---
title: Komatsu-Style Extended Gevrey Classes
url: https://www.emergentmind.com/topics/komatsu-style-extended-gevrey-classes
type: topic
---

# Komatsu-Style Extended Gevrey Classes

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 [1605.06910] [2209.00457] [2404.17366]. 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 [1901.00698] [1902.09206].

## 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
\[
(M.3)\quad \sum_{q=p+1}^{\infty}\frac{M_{q-1}}{M_q}\le A\,p\,\frac{M_p}{M_{p+1}},
\]
together with weaker variants \((M.2)'\) and \((M.3)'\) used in several extended-Gevrey treatments [2404.17366].

For the extended Gevrey sequence \(M_p^{\tau,\sigma}=p^{\tau p^\sigma}\), 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 \(T,\sigma\) the spaces \(E_{\tau,\sigma,h}(K)\) fail Komatsu’s \((M.2)\) in general, and that one regains a satisfactory ultradifferentiable theory by passing to unions or intersections over the matrix parameter [2209.00457]. Other expositions record weakened estimates such as \((M.2)'\) or power-type bounds of the form
\[
M_{p+q}^{\tau,\sigma}\le C^{\,p^\sigma+q^\sigma}M_p^{\tau,\sigma}M_q^{\tau,\sigma},
\]
which are sufficient for many continuity and microlocal arguments [1605.06910] [2404.17366].

The sequence \(p^{\tau p^\sigma}\) is an extension of the Gevrey scale in a precise limiting sense. As \(\sigma\downarrow 1\), one has \(M_p\approx p^{\tau p}\sim (p!)^\tau\) by Stirling’s formula, so the classes recover the classical Gevrey-\(\tau\) scale; as \(\tau\downarrow 0\), \(M_p\to 1\), and the corresponding class approaches \(C^\infty\) [2209.00457]. For every fixed \(\sigma>1\), there are continuous dense embeddings
\[
G^{s}(\mathbb R^d)\subset \bigcup_{T>0}\mathcal E_{\{T,\sigma\}}(\mathbb R^d)\subset C^\infty(\mathbb R^d),
\]
and the first inclusion is strict [2404.17366].

## 2. Local norms and global Roumieu–Beurling classes

Let \(U\subset \mathbb R^d\) be open, \(K\Subset U\) compact, and \(h>0\). In the normalization used by Teofanov–Tomić, the basic Komatsu–Carleman seminorm is
\[
\|\varphi\|_{\tau,\sigma,h,K}
=\sup_{x\in K,\ p\in\mathbb N_0,\ |\alpha|=p}
\frac{|D^\alpha\varphi(x)|}{h^{p^\sigma}\,p^{\tau p^\sigma}}.
\]
The corresponding Banach space \(E_{\tau,\sigma,h}(K)\) consists of all \(\varphi\in C^\infty(K)\) for which this seminorm is finite [2209.00457].

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

| Type | Local condition | Global realization |
|---|---|---|
| Roumieu | there exists \(h>0\) such that \(\|\varphi\|_{\tau,\sigma,h,K}<\infty\) | \(\varinjlim_{K\Subset U}\varinjlim_{h\to\infty}E_{\tau,\sigma,h}(K)\) |
| Beurling | for every \(h>0\), \(\|\varphi\|_{\tau,\sigma,h,K}<\infty\) | \(\varinjlim_{K\Subset U}\varprojlim_{h\to0^+}E_{\tau,\sigma,h}(K)\) |

This is the basic Komatsu-style distinction. In alternative notation one writes \(\mathcal E_{\{T,\sigma\}}(\Omega)\) and \(\mathcal E_{(T,\sigma)}(\Omega)\), with the local derivative bounds
\[
\sup_{x\in K}|D^\alpha f(x)|\le C\,h^{|\alpha|}\,M^{[T,\sigma]}_{|\alpha|}
\]
for Roumieu type, and the same bound for every \(h>0\) in the Beurling case [2404.17366].

Several structural consequences are already visible at this level. The classes are nontrivial because \((M.3)'\) holds, and compactly supported cutoffs exist in the non-quasianalytic regimes under discussion. In fact, for each \(a>0\) there exists a compactly supported \(\chi\in\mathcal E_{\{T,\sigma\}}(\mathbb R^d)\) with \(\int\chi=1\) and \(\chi\notin G^s\) for every \(s>1\), which explicitly shows that extended Gevrey regularity strictly contains all classical Gevrey classes [2404.17366].

