We give a simple construction of the log-convex minorant of a sequence $\{M_\alpha\}_{\alpha\in\mathbb N_0^d}$ and consequently extend to the $d$-dimensional case the well-known formula that relates a log-convex sequence $\{M_p\}_{p\in\mathbb N_0}$ to its associated function $\omega_M$, that is $M_p=\sup_{t>0}t^p\exp(-\omega_M(t))$. We show that in the more dimensional anisotropic case the classical log-convex condition $M_\alpha^2\leq M_{\alpha-e_j}M_{\alpha+e_j}$ 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.

Construction of the log-convex minorant of a sequence $\{M_\alpha\}_{\alpha\in\mathbb N_0^d}$

Oliaro Alessandro;
In corso di stampa

Abstract

We give a simple construction of the log-convex minorant of a sequence $\{M_\alpha\}_{\alpha\in\mathbb N_0^d}$ and consequently extend to the $d$-dimensional case the well-known formula that relates a log-convex sequence $\{M_p\}_{p\in\mathbb N_0}$ to its associated function $\omega_M$, that is $M_p=\sup_{t>0}t^p\exp(-\omega_M(t))$. We show that in the more dimensional anisotropic case the classical log-convex condition $M_\alpha^2\leq M_{\alpha-e_j}M_{\alpha+e_j}$ 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.
In corso di stampa
1
24
https://arxiv.org/abs/2401.11245
Regularization of sequences, log-convex sequences, ultradifferentiable functions, matrix weights
Boiti Chiara; Jornet David; Oliaro Alessandro; Schindl Gerhard
File in questo prodotto:
Non ci sono file associati a questo prodotto.

I documenti in IRIS sono protetti da copyright e tutti i diritti sono riservati, salvo diversa indicazione.

Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/2318/2029811
Citazioni
  • ???jsp.display-item.citation.pmc??? ND
  • Scopus ND
  • ???jsp.display-item.citation.isi??? ND
social impact