Let -div be a second order elliptic operator with real, symmetric, bounded measurable coefficients on or on a bounded Lipschitz domain subject to Dirichlet boundary condition. For any fixed , a necessary and sufficient condition is obtained for the boundedness of the Riesz transform on the space. As an application, for , we establish the boundedness of Riesz transforms on Lipschitz domains for operators with coefficients. The range of is sharp. The closely related boundedness of on weighted spaces is also studied.
Soit -div un opérateur elliptique du second ordre à coefficients réels mesurables bornés symétriques sur ou sur un domaine à bord Lipschitzien, soumis à une condition au bord de type Dirichlet. Pour tout , nous obtenons une condition nécessaire et suffisante pour que la transformée de soit bornée sur l’espace . A titre d’application, nous établissons pour , le caractère borné en norme des transformées de Riez d’opérateurs à coefficients sur les domaines à bord Lipschitzien. L’intervalle obtenu pour est optimal. Nous étudions également si est borné dans les espaces à poids.
Keywords: Riesz transform, elliptic operator, Lipschitz domain
Mot clés : transformées de Riesz, opérateur elliptique, domaine à bord Lipschitzien
Shen, Zhongwei 1
@article{AIF_2005__55_1_173_0, 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/} }
TY - JOUR AU - Shen, Zhongwei TI - Bounds of Riesz Transforms on $L^p$ Spaces for Second Order Elliptic Operators JO - Annales de l'Institut Fourier PY - 2005 SP - 173 EP - 197 VL - 55 IS - 1 PB - Association des Annales de l’institut Fourier UR - https://aif.centre-mersenne.org/articles/10.5802/aif.2094/ DO - 10.5802/aif.2094 LA - en ID - AIF_2005__55_1_173_0 ER -
%0 Journal Article %A Shen, Zhongwei %T Bounds of Riesz Transforms on $L^p$ Spaces for Second Order Elliptic Operators %J Annales de l'Institut Fourier %D 2005 %P 173-197 %V 55 %N 1 %I Association des Annales de l’institut Fourier %U https://aif.centre-mersenne.org/articles/10.5802/aif.2094/ %R 10.5802/aif.2094 %G en %F AIF_2005__55_1_173_0
Shen, Zhongwei. Bounds of Riesz Transforms on $L^p$ Spaces for Second Order Elliptic Operators. Annales de l'Institut Fourier, Volume 55 (2005) no. 1, pp. 173-197. doi : 10.5802/aif.2094. https://aif.centre-mersenne.org/articles/10.5802/aif.2094/
[1] On necessary and sufficient conditions for estimates of Riesz transform associated to elliptic operators on and related estimates (2004) (Preprint) | Zbl
[2] Riesz transforms on manifolds and heat kernel regularity (to appear in Annales de l'École Normale Supérieure de Paris) | Zbl
[3] Observation on 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] Square root problem for divergence operators and related topics (Astérisque), Volume 249 (1998) | Zbl
[5] Square roots of elliptic second order divergence operators on strongly Lipschitz domains: theory, Math. Ann., Volume 320 (2001), pp. 577-623 | DOI | MR | Zbl
[6] On estimates for elliptic equations in divergence form, Comm. Pure App. Math., Volume 51 (1998), pp. 1-21 | DOI | MR | Zbl
[7] Riesz transforms for , Trans. Amer. Math. Soc., Volume 351 (1999), pp. 1151-1169 | DOI | MR | Zbl
[8] Hardy spaces and the Neumann problem in for Laplace's equation in Lipschitz domains, Ann. of Math., Volume 125 (1987), pp. 437-466 | DOI | MR | Zbl
[9] Heat Kernels and Spectral Theory, Cambridge University Press, 1989 | MR | Zbl
[10] Fourier Analysis, Graduate Studies in Math., 29, Amer. Math. Soc., 2000 | Zbl
[11] estimates for divergence form elliptic equations with discontinuous coefficients (Boll. Un. Mat. Ital. A), Volume 10 (1996), pp. 409-420 | Zbl
[12] Multiple Integrals in the Calculus of Variations and Non-Linear Elliptic Systems, 105, Princeton Univ. Press, 1983 | Zbl
[13] The inhomogeneous Dirichlet problem in Lipschitz domains, J. Funct. Anal., Volume 130 (1995), pp. 161-219 | DOI | MR | Zbl
[14] Harmonic Analysis Techniques for Second Order Elliptic Boundary Value Problems (Regional Conf. Series in Math.), Volume 83 (1994) | Zbl
[15] An 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] Factorization theory and the weights, Amer. J. Math., Volume 106 (1984), pp. 533-547 | DOI | MR | Zbl
[17] The Dirichlet problem for elliptic systems on Lipschitz domains (2004) (Preprint)
[18] Singular Integrals and Differentiability Properties of Functions, Princeton University Press, 1970 | MR | Zbl
[19] 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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