A counterexample to the \(L^ 2\) estimate \(\| XYu \|\leq C(\| X^ k u\|+ \| Y^ k u\|+ \| u\|)\)
type: Item
instance of: scholarly article
title: A counterexample to the \(L^ 2\) estimate \(\| XYu \|\leq C(\| X^ k u\|+ \| Y^ k u\|+ \| u\|)\)
zbMATH Open document ID: 0835.47037
DOI: 10.1007/BF01446286
author: Jarosław Wróblewski
published in: Mathematische Annalen
publication date: 1996-02-01T00:00:00Z
full work available at URL: https://eudml.org/doc/165396
review text: An example of two \(C^\infty\) vector fields \(X\), \(Y\) on \(\mathbb{R}^2\) is constructed with the property that for any natural number \(k\) none of the following estimates holds. \[ |XY u|\leq C(|X^k u|+ |Y^k u|+ |u|),\;|(XY+ YX) u|\leq C(|X^k u|+ |Y^k u|+ |u|), \] \[ |[X, Y] u|\leq C(|X^k u|+ |Y^k u|+|u|). \] It is shown that the above estimates fail not only in \(L^2\), but in any \(L^p\), \(1\leq p\leq \infty\) and cannot be saved by adding terms of the form \(|Y^k X^\ell u|\) to the right-hand side. Finally, in the given example the operator \(X^2+ Y^2\) is hypoelliptic. A small modification of the example produces two vector fields in \(\mathbb{R}^3\) generating a pseudoconvex CR-structure.
Mathematics Subject Classification ID: 47F05
Mathematics Subject Classification ID: 46E30
zbMATH DE Number: 840215
zbMATH Keywords: \(L^ 2\) estimate
zbMATH Keywords: pseudoconvex CR-structure
MaRDI profile type: Publication
cites work: Fundamental solutions for second order subelliptic operators
cites work: Q5677870
cites work: Hypoelliptic differential operators and nilpotent groups
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description: scientific article; zbMATH DE number 840215