×

zbMATH — the first resource for mathematics

Bounds of Riesz transforms on \(L^p\) spaces for second order elliptic operators. (English) Zbl 1068.47058
The second-order elliptic operator of divergence form \[ {\mathcal L}= -\text{div}(A(x)\nabla)\quad\text{on }\Omega= \mathbb{R}^n, \] is considered on a bounded open set of \(\mathbb{R}^n\). In the case of bounded domains, a Dirichlet condition \(u= 0\) is imposed on \(\partial\Omega\). Assuming that the differential operator satisfies certain conditions regarding its coefficients, where the Riesz transform \(\nabla({\mathcal L})^{-1/2}\) is bounded on \(L^p(\Omega)\) for \(1< p< 2+\varepsilon\), the boundedness of Riesz transforms is established on Lipschitz domains for an optimal range of \(p\).

MSC:
47F05 General theory of partial differential operators (should also be assigned at least one other classification number in Section 47-XX)
35J15 Second-order elliptic equations
35J25 Boundary value problems for second-order elliptic equations
42B20 Singular and oscillatory integrals (Calderón-Zygmund, etc.)
PDF BibTeX XML Cite
Full Text: DOI Numdam EuDML arXiv
References:
[1] On necessary and sufficient conditions for \(L^p\) estimates of Riesz transform associated to elliptic operators on \({\Bbb R}^n\) and related estimates, (2004) · Zbl 1221.42022
[2] Riesz transforms on manifolds and heat kernel regularity · Zbl 1086.58013
[3] Observation on \(W^{1,p}\) estimates for divergence elliptic equations with VMO coefficients, Boll. Unione Mat. Ital. Sez. B Artic. Ric. Mat., 5, 7, 487-509, (2002) · Zbl 1173.35419
[4] Square root problem for divergence operators and related topics, 249, (1998), Soc. Math. France · Zbl 0909.35001
[5] Square roots of elliptic second order divergence operators on strongly Lipschitz domains:\( L^p\) theory, Math. Ann., 320, 577-623, (2001) · Zbl 1161.35350
[6] On \(W^{1,p}\) estimates for elliptic equations in divergence form, Comm. Pure App. Math., 51, 1-21, (1998) · Zbl 0906.35030
[7] Riesz transforms for \(1\le p\le 2,\) Trans. Amer. Math. Soc., 351, 1151-1169, (1999) · Zbl 0973.58018
[8] Hardy spaces and the Neumann problem in \(L^p\) for Laplace’s equation in Lipschitz domains, Ann. of Math., 125, 437-466, (1987) · Zbl 0658.35027
[9] Heat Kernels and Spectral Theory, (1989), Cambridge University Press · Zbl 0699.35006
[10] Fourier Analysis, 29, (2000), Amer. Math. Soc. · Zbl 0969.42001
[11] \(L^p\) estimates for divergence form elliptic equations with discontinuous coefficients, 10, 409-420, (1996) · Zbl 0865.35048
[12] Multiple Integrals in the Calculus of Variations and Non-Linear Elliptic Systems, 105, (1983), Princeton Univ. Press · Zbl 0516.49003
[13] The inhomogeneous Dirichlet problem in Lipschitz domains, J. Funct. Anal., 130, 161-219, (1995) · Zbl 0832.35034
[14] Harmonic analysis techniques for second order elliptic boundary value problems, 83, (1994), Amer. Math. Soc. · Zbl 0812.35001
[15] An \(L^p\) estimate for the gradient of solutions of second order elliptic divergence equations, Ann. Sc. Norm. Sup. Pisa, 17, 189-206, (1963) · Zbl 0127.31904
[16] Factorization theory and the \(A_p\) weights, Amer. J. Math., 106, 533-547, (1984) · Zbl 0558.42012
[17] The \(L^p\) Dirichlet problem for elliptic systems on Lipschitz domains, (2004)
[18] Singular Integrals and Differentiability Properties of Functions, (1970), Princeton University Press · Zbl 0207.13501
[19] A geometric approach to the Calderón-Zygmund estimates, Acta Math. Sinica (Engl. Ser.), 19, 381-396, (2003) · Zbl 1026.31003
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.