Bounds of Riesz Transforms on $L^p$ Spaces for Second Order Elliptic Operators

Shen, Zhongwei

doi:10.5802/aif.2094

Bounds of Riesz Transforms on

L^{p}

Spaces for Second Order Elliptic Operators
[Bornes

L^{p}

des transformées de Riesz des opérateurs elliptiques du second ordre]

Shen, Zhongwei ¹

¹ University of Kentucky, Department of Mathematics, Lexington, KY 40506 (USA)

Annales de l'Institut Fourier, Tome 55 (2005) no. 1, pp. 173-197.

Résumé
Abstract

Soit $ℒ =$ -div $(A (x) \nabla)$ un opérateur elliptique du second ordre à coefficients réels mesurables bornés symétriques sur $ℝ^{n}$ ou sur un domaine à bord Lipschitzien, soumis à une condition au bord de type Dirichlet. Pour tout $p > 2$ , nous obtenons une condition nécessaire et suffisante pour que la transformée de $\nabla {(ℒ)}^{- 1 / 2}$ soit bornée sur l’espace $L^{p}$ . A titre d’application, nous établissons pour $1 < p < 3 + ϵ$ , le caractère borné en norme $L^{p}$ des transformées de Riez d’opérateurs à coefficients $V M O$ sur les domaines à bord Lipschitzien. L’intervalle obtenu pour $p$ est optimal. Nous étudions également si $\nabla {(ℒ)}^{- 1 / 2}$ est borné dans les espaces $L^{2}$ à poids.

Let $ℒ =$ -div $(A (x) \nabla)$ be a second order elliptic operator with real, symmetric, bounded measurable coefficients on $ℝ^{n}$ or on a bounded Lipschitz domain subject to Dirichlet boundary condition. For any fixed $p > 2$ , a necessary and sufficient condition is obtained for the boundedness of the Riesz transform $\nabla {(ℒ)}^{- 1 / 2}$ on the $L^{p}$ space. As an application, for $1 < p < 3 + ϵ$ , we establish the $L^{p}$ boundedness of Riesz transforms on Lipschitz domains for operators with $V M O$ coefficients. The range of $p$ is sharp. The closely related boundedness of $\nabla {(ℒ)}^{- 1 / 2}$ on weighted $L^{2}$ spaces is also studied.

MR Zbl

DOI : 10.5802/aif.2094

Classification : 32J15, 35J25, 42B20
Keywords: Riesz transform, elliptic operator, Lipschitz domain
Mot clés : transformées de Riesz, opérateur elliptique, domaine à bord Lipschitzien

Affiliations des auteurs :

Shen, Zhongwei ¹

¹ University of Kentucky, Department of Mathematics, Lexington, KY 40506 (USA)

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     author = {Shen, Zhongwei},
     title = {Bounds of {Riesz} {Transforms} on $L^p$ {Spaces} for {Second} {Order} {Elliptic} {Operators}},
     journal = {Annales de l'Institut Fourier},
     pages = {173--197},
     publisher = {Association des Annales de l{\textquoteright}institut Fourier},
     volume = {55},
     number = {1},
     year = {2005},
     doi = {10.5802/aif.2094},
     zbl = {1068.47058},
     mrnumber = {2141694},
     language = {en},
     url = {https://aif.centre-mersenne.org/articles/10.5802/aif.2094/}
}

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Shen, Zhongwei. Bounds of Riesz Transforms on $L^p$ Spaces for Second Order Elliptic Operators. Annales de l'Institut Fourier, Tome 55 (2005) no. 1, pp. 173-197. doi : 10.5802/aif.2094. https://aif.centre-mersenne.org/articles/10.5802/aif.2094/

Bibliographie
Cité par

[1] P. Auscher On necessary and sufficient conditions for $L^{p}$ estimates of Riesz transform associated to elliptic operators on $ℝ^{n}$ and related estimates (2004) (Preprint) | Zbl

[2] P. Aucher; T. Coulhon; X.T. Duong; S. Hofmann Riesz transforms on manifolds and heat kernel regularity (to appear in Annales de l'École Normale Supérieure de Paris) | Zbl

[3] P. Auscher; M. Qafsaoui Observation on $W^{1, p}$ estimates for divergence elliptic equations with VMO coefficients, Boll. Unione Mat. Ital. Sez. B Artic. Ric. Mat., Volume 5 (2002) no. 7, pp. 487-509 | MR | Zbl

[4] P. Auscher; Ph. Tchamitchian Square root problem for divergence operators and related topics (Astérisque), Volume 249 (1998) | Zbl

[5] P. Auscher; Ph. Tchamitchian Square roots of elliptic second order divergence operators on strongly Lipschitz domains: $L^{p}$ theory, Math. Ann., Volume 320 (2001), pp. 577-623 | DOI | MR | Zbl

[6] L.A. Caffarelli; I. Peral On $W^{1, p}$ estimates for elliptic equations in divergence form, Comm. Pure App. Math., Volume 51 (1998), pp. 1-21 | DOI | MR | Zbl

[7] T. Coulhon; X.T. Duong Riesz transforms for $1 \leq p \leq 2$ , Trans. Amer. Math. Soc., Volume 351 (1999), pp. 1151-1169 | DOI | MR | Zbl

[8] B. Dahlberg; C. Kenig Hardy spaces and the Neumann problem in $L^{p}$ for Laplace's equation in Lipschitz domains, Ann. of Math., Volume 125 (1987), pp. 437-466 | DOI | MR | Zbl

[9] E. Davies Heat Kernels and Spectral Theory, Cambridge University Press, 1989 | MR | Zbl

[10] J. Duoandikoetxea Fourier Analysis, Graduate Studies in Math., 29, Amer. Math. Soc., 2000 | Zbl

[11] G. Di Fazio $L^{p}$ estimates for divergence form elliptic equations with discontinuous coefficients (Boll. Un. Mat. Ital. A), Volume 10 (1996), pp. 409-420 | Zbl

[12] F. Giaquinta Multiple Integrals in the Calculus of Variations and Non-Linear Elliptic Systems, 105, Princeton Univ. Press, 1983 | Zbl

[13] D. Jerison; C. Kenig The inhomogeneous Dirichlet problem in Lipschitz domains, J. Funct. Anal., Volume 130 (1995), pp. 161-219 | DOI | MR | Zbl

[14] C. Kenig Harmonic Analysis Techniques for Second Order Elliptic Boundary Value Problems (Regional Conf. Series in Math.), Volume 83 (1994) | Zbl

[15] N. Meyers An $L^{p}$ estimate for the gradient of solutions of second order elliptic divergence equations, Ann. Sc. Norm. Sup. Pisa, Volume 17 (1963), pp. 189-206 | Numdam | MR | Zbl

[16] J.L. Rubio de Francia Factorization theory and the $A_{p}$ weights, Amer. J. Math., Volume 106 (1984), pp. 533-547 | DOI | MR | Zbl

[17] Z. Shen The $L^{p}$ Dirichlet problem for elliptic systems on Lipschitz domains (2004) (Preprint)

[18] E. Stein Singular Integrals and Differentiability Properties of Functions, Princeton University Press, 1970 | MR | Zbl

[19] L. Wang A geometric approach to the Calderón-Zygmund estimates, Acta Math. Sinica (Engl. Ser.), Volume 19 (2003), pp. 381-396 | DOI | MR | Zbl

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