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

A major development is the reinterpretation of these classes by weighted matrices. For fixed \((\tau,\sigma,h)\), one introduces the associated function
\[
T_{\tau,\sigma,h}(k)
:=\sup_{p\in\mathbb N_0}
\Bigl\{
\ln^+\!\Bigl(h^{p^\sigma}|k|^p/M_p^{T}\Bigr)
\Bigr\},
\qquad k\in\mathbb R^d,
\]
and then identifies \(T_{\tau,\sigma,h}(|k|)\) with a standard Braun–Meise–Taylor weight function \(\omega\). Writing \(\omega^*\) for the Young–Fenchel conjugate,
\[
\omega^*(p)=\sup_{t>0}\{p\,t-\omega(e^t)\},
\]
one obtains the alternative Banach spaces
\[
B_{\tau,\sigma,H}(K)
=
\Bigl\{
\varphi\in C^\infty(K):
\sup_{p,\ |\alpha|=p}\sup_{x\in K}
|D^\alpha\varphi(x)|\,e^{-H\omega^*(p)}<\infty
\Bigr\}.
\]
Their Roumieu and Beurling limits are formed exactly as for the \(E\)-spaces, now with respect to the matrix parameter \(H\) [2209.00457].

The central equivalence theorem of Teofanov–Tomić identifies the Komatsu-style and matrix-style descriptions. First, for each fixed \((\tau,\sigma,h,T)\), the function \(k\mapsto T_{\tau,\sigma,h}(k)\) is equivalent, in both growth directions, to a standard Braun–Meise–Taylor weight function. Second, the two weight matrices
\[
M^{(1)}=\{M^T\}_{T>0},\qquad M_p^T=p^{\tau p^\sigma},
\]
and
\[
M^{(2)}=\{e^{H\omega^*(p)}\}_{H>0}
\]
are equivalent in the sense of mutual majorization by suitable constants [2209.00457].

The space-level consequence is asymmetric. In the Beurling case,
\[
E_{(\tau,\sigma)}(U)=B_{(\tau,\sigma)}(U),
\]
while in the Roumieu case
\[
B_{\{\tau,\sigma\}}(U)\supsetneq E_{\{\tau,\sigma\}}(U).
\]
Accordingly, the matrix formalism gives an exact identification in the Beurling setting, but only a strict enlargement in the Roumieu setting [2209.00457].

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 \(p^{\tau p^\sigma}\) 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 [2209.00457].

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

To any weight sequence \(M\) one attaches the associated function
\[
T_M(h)=\sup_{p\ge0}\ln\frac{h^p}{M_p},
\qquad h>0,
\]
equivalently the Carleman-associated function \(p_M(h)=\exp(-T_M(h))\) [2404.17366]. For the extended Gevrey sequence \(M_p^{[T,\sigma]}=p^{T p^\sigma}\), the associated function admits a sharp asymptotic description in terms of the principal branch of the Lambert \(W\)-function:
\[
T_{[T,\sigma]}(h)\asymp
(\ln h)^{1/(\sigma-1)}
\,W(\ln h)^{-(\sigma-2)/(\sigma-1)},
\qquad h\to\infty,
\]
which is one of the characteristic signatures of the theory [2404.17366].

The Lambert \(W\)-function also enters the sharp Paley–Wiener theory for compactly supported extended Gevrey functions. Pilipović–Teofanov–Tomić introduced an associated function for \(M_p=p^{\tau p^\sigma}\), derived its sharp asymptotics in terms of \(W\), and used these estimates to prove a Paley–Wiener theorem in extended Gevrey regularity [1901.00698]. In the STFT-based treatment, if \(K\subset\mathbb R^d\) is compact and \(f\in D_{\tau,\sigma}(K)\), then for \(\sigma\in(1,2)\) the Fourier–Laplace transform extends to an entire function satisfying
\[
|\widehat f(z)|
\le
A\exp\bigl\{-T_{\tau,\sigma,B}(|\operatorname{Re}z|)+H_K(\operatorname{Im}z)\bigr\},
\]
and conversely any entire function satisfying this estimate arises from a unique \(f\in D_{\tau,\sigma}(K)\) [1902.09206].

A complementary description is furnished by the short-time Fourier transform. For a fixed nonzero window \(\phi\in S^{(1)}(\mathbb R^d)\), Teofanov–Tomić proved that a smooth function \(f\) belongs to \(E_{\tau,\sigma}(\mathbb R^d)\) if and only if its STFT satisfies
\[
|V_\phi f(x,\xi)|\le C\exp\{-T_{\tau,\sigma,h}(|\xi|)\},
\]
uniformly in \(x\), for suitable \(C,h>0\) [1902.09206]. 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, \((x_0,\xi_0)\notin \WF_{\{T,\sigma\}}(u)\) if there exist a cutoff \(\varphi\in\mathcal D_{\{T,\sigma\}}\) with \(\varphi\equiv1\) near \(x_0\), a conic neighborhood \(\Gamma\ni\xi_0\), and constants \(C,h>0\) such that
\[
|\widehat{\varphi u}(\xi)|
\le
C\exp\bigl(-T_{[T,\sigma]}(h,|\xi|)\bigr),
\qquad \xi\in\Gamma,
\]
and there is an equivalent STFT characterization using \(V_g u\) for windows \(g\in\mathcal D_{\{T,\sigma\}}\) [2404.17366].

