Komatsu-Style Extended Gevrey Classes

• Komatsu-style extended Gevrey classes are defined by the sequence Mp = p^(τ p^σ), establishing a smoothness scale strictly between classical Gevrey classes and C∞.
• The framework uses local Banach and global Roumieu–Beurling constructions alongside weighted matrix reformulations and associated functions involving the Lambert W-function.
• These classes support robust algebraic operations and microlocal propagation results, ensuring operator continuity and precise control of singularities in PDE applications.

Komatsu-style extended Gevrey classes are ultradifferentiable classes generated by the two-parameter sequence

$$M_p^{\tau,\sigma}=p^{\tau p^\sigma},\qquad \tau>0,\ \sigma>1,$$

or, in equivalent notation used in part of the literature, $M_p^{[T,\sigma]}=p^{T p^\sigma}$. They were introduced to describe regularity strictly weaker than any classical Gevrey class but still stronger than unrestricted $C^\infty$ smoothness, and they now appear in Komatsu-type, weighted-matrix, STFT, microlocal, Gelfand–Shilov, and wavelet frameworks (Teofanov et al., 2016, Teofanov et al., 2022, Teofanov et al., 2024). For fixed $\sigma>1$, the resulting scales refine the interval between $\bigcup_{s>1}G^s$ and $C^\infty$, while preserving many of the structural features expected of non-quasianalytic ultradifferentiable theories (Pilipović et al., 2019, Teofanov et al., 2019).

1. Position within Komatsu’s ultradifferentiable framework

Komatsu’s classical framework starts from a positive sequence $M=(M_p)_{p\in\mathbb N_0}$, $M_0=1$, and imposes log-convexity, moderate growth, and strong non-quasianalyticity. In the notation recorded in later surveys, these are the conditions

$(M.1)\quad M_p^2\le M_{p-1}M_{p+1},$

$(M.2)\quad M_{p+q}\le A\,B^{p+q}M_pM_q,$

and

$M_p^{[T,\sigma]}=p^{T p^\sigma}$0

together with weaker variants $M_p^{[T,\sigma]}=p^{T p^\sigma}$1 and $M_p^{[T,\sigma]}=p^{T p^\sigma}$2 used in several extended-Gevrey treatments (Teofanov et al., 2024).

For the extended Gevrey sequence $M_p^{[T,\sigma]}=p^{T p^\sigma}$3, the literature agrees on log-convexity and non-quasianalyticity, but it distinguishes between the fixed-sequence and matrix viewpoints. Teofanov–Tomić emphasize that for fixed $M_p^{[T,\sigma]}=p^{T p^\sigma}$4 the spaces $M_p^{[T,\sigma]}=p^{T p^\sigma}$5 fail Komatsu’s $M_p^{[T,\sigma]}=p^{T p^\sigma}$6 in general, and that one regains a satisfactory ultradifferentiable theory by passing to unions or intersections over the matrix parameter (Teofanov et al., 2022). Other expositions record weakened estimates such as $M_p^{[T,\sigma]}=p^{T p^\sigma}$7 or power-type bounds of the form

$M_p^{[T,\sigma]}=p^{T p^\sigma}$8

which are sufficient for many continuity and microlocal arguments (Teofanov et al., 2016, Teofanov et al., 2024).

The sequence $M_p^{[T,\sigma]}=p^{T p^\sigma}$9 is an extension of the Gevrey scale in a precise limiting sense. As $C^\infty$0, one has $C^\infty$1 by Stirling’s formula, so the classes recover the classical Gevrey-$C^\infty$2 scale; as $C^\infty$3, $C^\infty$4, and the corresponding class approaches $C^\infty$5 (Teofanov et al., 2022). For every fixed $C^\infty$6, there are continuous dense embeddings

$C^\infty$7

and the first inclusion is strict (Teofanov et al., 2024).

2. Local norms and global Roumieu–Beurling classes

Let $C^\infty$8 be open, $C^\infty$9 compact, and $\sigma>1$0. In the normalization used by Teofanov–Tomić, the basic Komatsu–Carleman seminorm is

$\sigma>1$1

The corresponding Banach space $\sigma>1$2 consists of all $\sigma>1$3 for which this seminorm is finite (Teofanov et al., 2022).

From these local Banach spaces one forms the two standard global classes.

