Non-Homogeneous Carleman Classes

• Non-homogeneous Carleman classes are function spaces defined by allowing distinct weight sequences to control derivative growth, generalizing the classical Gevrey and Carleman frameworks.
• They enable refined analysis in summability methods, multisummability, and the study of differential and difference equations, offering tools beyond traditional analytic regimes.
• The framework supports operator theory and PDE applications through explicit constructions like optimal flat functions, Borel-map inversion, and tailored extension operators.

Non-homogeneous Carleman classes generalize the classical Carleman and Denjoy–Carleman classes of functions by relaxing the requirement that smoothness and growth are governed by a single homogeneous sequence of weights. Instead, these classes permit distinct, potentially inhomogeneous, control of the derivatives or other generalized growth features, enabling a richer analysis of function and series spaces arising in analytic, asymptotic, and summability problems. Their development has been pivotal in advancing multisummation theory, generalized Laplace/Borel transforms, and the resolution of complex analytic and differential-analytic structures beyond the Gevrey setting.

1. Foundational Concepts: Carleman and Denjoy–Carleman Classes

The classical Carleman class $C^M(U)$, associated with a weight sequence $M = (M_n)_{n\geq 0}$, comprises $C^\infty$ functions $f$ on $U$ satisfying, locally, $|f^{(n)}(x)| \leq A B^n M_n$ for all $n\geq 0$. The Denjoy–Carleman classes are distinguished by the log-convexity of $M$—$M_n^2 \leq M_{n-1}M_{n+1}$—and further, the quasianalyticity criterion relates to the divergence of $\sum M_n/M_{n+1}$ (Buhovsky et al., 2018). In the classical “homogeneous” setting, $M$ is typically of the form $M_n = (n!)^\alpha$, corresponding to Gevrey classes.

Non-homogeneous Carleman classes break this uniformity, permitting more intricate structure in the control sequences and hence supporting a broader range of growth phenomena.

2. Non-Homogeneous Carleman Classes: Definitions and Prototypes

Let $M = (M_n)$ and $N = (N_n)$ be sequences of positive numbers. For an interval $I \subset \mathbb{R}$ and parameter $\eta > 0$, the non-homogeneous Carleman class $B_{\eta}(M, N; I)$ is defined by

• $f \in C^\infty(I)$ such that for every compact $J \subset I$, there exists $C>0$ with
  • $$\forall n\geq 0, \ \forall x \in J,\ |f^{(n)}(x)| \leq C^{n+1} M_n$$
  • $\forall n\geq 0, \ \forall \xi$ with $$e^\xi \in J, \ |d^n/d\xi^n f(e^\xi)| \leq C \eta^n N_n$$

The first inequality enforces Carleman-type control on the derivatives, while the second tracks the growth after logarithmic reparametrization, introducing genuine non-homogeneity (Kiro, 11 Jan 2026). If $N_n = M_n$ and $\eta = 1$, the classical homogeneous Carleman class $C^M$ is recovered.

Prototypical non-homogeneous classes include:

• Mixed (log-)Gevrey weights: $$M_n = (n!)^\alpha \prod_{m=0}^{n-1} (\log(e+m))^\beta$$ for $\alpha > 0$, $\beta \in \mathbb{R}$
• Level “$1+$” difference equation sequences: $M_n = \prod_{m=0}^{n-1} \log(e+m)$

These encode growth rates not captured by Gevrey regularity and are crucial in the study of “irregular” singularities of differential or difference equations (Jiménez-Garrido et al., 2018).

3. Structural Properties and Quasianalyticity

Non-homogeneous Carleman classes retain several critical properties from the classical theory, provided the underlying sequences satisfy log-convexity and strong non-quasianalyticity (SNQ):

• SNQ condition: $\exists B > 0$, $$\sum_{q=p}^\infty M_q / ((q+1) M_{q+1}) \leq B M_p / M_{p+1}$$
• Derivative closure: If the sequence is derivation-closed $(\exists D>0 : M_{p+1} \leq D^{p+1} M_p)$, these classes are stable under differentiation (Jiménez-Garrido et al., 2022).

Quasianalyticity in non-homogeneous Carleman classes continues to be governed by the associated proximate order $d(r) = \log M(r)/\log r$, which, under mild hypotheses, is itself a proximate order (piecewise continuous, tends to a finite limit, and with $r d'(r) \log r \to 0$) (Sanz, 2014). The critical threshold for quasianalyticity is given by

$$A_M(S_y) \text{ is quasianalytic} \iff y > \omega(M)$$

where $$\omega(M) = \inf\{y > 0 : A_M(S_y) \text{ is quasianalytic} \}$$ (Sanz, 2014, Lastra et al., 2014).

In the non-homogeneous setting, classes exhibit a rich phase transition structure analogous to, but more intricate than, the Gevrey case. For instance, for certain non-homogeneous $M$, flat functions (those with zero Taylor expansion) exist below the critical opening, and the Borel map is invertible (i.e., surjective) there.

