu side (Dasgupta et al., 2012).

A generalized Beurling analogue beyond the classical Gevrey sequence is given by the extended Gevrey classes

$X$5

with Beurling class

$X$6

defined by the requirement that for every compact $X$7 and every $X$8, there exists $X$9 such that

$\{m_n\}_{n=0}^\infty$0

Here again the Beurling feature is the universal quantifier in $\{m_n\}_{n=0}^\infty$1 (Teofanov et al., 2024).

2. Global Fourier and spectral descriptions

On compact Lie groups the Beurling quantifier becomes a global Fourier-decay condition. For $\{m_n\}_{n=0}^\infty$2,

$\{m_n\}_{n=0}^\infty$3

for all $\{m_n\}_{n=0}^\infty$4. The corresponding ultradistribution dual is characterized by the reversed quantifier: $\{m_n\}_{n=0}^\infty$5 The same pattern extends to compact homogeneous spaces after restriction to class-I representations with respect to the subgroup $\{m_n\}_{n=0}^\infty$6 (Dasgupta et al., 2012).

On the torus, Fourier-defined Gevrey spaces make the Beurling/Roumieu split especially transparent. For the Gevrey weight $\{m_n\}_{n=0}^\infty$7, $\{m_n\}_{n=0}^\infty$8,

$\{m_n\}_{n=0}^\infty$9

The projective limit

$C^\infty(A):=\bigcap_{n=0}^{\infty}D(A^n),$0

is identified as a Beurling-type Gevrey class, while

$C^\infty(A):=\bigcap_{n=0}^{\infty}D(A^n),$1

is its Roumieu counterpart. In this setting the linear cohomological equation is solvable in $C^\infty(A):=\bigcap_{n=0}^{\infty}D(A^n),$2 if and only if the frequency vector belongs to the arithmetic class $C^\infty(A):=\bigcap_{n=0}^{\infty}D(A^n),$3, and the nonlinear torus vector-field linearization problem satisfies the same projective-limit criterion (Bounemoura, 2020).

These Fourier descriptions are complemented by operator-spectrum descriptions. For scalar type spectral operators, ultradifferentiability can be converted into domain conditions for functions of the operator: $C^\infty(A):=\bigcap_{n=0}^{\infty}D(A^n),$4 where

$C^\infty(A):=\bigcap_{n=0}^{\infty}D(A^n),$5

This converts Beurling growth of iterates $C^\infty(A):=\bigcap_{n=0}^{\infty}D(A^n),$6 into projective domain membership for $C^\infty(A):=\bigcap_{n=0}^{\infty}D(A^n),$7 (Markin, 2016).

3. Semigroups and scalar type spectral operators

For a $C^\infty(A):=\bigcap_{n=0}^{\infty}D(A^n),$8-semigroup $C^\infty(A):=\bigcap_{n=0}^{\infty}D(A^n),$9, Beurling ultradifferentiability means that every orbit belongs to

$$C_{\{m_n\}}(A):=\left\{f\in C^\infty(A)\mid \exists \alpha>0\ \exists c>0:\ \|A^n f\|\le c\alpha^n m_n,\ n\in\mathbb Z_+\right\},$$0

In the scalar type spectral setting, the main generation theorem states that if $$C_{\{m_n\}}(A):=\left\{f\in C^\infty(A)\mid \exists \alpha>0\ \exists c>0:\ \|A^n f\|\le c\alpha^n m_n,\ n\in\mathbb Z_+\right\},$$1 satisfies the conditions $$C_{\{m_n\}}(A):=\left\{f\in C^\infty(A)\mid \exists \alpha>0\ \exists c>0:\ \|A^n f\|\le c\alpha^n m_n,\ n\in\mathbb Z_+\right\},$$2 and $$C_{\{m_n\}}(A):=\left\{f\in C^\infty(A)\mid \exists \alpha>0\ \exists c>0:\ \|A^n f\|\le c\alpha^n m_n,\ n\in\mathbb Z_+\right\},$$3, then a scalar type spectral operator $$C_{\{m_n\}}(A):=\left\{f\in C^\infty(A)\mid \exists \alpha>0\ \exists c>0:\ \|A^n f\|\le c\alpha^n m_n,\ n\in\mathbb Z_+\right\},$$4 generates a $$C_{\{m_n\}}(A):=\left\{f\in C^\infty(A)\mid \exists \alpha>0\ \exists c>0:\ \|A^n f\|\le c\alpha^n m_n,\ n\in\mathbb Z_+\right\},$$5-semigroup if and only if, for any $$C_{\{m_n\}}(A):=\left\{f\in C^\infty(A)\mid \exists \alpha>0\ \exists c>0:\ \|A^n f\|\le c\alpha^n m_n,\ n\in\mathbb Z_+\right\},$$6, there is an $$C_{\{m_n\}}(A):=\left\{f\in C^\infty(A)\mid \exists \alpha>0\ \exists c>0:\ \|A^n f\|\le c\alpha^n m_n,\ n\in\mathbb Z_+\right\},$$7 such that

$$C_{\{m_n\}}(A):=\left\{f\in C^\infty(A)\mid \exists \alpha>0\ \exists c>0:\ \|A^n f\|\le c\alpha^n m_n,\ n\in\mathbb Z_+\right\},$$8