Type	Local condition	Global realization
Roumieu	there exists $\sigma>1$4 such that $\sigma>1$5	$\sigma>1$6
Beurling	for every $\sigma>1$7, $\sigma>1$8	$\sigma>1$9

This is the basic Komatsu-style distinction. In alternative notation one writes $\bigcup_{s>1}G^s$0 and $\bigcup_{s>1}G^s$1, with the local derivative bounds

$\bigcup_{s>1}G^s$2

for Roumieu type, and the same bound for every $\bigcup_{s>1}G^s$3 in the Beurling case (Teofanov et al., 2024).

Several structural consequences are already visible at this level. The classes are nontrivial because $\bigcup_{s>1}G^s$4 holds, and compactly supported cutoffs exist in the non-quasianalytic regimes under discussion. In fact, for each $\bigcup_{s>1}G^s$5 there exists a compactly supported $\bigcup_{s>1}G^s$6 with $\bigcup_{s>1}G^s$7 and $\bigcup_{s>1}G^s$8 for every $\bigcup_{s>1}G^s$9, which explicitly shows that extended Gevrey regularity strictly contains all classical Gevrey classes (Teofanov et al., 2024).

3. Weighted matrices and the Komatsu–Braun–Meise–Taylor correspondence

A major development is the reinterpretation of these classes by weighted matrices. For fixed $C^\infty$0, one introduces the associated function

$C^\infty$1

and then identifies $C^\infty$2 with a standard Braun–Meise–Taylor weight function $C^\infty$3. Writing $C^\infty$4 for the Young–Fenchel conjugate,

$C^\infty$5

one obtains the alternative Banach spaces

$C^\infty$6

Their Roumieu and Beurling limits are formed exactly as for the $C^\infty$7-spaces, now with respect to the matrix parameter $C^\infty$8 (Teofanov et al., 2022).

The central equivalence theorem of Teofanov–Tomić identifies the Komatsu-style and matrix-style descriptions. First, for each fixed $C^\infty$9, the function $M=(M_p)_{p\in\mathbb N_0}$0 is equivalent, in both growth directions, to a standard Braun–Meise–Taylor weight function. Second, the two weight matrices

$M=(M_p)_{p\in\mathbb N_0}$1

and

$M=(M_p)_{p\in\mathbb N_0}$2

are equivalent in the sense of mutual majorization by suitable constants (Teofanov et al., 2022).

The space-level consequence is asymmetric. In the Beurling case,

$M=(M_p)_{p\in\mathbb N_0}$3

while in the Roumieu case

$M=(M_p)_{p\in\mathbb N_0}$4

Accordingly, the matrix formalism gives an exact identification in the Beurling setting, but only a strict enlargement in the Roumieu setting (Teofanov et al., 2022).

This point is one of the main terminological subtleties of the subject. The label “Komatsu-style” refers to the derivative-growth format and the associated Carleman construction, but the fixed sequence $M=(M_p)_{p\in\mathbb N_0}$5 does not behave like a classical strongly regular Komatsu sequence in every respect. The weighted-matrix reformulation is therefore not merely cosmetic; it resolves the moderate-growth defect by passing from a single sequence to an equivalent family of admissible weights (Teofanov et al., 2022).

4. Associated functions, Fourier decay, and time–frequency characterizations

To any weight sequence $M=(M_p)_{p\in\mathbb N_0}$6 one attaches the associated function

$M=(M_p)_{p\in\mathbb N_0}$7

equivalently the Carleman-associated function $M=(M_p)_{p\in\mathbb N_0}$8 (Teofanov et al., 2024). For the extended Gevrey sequence $M=(M_p)_{p\in\mathbb N_0}$9, the associated function admits a sharp asymptotic description in terms of the principal branch of the Lambert $M_0=1$0-function: $M_0=1$1 which is one of the characteristic signatures of the theory (Teofanov et al., 2024).

The Lambert $M_0=1$2-function also enters the sharp Paley–Wiener theory for compactly supported extended Gevrey functions. Pilipović–Teofanov–Tomić introduced an associated function for $M_0=1$3, derived its sharp asymptotics in terms of $M_0=1$4, and used these estimates to prove a Paley–Wiener theorem in extended Gevrey regularity (Pilipović et al., 2019). In the STFT-based treatment, if $M_0=1$5 is compact and $M_0=1$6, then for $M_0=1$7 the Fourier–Laplace transform extends to an entire function satisfying

$M_0=1$8

and conversely any entire function satisfying this estimate arises from a unique $M_0=1$9 (Teofanov et al., 2019).

