Spaces of anisotropic ultradifferentiable functions and local solvability for semilinear partial differential equations

Cadeddu, Lucio; Gramchev, Todor

doi:10.1080/10652460802564902

“…by standard inequalities for the factorial. But with true multi-dimensional weights one may also consider functions with anisotropic differentiability properties -see for example [7] and Sect. 6.…”

Section: Ultradifferentiable Functions and Mapsmentioning

confidence: 99%

“…Here, m w = (w k m k ) k≥0 is log-convex, if the logs of the weights are convex with respect to k. Strongly non-analytic weights are defined in Sect. 7. Both conditions essentially amount to weak growth conditions.…”

mentioning

confidence: 99%

On the Siegel-Sternberg Linearization Theorem

Pöschel

¹

2021

J Dyn Diff Equat

3

0

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We establish a general version of the Siegel-Sternberg linearization theorem for ultradiffentiable maps which includes the analytic case, the smooth case and the Gevrey case. It may be regarded as a small divisior theorem without small divisor conditions. Along the way we give an exact characterization of those classes of ultradifferentiable maps which are closed under composition, and reprove regularity results for solutions of ode's and pde's.We consider the problem of linearizing a smooth map g in the neighbourhood of a fixed point. Placing this fixed point at the origin we write g = Λ +ĝ, where Λ denotes its linear part andĝ comprises its nonlinear terms. We then look for a diffeomorphism ϕ = id +φ around the origin such thatIt is well known that any solution to this problem depends on the eigenvalues of its linearization. Let g be a map in s-dimensional space, and let λ 1 , . ., λ s be the eigenvalues of Λ. Within the category of formal power series there is always a formal solution to this problem, if the infinitely many nonresonance conditionsare satisfied, where for k ∈ {0, 1, .

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“…by standard inequalities for the factorial. But with true multi-dimensional weights one may also consider functions with anisotropic differentiability properties -see for example [6] and section 6.…”

Section: Ultradifferentiable Functions and Mapsmentioning

confidence: 99%

On the Siegel-Sternberg linearization theorem

Pöschel¹

2017

Preprint

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We establish a general version of the Siegel-Sternberg linearization theorem for ultradiffentiable maps which includes the analytic case, the smooth case and the Gevrey case. It may be regarded as a small divisior theorem without small divisor conditions. Along the way we give an exact characterization of those classes of ultradifferentiable maps which are closed under composition, and reprove regularity results for solutions of ode's and pde's. This will open up new directions in kam-theory and other applications of ultradifferentiable functions.

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Construction of the log‐convex minorant of a sequence {Mα}α∈N0d$\lbrace M_\alpha \rbrace _{\alpha \in \mathbb {N}_0^d}$

Boiti,

Jornet,

Oliaro

et al. 2024

Mathematische Nachrichten

0

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We give a simple construction of the log‐convex minorant of a sequence and consequently extend to the ‐dimensional case the well‐known formula that relates a log‐convex sequence to its associated function , that is, . We show that in the more dimensional anisotropic case the classical log‐convex condition is not sufficient: convexity as a function of more variables is needed (not only coordinate‐wise). We finally obtain some applications to the inclusion of spaces of rapidly decreasing ultradifferentiable functions in the matrix weighted setting.

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Spaces of anisotropic ultradifferentiable functions and local solvability for semilinear partial differential equations

Cited by 3 publications

References 13 publications

On the Siegel-Sternberg Linearization Theorem

On the Siegel-Sternberg Linearization Theorem

On the Siegel-Sternberg linearization theorem

Construction of the log‐convex minorant of a sequence {Mα}α∈N0d$\lbrace M_\alpha \rbrace _{\alpha \in \mathbb {N}_0^d}$

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