4. Kernel Functions, Laplace/Borel Transforms, and Summability

For strongly regular $M$ admitting a nonzero proximate order, Laplace-like and Borel-like kernels $(e, E)$ can be constructed. The analytic Laplace transform associated to such $e$ is

$$T_{e,\tau} f(z) := \int_0^{\infty e^{i\tau}} e(u/z) f(u) \frac{du}{u}$$

with $e$ decaying as $\exp(-W_M(|z|/k))$, $W_M$ the generalized associated function. These transforms enable:

• Summability of formal power series in directions prescribed by the proximate order
• Iterated acceleration and multisummability procedures, with each level corresponding to a distinct, non-equivalent weight sequence $M_j$ with ordered growth indices $\omega(M_1) > \omega(M_2) > \cdots$ (Jiménez-Garrido et al., 2018, Lastra et al., 2014)

In this non-homogeneous regime, Borel–Laplace theory extends to capture multiple growth scales, and formal series which are summable with respect to multiple such sequences may be reconstructed via iterated Laplace-type operators:

$$S_{M_1,\ldots,M_n}[\hat f] = L_1 \circ (A_{1,2} \circ L_2) \circ \cdots \circ (A_{n-1,n} \circ L_n) (B_n \circ \cdots \circ B_1(\hat f))$$

where each $A_{j,j+1}$ is an acceleration operator connecting the growth scales (Jiménez-Garrido et al., 2018). Cohomological techniques, based on the sheaf of sectorial ultraholomorphic germs, yield uniqueness and existence results for multisummability (Jiménez-Garrido et al., 2018).

5. Applications and Operator Theory

Non-homogeneous Carleman classes have broad applications:

• Moment summation methods: Classes $B_\eta(M, N; I)$ precisely characterize the image of spaces of smooth functions under generalized Laplace transforms, allowing for fine control of summability for both homogeneous and non-homogeneous scales (Kiro, 11 Jan 2026).
• Partial differential equations: They facilitate the study of equations with coefficient or solution regularity beyond the analytic or Gevrey field, including cases with "subelliptic" or "partial Carleman" estimates (Albano et al., 2021).
• Euler-type and difference equations: By constructing $\Gamma$-Euler operators and Laplace/Borel transforms governed by non-homogeneous sequences, one obtains explicit solvability and summability results for generalized equations (Kiro, 11 Jan 2026).
• Carleman–Sobolev spaces: For small exponents $p<1$, the interplay between $L^p$ norms and Carleman weights gives rise to non-homogeneous regularity spaces, with sharp embedding theorems mirroring the Carleman–Denjoy–Carleman criteria (Behm et al., 2014).

In all these cases, the flexibility of non-homogeneous classes enables analysis of growth and summability properties not accessible in the homogeneous theory.

6. Explicit Construction of Flat and Extension Functions

Optimal flat functions—nontrivial elements of ultraholomorphic classes with vanishing Taylor expansion—are explicitly constructed using harmonic extensions of growth functions (e.g., Poisson extensions of $\omega_M$, the logarithm of the weight) and ramification techniques (Jiménez-Garrido et al., 2022):

• For regular (in Dyn'kin’s sense) $M$ and sectors $S(\theta)$ with $\theta < \pi\gamma(M)$, there exists an optimal $\{M\}$-flat function $G$, majorized below and above by the fundamental function $h_M$
• Applicable in explicit cases such as $q$-Gevrey or mixed polynomial-exponential weights

These flat functions provide the kernel for Borel–Laplace inversion formulas, leading to the construction of linear extension operators—right inverses of the Borel map—that extend coefficient sequences to functions with prescribed $M$-growth (Jiménez-Garrido et al., 2022, Sanz, 2014). The existence of such operators is central to the surjectivity of the Borel map and the construction of solutions with given asymptotics.

7. Interpolation, Power Substitution, and Inhomogeneous Derivative Control

Non-homogeneous Carleman theory addresses subtle questions of interpolation, sparse control, and power substitution:

• Derivative interpolation: If only a sparse set of derivatives of $f$ obey Carleman bounds (at indices $d_n$), one can deduce full membership in $C^M$ if the relative gaps $d_{n+1}/d_n$ are uniformly bounded—a phenomenon intrinsic to inhomogeneous inductive regimes in PDE (Albano et al., 2021).
• Power substitution: Carleman classes are stable under power substitutions, with the induced class determined by a modified weight sequence reflecting the non-homogeneous nature of the transformation (Buhovsky et al., 2018).
• Carleman–Sobolev analogues: Infinite order Sobolev spaces with Carleman weights can be fully classifiable as $C^\infty$ iff a certain product condition on the weights is finite; otherwise, the space collapses onto $L^p(\mathbb{R})$ (Behm et al., 2014).

Comprehensive tabulation of key non-homogeneous Carleman classes and properties:

Class/Sequence	Proximate Order	Quasianalyticity Criterion
Gevrey $(n!)^\alpha$	$\rho(t)=\alpha$	$\sum n!^\alpha / (n+1)!^\alpha$ divergent
Log-Gevrey	$\alpha + \beta/\log t$	as above with log factors
Level "1+," e.g. $\prod_{m=0}^{n-1}\log(e+m)$	$1 + 1/\log t$	SNQ via index $\gamma(M) > 0$
Power-substituted $(M^{(k)}_n)$	Modified from $M_n$	Inherits from base or majorant

• Flat functions in Carleman ultraholomorphic classes via proximate orders (Sanz, 2014)
• Summability in general Carleman ultraholomorphic classes (Lastra et al., 2014)
• Multisummability in Carleman ultraholomorphic classes by means of nonzero proximate orders (Jiménez-Garrido et al., 2018)
• An interpolation problem in the Denjoy-Carleman classes (Albano et al., 2021)
• Moment Summation Methods and Non-Homogeneous Carleman Classes (Kiro, 11 Jan 2026)
• Optimal flat functions in Carleman-Roumieu ultraholomorphic classes in sectors (Jiménez-Garrido et al., 2022)
• Carleman-Sobolev classes for small exponents (Behm et al., 2014)
• Power substitution in quasianalytic Carleman classes (Buhovsky et al., 2018)