A complementary description is furnished by the short-time Fourier transform. For a fixed nonzero window $(M.1)\quad M_p^2\le M_{p-1}M_{p+1},$0, Teofanov–Tomić proved that a smooth function $(M.1)\quad M_p^2\le M_{p-1}M_{p+1},$1 belongs to $(M.1)\quad M_p^2\le M_{p-1}M_{p+1},$2 if and only if its STFT satisfies

$(M.1)\quad M_p^2\le M_{p-1}M_{p+1},$3

uniformly in $(M.1)\quad M_p^2\le M_{p-1}M_{p+1},$4, for suitable $(M.1)\quad M_p^2\le M_{p-1}M_{p+1},$5 (Teofanov et al., 2019). This yields both a global regularity criterion and a microlocal localization principle.

Wave-front sets are defined by the same decay pattern after localization. In the Fourier formulation, $(M.1)\quad M_p^2\le M_{p-1}M_{p+1},$6 if there exist a cutoff $(M.1)\quad M_p^2\le M_{p-1}M_{p+1},$7 with $(M.1)\quad M_p^2\le M_{p-1}M_{p+1},$8 near $(M.1)\quad M_p^2\le M_{p-1}M_{p+1},$9, a conic neighborhood $(M.2)\quad M_{p+q}\le A\,B^{p+q}M_pM_q,$0, and constants $(M.2)\quad M_{p+q}\le A\,B^{p+q}M_pM_q,$1 such that

$(M.2)\quad M_{p+q}\le A\,B^{p+q}M_pM_q,$2

and there is an equivalent STFT characterization using $(M.2)\quad M_{p+q}\le A\,B^{p+q}M_pM_q,$3 for windows $(M.2)\quad M_{p+q}\le A\,B^{p+q}M_pM_q,$4 (Teofanov et al., 2024).

5. Algebraic stability, superposition, and microlocal propagation

Despite the failure of the full Komatsu $(M.2)\quad M_{p+q}\le A\,B^{p+q}M_pM_q,$5 for fixed $(M.2)\quad M_{p+q}\le A\,B^{p+q}M_pM_q,$6, the extended Gevrey classes retain a substantial algebraic calculus. Teofanov–Tomić–Žigić show that $(M.2)\quad M_{p+q}\le A\,B^{p+q}M_pM_q,$7 and $(M.2)\quad M_{p+q}\le A\,B^{p+q}M_pM_q,$8 are algebras under pointwise product and are invariant under finite-order partial derivatives (Teofanov et al., 2024). They also prove inverse-closedness in the Beurling setting: if $(M.2)\quad M_{p+q}\le A\,B^{p+q}M_pM_q,$9 never vanishes, then $M_p^{[T,\sigma]}=p^{T p^\sigma}$00 (Teofanov et al., 2024).

A stronger structural statement is closure under superposition. Pilipović–Teofanov–Tomić prove that if

$M_p^{[T,\sigma]}=p^{T p^\sigma}$01

then

$M_p^{[T,\sigma]}=p^{T p^\sigma}$02

The proof uses the Faà di Bruno formula together with a modified Faà di Bruno property for the rescaled sequence $M_p^{[T,\sigma]}=p^{T p^\sigma}$03 (Pilipović et al., 2016). This superposition theorem immediately yields inverse-closedness in the Roumieu case for nonvanishing functions (Pilipović et al., 2016).

The operator theory is similarly robust. Teofanov–Tomić show that many constant-coefficient ultradifferential operators of infinite order act continuously on $M_p^{[T,\sigma]}=p^{T p^\sigma}$04 provided the coefficients satisfy bounds of the form

$M_p^{[T,\sigma]}=p^{T p^\sigma}$05

for suitable $M_p^{[T,\sigma]}=p^{T p^\sigma}$06 (Teofanov et al., 2016). This is important because it demonstrates that the loss of classical moderate growth does not destroy operator continuity for the classes of interest.

