Hostname: page-component-cd9895bd7-gxg78 Total loading time: 0 Render date: 2024-12-24T13:11:14.711Z Has data issue: false hasContentIssue false
  • English
  • Français

Sobolev Algebras Through a ‘Carré Du Champ’ Identity

Part of: Abstract harmonic analysis Linear function spaces and their duals Groups and semigroups of linear operators, their generalizations and applications

Published online by Cambridge University Press:  24 July 2018

Frédéric Bernicot  and
Dorothee Frey
Frédéric Bernicot
Affiliation:
CNRS – Université de Nantes, Laboratoire Jean Leray, 2 rue de la Houssinière, 44322 Nantes cedex 3, France ([email protected])
Dorothee Frey
Affiliation:
Delft Institute of Applied Mathematics, Delft University of Technology, PO Box 5031, 2600 GA Delft, The Netherlands ([email protected])
Get access Rights & Permissions [Opens in a new window]

Abstract

We consider abstract Sobolev spaces of Bessel-type associated with an operator. In this work, we pursue the study of algebra properties of such functional spaces through the corresponding semigroup. As a follow-up to our previous work, we show that by making use of the property of a ‘carré du champ’ identity, this algebra property holds in a wider range than previously shown.

Keywords

Sobolev spacealgebra propertyheat semigroup

MSC classification

Primary: 46E35: Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems
Secondary: 43A85: Analysis on homogeneous spaces 47D06: One-parameter semigroups and linear evolution equations
Type
Research Article
Information
Proceedings of the Edinburgh Mathematical Society , Volume 61 , Issue 4 , November 2018 , pp. 1041 - 1054
Copyright
Copyright © Edinburgh Mathematical Society 2018 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

1.Auscher, P., On necessary and sufficient conditions for L p-estimates of Riesz transforms associated to elliptic operators on ℝn and related estimates, Mem. Amer. Math. Soc. 186 (2007), 871.Google Scholar
2.Auscher, P., Change of angle in tent spaces, C. R. Acad. Sci. Paris, Ser. I 349 (2011), 297–301.Google Scholar
3.Auscher, P., Hofmann, S. and Martell, J.-M., Vertical versus conical square functions, Trans. Amer. Math. Soc. 364(10) (2012), 5469–5489.Google Scholar
4.Bernicot, F., Coulhon, T. and Frey, D., Sobolev algebras through heat kernel estimates, J. Éc. Polytech. Math. 3 (2016), 99–161.Google Scholar
5.Blunck, S. and Kunstmann, P. C., Generalized Gaussian estimates and the Legendre transform, J. Operator Theory 53(2) (2005), 351–365.Google Scholar
6.Coifman, R. R. and Meyer, Y., Au-delà des opérateurs pseudo-différentiels, Astérisque, 57, Société Math. de France (1978).Google Scholar
7.Duong, X.-T. and Ouhabaz, El M., Gaussian upper bounds for heat kernels of a class of nondivergence operators. International Conference on Harmonic Analysis and Related Topics (Sydney, 2002), pp. 35–45, Volume 41 (Centre for Mathematics and its Applications, Australian National University, Canberra, Australia, 2003).Google Scholar
8.Frey, D., Paraproducts via H ∞-functional calculus, Rev. Matematica Iberoam. 29(2) (2013), 635–663.Google Scholar
9.Kato, T. and Ponce, G., Commutator estimates and the Euler and Navier-Stokes equations, Commun. Pure Appl. Math. 41 (1988), 891–907.Google Scholar
10.McIntosh, A. and Nahmod, A., Heat kernel estimates and functional calculi of −bΔ, Math. Scand. 87(2) (2000), 287–319.Google Scholar
11.Meyer, Y., Remarques sur un théorème de J. M. Bony, Rend. Circ. Mat. Palermo, II. Ser. 1 (1981), 1–20.Google Scholar
12.Stein, E. M., Interpolation of linear operators, Trans. Amer. Math. Soc. 83 (1956), 482–492.Google Scholar
13.Strichartz, R., Multipliers on fractional Sobolev spaces, J. Math. Mech. 16(9) (1967), 1031–1060.Google Scholar