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

Chiara Boiti, David Jornet, Alessandro Oliaro and Gerhard Schindl

Mathematische Nachrichten, 2025, vol. 298, issue 2, 456-477

Abstract: We give a simple construction of the log‐convex minorant of a sequence {Mα}α∈N0d$\lbrace M_\alpha \rbrace _{\alpha \in \mathbb {N}_0^d}$ and consequently extend to the d$d$‐dimensional case the well‐known formula that relates a log‐convex sequence {Mp}p∈N0$\lbrace M_p\rbrace _{p\in \mathbb {N}_0}$ to its associated function ωM$\omega _M$, that is, Mp=supt>0tpexp(−ωM(t))$M_p=\sup _{t>0}t^p\exp (-\omega _M(t))$. We show that in the more dimensional anisotropic case the classical log‐convex condition Mα2≤Mα−ejMα+ej$M_\alpha ^2\le M_{\alpha -e_j}M_{\alpha +e_j}$ 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.

Date: 2025
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