On an extension of a bilinear functional on $ {\mathrm{L}}^{p}({\mathbf{R}}^{d})\otimes E$ to a Bochner space with an application to velocity averaging

Lazar, Martin; Mitrović, Darko

doi:10.1016/j.crma.2013.04.013

Partial Differential Equations/Functional Analysis

On an extension of a bilinear functional on

L^{p} (R^{d}) \otimes E

to a Bochner space with an application to velocity averaging
[Sur une extension dʼune fonctionnelle bilinéaire sur

L^{p} (R^{d}) \otimes E

aux espaces du Bochner avec une application sur la moyennisation en vitesse]

Présenté par : Yves Meyer

Lazar, Martin ^1,² ; Mitrović, Darko ³

¹ University of Dubrovnik, Department of Electrical Engineering and Computing, Dubrovnik, Croatia
² BCAM – Basque Center for Applied Mathematics, Bilbao, Spain
³ University of Montenegro, Faculty of Mathematics, Podgorica, Montenegro

Comptes Rendus. Mathématique, Tome 351 (2013) no. 7-8, pp. 261-264.

Résumé
Abstract

Nous examinons les conditions nécessaires et suffisantes pour quʼune fonctionelle bilinéaire continue sur $L^{p} (R^{d}) \otimes E$ , $p > 1$ , E étant un espace de Banach séparable, peut être étendue à une fonctionnelle linaire sur $L^{p} (R^{d}; E)$ . Lʼextension permet une généralisation de lʼH-distribution, qui fournit lʼamélioration dʼun résultat de moyennisation en vitesse (hétèrogène) sur le cadre $L^{p}$ pour tout $p > 1$ .

We examine necessary and sufficient conditions under which a continuous bilinear functional B on $L^{p} (R^{d}) \otimes E$ , $p > 1$ , E being a separable Banach space, can be continuously extended to a linear functional on $L^{p} (R^{d}; E)$ . The extension enables a generalisation of the H-distribution concept, allowing us to obtain a (heterogeneous) velocity averaging result in the $L^{p}$ framework for any $p > 1$ .

Reçu le : 2013-02-22
Accepté le : 2013-04-17
Publié le : 2013-04-01

DOI : 10.1016/j.crma.2013.04.013

Affiliations des auteurs :

Lazar, Martin ^{1, 2} ; Mitrović, Darko ³

@article{CRMATH_2013__351_7-8_261_0,
     author = {Lazar, Martin and Mitrovi\'c, Darko},
     title = {On an extension of a bilinear functional on $ {\mathrm{L}}^{p}({\mathbf{R}}^{d})\otimes E$ to a {Bochner} space with an application to velocity averaging},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {261--264},
     publisher = {Elsevier},
     volume = {351},
     number = {7-8},
     year = {2013},
     doi = {10.1016/j.crma.2013.04.013},
     language = {en},
     url = {http://www.numdam.org/articles/10.1016/j.crma.2013.04.013/}
}

TY  - JOUR
AU  - Lazar, Martin
AU  - Mitrović, Darko
TI  - On an extension of a bilinear functional on $ {\mathrm{L}}^{p}({\mathbf{R}}^{d})\otimes E$ to a Bochner space with an application to velocity averaging
JO  - Comptes Rendus. Mathématique
PY  - 2013
SP  - 261
EP  - 264
VL  - 351
IS  - 7-8
PB  - Elsevier
UR  - http://www.numdam.org/articles/10.1016/j.crma.2013.04.013/
DO  - 10.1016/j.crma.2013.04.013
LA  - en
ID  - CRMATH_2013__351_7-8_261_0
ER  -

%0 Journal Article
%A Lazar, Martin
%A Mitrović, Darko
%T On an extension of a bilinear functional on $ {\mathrm{L}}^{p}({\mathbf{R}}^{d})\otimes E$ to a Bochner space with an application to velocity averaging
%J Comptes Rendus. Mathématique
%D 2013
%P 261-264
%V 351
%N 7-8
%I Elsevier
%U http://www.numdam.org/articles/10.1016/j.crma.2013.04.013/
%R 10.1016/j.crma.2013.04.013
%G en
%F CRMATH_2013__351_7-8_261_0

Lazar, Martin; Mitrović, Darko. On an extension of a bilinear functional on $ {\mathrm{L}}^{p}({\mathbf{R}}^{d})\otimes E$ to a Bochner space with an application to velocity averaging. Comptes Rendus. Mathématique, Tome 351 (2013) no. 7-8, pp. 261-264. doi : 10.1016/j.crma.2013.04.013. http://www.numdam.org/articles/10.1016/j.crma.2013.04.013/

Bibliographie
Cité par

[1] Agoshkov, V.I. Spaces of functions with differential-difference characteristics and smoothness of solutions of the transport equation, Sov. Math. Dokl., Volume 29 (1984), pp. 662-666

[2] Antonić, N.; Lazar, M. H-measures and variants applied to parabolic equations, J. Math. Anal. Appl., Volume 343 (2008), pp. 207-225

[3] Antonić, N.; Mitrović, D. H-distributions: an extension of H-measures to an $L^{p} - L^{q}$ setting, Abstr. Appl. Anal., Volume 2011 (2011) (12 pp)

[4] Benson, D.; Schumer, R.; Wheatcraft, S.; Meerschaert, M. Fractional dispersion, Lévy motion, and the MADE tracer tests, Transp. Porous Media, Volume 42 (2001), pp. 211-240

[5] Cifani, S.; Jakobsen, E.R. Entropy solution theory for fractional degenerate convection–diffusion equations, Ann. Inst. Henri Poincaré, Anal. Non Linéaire, Volume 28 (2011), pp. 413-441

[6] DiPerna, R.J.; Lions, P.L. Global solutions of Boltzmann equations and the entropy inequality, Arch. Ration. Mech. Anal., Volume 114 (1991), pp. 47-55

[7] DiPerna, R.J.; Lions, P.L.; Meyer, Y. $L^{p}$ regularity of velocity averages, Ann. Inst. Henri Poincaré, Anal. Non Linéaire, Volume 8 (1991), pp. 271-287

[8] Gérard, P. Microlocal defect measures, Commun. Partial Differ. Equ., Volume 16 (1991), pp. 1761-1794

[9] Grafakos, L. Classical Fourier Analysis, Grad. Texts in Math., vol. 249, Springer Science and Business Media, LLC, 2008

[10] Lazar, M.; Mitrović, D. The velocity averaging for a heterogeneous heat type equation, Math. Commun., Volume 16 (2011), pp. 271-282

[11] Lazar, M.; Mitrović, D. Velocity averaging – a general framework, Dyn. Partial Differ. Equ., Volume 3 (2012), pp. 239-260

[12] Lions, P.L.; Perthame, B.; Tadmor, E. A kinetic formulation of multidimensional scalar conservation law and related equations, J. Am. Math. Soc., Volume 7 (1994), pp. 169-191

[13] Perthame, B.; Souganidis, P. A limiting case for velocity averaging, Ann. Sci. Éc. Norm. Super., Volume 4 (1998), pp. 591-598

[14] Tao, T.; Tadmor, E. Velocity averaging, kinetic formulations, and regularizing effects in quasi-linear partial differential equations, Commun. Pure Appl. Math., Volume 60 (2007), pp. 1488-1521

[15] Tartar, L. H-measures, a new approach for studying homogenisation, oscillation and concentration effects in PDEs, Proc. R. Soc. Edinb. A, Volume 115 (1990), pp. 193-230

Cité par Sources :