The Beurling feature is again the universal quantifier: for every $$C_{\{m_n\}}(A):=\left\{f\in C^\infty(A)\mid \exists \alpha>0\ \exists c>0:\ \|A^n f\|\le c\alpha^n m_n,\ n\in\mathbb Z_+\right\},$$9 there must exist a corresponding $$C_{(m_n)}(A):=\left\{f\in C^\infty(A)\mid \forall \alpha>0\ \exists c>0:\ \|A^n f\|\le c\alpha^n m_n,\ n\in\mathbb Z_+\right\}.$$0. The paper contrasts this with the Roumieu criterion, where one only asks for the existence of one pair $$C_{(m_n)}(A):=\left\{f\in C^\infty(A)\mid \forall \alpha>0\ \exists c>0:\ \|A^n f\|\le c\alpha^n m_n,\ n\in\mathbb Z_+\right\}.$$1 (Markin, 2016).

For the Gevrey sequence $$C_{(m_n)}(A):=\left\{f\in C^\infty(A)\mid \forall \alpha>0\ \exists c>0:\ \|A^n f\|\le c\alpha^n m_n,\ n\in\mathbb Z_+\right\}.$$2, the decisive regime is $$C_{(m_n)}(A):=\left\{f\in C^\infty(A)\mid \forall \alpha>0\ \exists c>0:\ \|A^n f\|\le c\alpha^n m_n,\ n\in\mathbb Z_+\right\}.$$3. In that case the spectral condition becomes

$$C_{(m_n)}(A):=\left\{f\in C^\infty(A)\mid \forall \alpha>0\ \exists c>0:\ \|A^n f\|\le c\alpha^n m_n,\ n\in\mathbb Z_+\right\}.$$4

again with

$$C_{(m_n)}(A):=\left\{f\in C^\infty(A)\mid \forall \alpha>0\ \exists c>0:\ \|A^n f\|\le c\alpha^n m_n,\ n\in\mathbb Z_+\right\}.$$5

This is the Gevrey–Beurling generation criterion for scalar type spectral operators. The same paper notes that $$C_{(m_n)}(A):=\left\{f\in C^\infty(A)\mid \forall \alpha>0\ \exists c>0:\ \|A^n f\|\le c\alpha^n m_n,\ n\in\mathbb Z_+\right\}.$$6 may be replaced by $$C_{(m_n)}(A):=\left\{f\in C^\infty(A)\mid \forall \alpha>0\ \exists c>0:\ \|A^n f\|\le c\alpha^n m_n,\ n\in\mathbb Z_+\right\}.$$7, so that $$C_{(m_n)}(A):=\left\{f\in C^\infty(A)\mid \forall \alpha>0\ \exists c>0:\ \|A^n f\|\le c\alpha^n m_n,\ n\in\mathbb Z_+\right\}.$$8 is equivalent to $$C_{(m_n)}(A):=\left\{f\in C^\infty(A)\mid \forall \alpha>0\ \exists c>0:\ \|A^n f\|\le c\alpha^n m_n,\ n\in\mathbb Z_+\right\}.$$9 (Markin, 2016).

The low-order regime behaves differently. For $\alpha>0$0, the sequence $\alpha>0$1 does not satisfy $\alpha>0$2, so the theorem does not apply. For $\alpha>0$3, if a scalar type spectral operator generates an $\alpha>0$4-semigroup, then all orbits are entire vector functions; consequently the generator belongs to $\alpha>0$5, and the semigroup is uniformly continuous, indeed an entire semigroup of exponential type. The paper presents this as a collapse of the low-order Beurling theory to bounded generators (Markin, 2016).

A rapidly growing non-Gevrey example is also treated. For

$\alpha>0$6

the relevant comparison function is

$\alpha>0$7

and the generation criterion becomes

$\alpha>0$8

for every $\alpha>0$9, some $\alpha>0$0. This shows that the Beurling mechanism is not confined to factorial or power-like weights (Markin, 2016).

4. Extended Gevrey and generalized Beurling scales

A major generalization replaces the classical Gevrey sequence by

$\alpha>0$1

The corresponding Beurling spaces $\alpha>0$2 and compactly supported spaces $\alpha>0$3 are explicitly defined, and the paper proves continuous dense inclusions

$\alpha>0$4

It also records the existence of compactly supported elements in $\alpha>0$5 that do not belong to any Gevrey class, so the scale is strictly beyond classical Gevrey regularity (Teofanov et al., 2024).

These extended classes retain several Komatsu-type properties, including log-convexity and nonquasianalyticity, but they fail the standard moderate growth condition $\alpha>0$6. For that reason the paper places them outside the standard Komatsu framework and closer to generalized weight-sequence or weight-matrix theories (Teofanov et al., 2024).