Microlocally, the theory supports a full propagation-of-singularities statement. If $M_p^{[T,\sigma]}=p^{T p^\sigma}$07 is a differential operator with coefficients in $M_p^{[T,\sigma]}=p^{T p^\sigma}$08, then

$M_p^{[T,\sigma]}=p^{T p^\sigma}$09

and in the earlier formulation of Pilipović–Teofanov–Tomić, singularities in the characteristic set propagate along the null-bicharacteristic flow of the principal symbol (Pilipović et al., 2016, Teofanov et al., 2024). One concrete PDE application is the strictly hyperbolic Cauchy problem with coefficients that are Log-Lipschitz in time and spatially $M_p^{[T,\sigma]}=p^{T p^\sigma}$10-ultradifferentiable with $M_p^{[T,\sigma]}=p^{T p^\sigma}$11: well-posedness can be proved in $M_p^{[T,\sigma]}=p^{T p^\sigma}$12, while it fails in any smaller Gevrey class (Teofanov et al., 2024).

6. Later developments: duality, Gelfand–Shilov theory, and wavelets beyond Gevrey

The later literature extends the Komatsu-style classes in several directions. On the dual side, $M_p^{[T,\sigma]}=p^{T p^\sigma}$13 consists of ultradistributions $M_p^{[T,\sigma]}=p^{T p^\sigma}$14 satisfying estimates of the form

$M_p^{[T,\sigma]}=p^{T p^\sigma}$15

for suitable $M_p^{[T,\sigma]}=p^{T p^\sigma}$16 and all test functions $M_p^{[T,\sigma]}=p^{T p^\sigma}$17 (Teofanov et al., 2024). This dual framework is used in local and microlocal analysis.

A further extension is the Gelfand–Shilov theory associated with the weight matrix

$M_p^{[T,\sigma]}=p^{T p^\sigma}$18

Teofanov–Tomić–Žigić show that the corresponding Roumieu and Beurling Gelfand–Shilov spaces admit symmetric characterizations in terms of polynomial weights, derivative bounds, and Fourier transform estimates; they are Fourier invariant, nuclear, and stable under standard time–frequency representations such as the short-time Fourier transform, cross-Wigner distribution, Grossmann–Royer transform, and cross-ambiguity function (Teofanov et al., 17 Mar 2025).

Wavelet theory has provided especially concrete examples of “beyond Gevrey” behavior. In one construction, a smooth orthonormal wavelet $M_p^{[T,\sigma]}=p^{T p^\sigma}$19 is built so that both $M_p^{[T,\sigma]}=p^{T p^\sigma}$20 and $M_p^{[T,\sigma]}=p^{T p^\sigma}$21 belong to an extended Gevrey class $M_p^{[T,\sigma]}=p^{T p^\sigma}$22 for $M_p^{[T,\sigma]}=p^{T p^\sigma}$23, while lying beyond all classical Gevrey classes. The construction uses invariant cycles, extends the Lemarié–Meyer support of the low-pass filter $M_p^{[T,\sigma]}=p^{T p^\sigma}$24, and derives global decay estimates described by special functions involving the Lambert $M_p^{[T,\sigma]}=p^{T p^\sigma}$25-function (Tomić et al., 5 Dec 2025). The same paper states a uniform estimate, for suitable constants $M_p^{[T,\sigma]}=p^{T p^\sigma}$26 and $M_p^{[T,\sigma]}=p^{T p^\sigma}$27,

$M_p^{[T,\sigma]}=p^{T p^\sigma}$28

which makes explicit how extended Gevrey derivative growth and sub-exponential decay are linked (Tomić et al., 5 Dec 2025).

Two recurrent sources of confusion are clarified by these later developments. First, the extended classes are not merely Gevrey classes with a different parameter; for fixed $M_p^{[T,\sigma]}=p^{T p^\sigma}$29, they strictly contain every $M_p^{[T,\sigma]}=p^{T p^\sigma}$30 while remaining strictly smaller than $M_p^{[T,\sigma]}=p^{T p^\sigma}$31 (Teofanov et al., 2024). Second, the Komatsu-style sequence definition and the weighted-matrix definition do not coincide identically in all regimes: the Beurling classes agree, whereas the Roumieu matrix class is strictly larger than the corresponding inductive limit over the original extended Gevrey sequences (Teofanov et al., 2022). That distinction is now part of the standard structure theory of the subject.