## 5. Algebraic stability, superposition, and microlocal propagation

Despite the failure of the full Komatsu \((M.2)\) for fixed \(p^{\tau p^\sigma}\), the extended Gevrey classes retain a substantial algebraic calculus. Teofanov–Tomić–Žigić show that \(\mathcal E_{\{T,\sigma\}}(\mathbb R^d)\) and \(\mathcal E_{(T,\sigma)}(\mathbb R^d)\) are algebras under pointwise product and are invariant under finite-order partial derivatives [2404.17366]. They also prove inverse-closedness in the Beurling setting: if \(f\in\mathcal E_{(T,\sigma)}(\mathbb R^d)\) never vanishes, then \(1/f\in\mathcal E_{(T,\sigma)}(\mathbb R^d)\) [2404.17366].

A stronger structural statement is closure under superposition. Pilipović–Teofanov–Tomić prove that if
\[
f\in E^{\{\tau,\sigma\}}(U),\qquad g\in E^{\{\tau,\sigma\}}(V),\qquad g:V\to U,
\]
then
\[
f\circ g\in E^{\{\tau,\sigma\}}(V).
\]
The proof uses the Faà di Bruno formula together with a modified Faà di Bruno property for the rescaled sequence \(\widetilde M_p=M_p^{(\tau,\sigma)}/p!\) [1607.07176]. This superposition theorem immediately yields inverse-closedness in the Roumieu case for nonvanishing functions [1607.07176].

The operator theory is similarly robust. Teofanov–Tomić show that many constant-coefficient ultradifferential operators of infinite order act continuously on \(E_{\{\tau,\sigma\}}(\Omega)\) provided the coefficients satisfy bounds of the form
\[
|a_\alpha|\le A\frac{L^{|\alpha|}}{M_{|\alpha|}^{\tau,\sigma}},
\]
for suitable \(A,L>0\) [1605.06910]. 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 \(P(x,D)\) is a differential operator with coefficients in \(\mathcal E_{\{T,\sigma\}}\), then
\[
\WF_{\{T,\sigma\}}(Pu)\subset \WF_{\{T,\sigma\}}(u)
\subset \WF_{\{T,\sigma\}}(Pu)\cup \Char(P),
\]
and in the earlier formulation of Pilipović–Teofanov–Tomić, singularities in the characteristic set propagate along the null-bicharacteristic flow of the principal symbol [1607.07176] [2404.17366]. One concrete PDE application is the strictly hyperbolic Cauchy problem with coefficients that are Log-Lipschitz in time and spatially \(M_p\)-ultradifferentiable with \(M_p=p!^2\): well-posedness can be proved in \(\mathcal E_{\{T,2\}}\), while it fails in any smaller Gevrey class [2404.17366].

## 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, \((\mathcal E_{\{T,\sigma\}})'\) consists of ultradistributions \(u\) satisfying estimates of the form
\[
|\langle u,\varphi\rangle|
\le
C\sup_\alpha
\frac{\|D^\alpha\varphi\|_{L^\infty}}
{h^{|\alpha|}M^{[T,\sigma]}_{|\alpha|}},
\]
for suitable \(C,h>0\) and all test functions \(\varphi\in\mathcal E_{\{T,\sigma\}}\) [2404.17366]. This dual framework is used in local and microlocal analysis.

A further extension is the Gelfand–Shilov theory associated with the weight matrix
\[
\mathcal M^\sigma=\{M^{T,\sigma}:T>0\},\qquad M_p^{T,\sigma}=p^{T p^\sigma}.
\]
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 [2503.13634].

Wavelet theory has provided especially concrete examples of “beyond Gevrey” behavior. In one construction, a smooth orthonormal wavelet \(\psi\) is built so that both \(\psi\) and \(\widehat\psi\) belong to an extended Gevrey class \(\mathcal E_\sigma(\mathbb R)\) for \(\sigma>1\), while lying beyond all classical Gevrey classes. The construction uses invariant cycles, extends the Lemarié–Meyer support of the low-pass filter \(m_0\), and derives global decay estimates described by special functions involving the Lambert \(W\)-function [2512.05655]. The same paper states a uniform estimate, for suitable constants \(C,A,c>0\) and \(\alpha>0\),
\[
|\psi^{(n)}(x)|+|\widehat\psi^{(n)}(\xi)|
\le
C\,A^n\exp\bigl[-c\,n\,W(\alpha n)\bigr],
\]
which makes explicit how extended Gevrey derivative growth and sub-exponential decay are linked [2512.05655].

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 \(\sigma>1\), they strictly contain every \(G^s\) while remaining strictly smaller than \(C^\infty\) [2404.17366]. 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 [2209.00457]. That distinction is now part of the standard structure theory of the subject.

Source: https://www.emergentmind.com/topics/komatsu-style-extended-gevrey-classes