Their associated function is no longer power-like. For

$\alpha>0$7

the paper derives sharp asymptotics in terms of the Lambert $\alpha>0$8-function. This Lambert-$\alpha>0$9 behavior reappears in Fourier decay, in Paley–Wiener estimates, and in microlocal characterizations (Teofanov et al., 2024).

A related construction is the weight-matrix scale

$m_n=[n!]^\beta,$0

built from the same sequences $m_n=[n!]^\beta,$1. The wavelet paper emphasizes that this is not a standard Komatsu Beurling class attached to one fixed sequence, because the defining sequences fail Komatsu’s $m_n=[n!]^\beta,$2. Instead it is described as a weight-matrix class closely connected to Braun–Meise–Taylor theory, associated functions, and Paley–Wiener decay (Tomić et al., 5 Dec 2025).

In that framework one has a Fourier-side class $m_n=[n!]^\beta,$3 defined by

$m_n=[n!]^\beta,$4

where

$m_n=[n!]^\beta,$5

The Paley–Wiener theorem used there states that

$m_n=[n!]^\beta,$6

and conversely $m_n=[n!]^\beta,$7 implies $m_n=[n!]^\beta,$8. The same paper constructs an orthonormal wavelet $m_n=[n!]^\beta,$9 such that both $A$00 and $A$01 belong to $A$02 while lying outside all lower extended classes up to the threshold $A$03 (Tomić et al., 5 Dec 2025).

5. Microlocal and noncommutative realizations

Extended Gevrey regularity also has a microlocal formulation. For $A$04, the extended wave front set $A$05 is defined by the existence of a cutoff $A$06 and a conic neighborhood $A$07 such that

$A$08

The same paper gives a short-time Fourier transform characterization in terms of

$A$09

where

$A$10

This places Beurling-type regularity into the same microlocal framework as classical Gevrey wave front sets, but with logarithmic-Lambert decay replacing the stretched exponentials of the ordinary Gevrey scale (Teofanov et al., 2024).

A noncommutative realization appears for convolution operators on locally compact groups with a locally bounded length function $A$11. For $A$12 and $A$13, the weighted kernel space

$A$14

is explicitly described as Beurling type, because it is a projective limit over all $A$15. On the operator side, if $A$16 is the commutator derivation with the multiplication operator $A$17, the paper defines

$A$18

and in the group case

$A$19

This Gevrey–Beurling operator algebra is inverse-closed in $A$20, and if $A$21 with $A$22 is invertible in the unitized $A$23-pseudofunction algebra, then $A$24 belongs to the same Gevrey–Beurling operator algebra and satisfies explicit Gevrey seminorm estimates (Chen et al., 3 Jul 2026).

The same paper proves that, under $A$25 and unimodularity, $A$26 is a dense Fréchet subalgebra stable under holomorphic functional calculus, and the inclusion

$A$27

induces an isomorphism in topological $A$28-theory. This shows that Beurling-type subexponential decay and Gevrey commutator estimates can coexist with strong spectral-invariance properties in a genuinely noncommutative setting (Chen et al., 3 Jul 2026).

6. Ultrarolomorphic thresholds, PDE scope, and recurring boundaries

In ultraholomorphic theory, Beurling classes are built from

$A$29

and the asymptotic Borel map

$A$30

is studied on sectors $A$31. For the Gevrey sequence $A$32, the growth index satisfies $A$33. The paper proves that for regular sequences, and hence for Gevrey–Beurling classes, one has global extension operators and surjectivity for every

$A$34

while surjectivity implies

$A$35

Thus the classical Gevrey–Beurling Borel–Ritt threshold is recovered in sharp form (Jiménez-Garrido et al., 14 May 2025).

A recurring boundary in applications is that many PDE results are formulated on the Roumieu side, not the Beurling side. For weakly hyperbolic equations with Gevrey coefficients and real characteristic roots, one has Gevrey well-posedness for $A$36, and the note constructs examples showing that, in general, one cannot improve this to $A$37. The paper explicitly works with the standard Roumieu local Gevrey estimate and does not formulate the Beurling variant (Nishitani, 2022).

The same limitation is stated in more recent evolution-equation papers. For third-order degenerate evolution equations, the spaces $A$38, $A$39, and $A$40 are described as Roumieu-type, and the paper remarks that a Beurling analogue would require a projective condition such as

$A$41

Likewise, for variable-coefficient $A$42-evolution equations, the well-posedness theorem is proved in the inductive Gevrey-Sobolev scale $A$43, and the authors state that similar results may be obtained in the projective Gevrey space under suitable projective Gevrey estimates (Junior et al., 15 May 2026, Junior et al., 2024).

This division between projective Beurling scales and inductive Roumieu scales is one of the persistent structural facts of the subject. In definitions, Fourier decay, semigroup generation, ultraholomorphic extension, and operator algebra theory, Gevrey–Beurling classes are characterized by universal estimates over all radii or scales. In applications to PDE and dynamics, those universal estimates are often stronger than what current techniques directly yield.