Independence of derivatives in Carleman-Sobolev Classes for exponents $0<p<1$

TL;DR

This paper investigates the structure of Carleman-Sobolev classes for exponents 0<p<1, revealing a phase transition in the independence of derivatives depending on the weight sequence, with implications for understanding these function spaces.

Contribution

It proves a conjecture that under certain conditions, the Carleman-Sobolev classes exhibit complete independence of derivatives, showing a phase transition in their structure.

Findings

Derivatives are independent in certain weighted Sobolev spaces.

The space decomposes into a direct sum of L^p and a shifted space.

A phase transition occurs depending on the weight sequence condition.

Abstract

We continue the study of Carleman-Sobolev classes from previous joint work with G. Behm. We consider spaces denoted by $W_\mathcal{M}^p$, defined as abstract completions of sets of smooth functions with respect to a weighted Sobolev-flavoured norm involving derivatives of all orders. Previously we showed that these classes behaves very differently on two sides of a condition on the weight sequence $\mathcal{M}$. Here we prove a conjecture made in that paper; under some regularity assumptions on the weight, we show that on one side of the condition there will be a complete independence between derivatives, expressed as $$ W_\mathcal{M}^p\cong L^p\oplus W_{\mathcal{M}_1}^p $$ where $\mathcal{M}_1$ is the shifted sequence. On the other side, we already know that one can embed $W_\mathcal{M}^p$ into $C^{\infty}(\mathbb{R})$. Thus this is an instance of a kind of phase transition.