## Bounds of Riesz transforms on L p spaces for second order elliptic operators

Venue: | Ann. Inst. Fourier |

Citations: | 11 - 1 self |

### BibTeX

@ARTICLE{Shen_boundsof,

author = {Zhongwei Shen},

title = {Bounds of Riesz transforms on L p spaces for second order elliptic operators},

journal = {Ann. Inst. Fourier},

year = {},

pages = {173--197}

}

### OpenURL

### Abstract

Abstract. Let L = −div(A(x)∇) 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 ∇(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 MO coefficients. The range of p is sharp. The closely related boundedness of ∇(L) −1/2 on weighted L 2 spaces is also studied. 1.

### Citations

1373 |
Singular Integrals and Differentiability Properties of Functions
- Stein
- 1970
(Show Context)
Citation Context ...implies that ω ∈ Aq for some q < 2, and 1 ω ∈ A2. For the classical Riesz transform ∇(−∆) −1/2 , the boundedness on Lp (Rn) for 1 < p < ∞, and on L2 (Rn, ωdx) with ω ∈ A2(Rn ) is well known (see e.g. =-=[18,10]-=-). Remark 1.8. For a general second order elliptic operator L with real, symmetric, bounded measurable coefficients, ∇(L) −1/2 is bounded on L p for 1 < p < 2 + ε. The range of p was shown to be optim... |

334 |
Heat Kernels and Spectral Theory
- Davies
- 1989
(Show Context)
Citation Context ...timate for n ≥ 3: (2.10) |Γλ(x, y)| ≤ C e −c√ λ|x−y| · 1 , |x − y| n−2 which follows directly from the formula (L + λ) −1 = ∫ ∞ 0 e−λt e −tL dt and the well known upper bound for the heat kernel of L =-=[9]-=-. In the case n = 2, one needs to replace 1 |x−y| n−2 by | ln( √ λ|x − y|)| + 1. The rest of this section is devoted to the proof of the following theorem. We remark that estimates similar to (2.12)-(... |

215 | Multiple integrals in the calculus of variations and nonlinear elliptic systems - Giaquinta - 1983 |

104 |
Harmonic analysis techniques for second order elliptic boundary value problems
- Kenig
- 1994
(Show Context)
Citation Context ...(Ω) is an open interval. Let L = −∆ on a bounded Lipschitz domain Ω, subject to Dirichlet boundary condition. Using the solvability of the L2 regularity problem and the boundary Hölder estimates (see =-=[14]-=-), it is not hard to show that condition (i) in Theorem B holds for p = 3 if n ≥ 3, and for p = 4 in the case n = 2 (see Lemma 4.1). It follows that for n ≥ 3, the Riesz transform 3 +δ(Rn). ) with ∇(L... |

95 |
The inhomogeneous Dirichlet problem in Lipschitz domains
- Jerison, Kenig
- 1995
(Show Context)
Citation Context ...ed on Lp (Ω) for 1 < p < 3 + ε, and on L2 (Ω, dx) with ω ∈ A 4 ω If n = 2, ∇(L) −1/2 is bounded on Lp (Ω) for 1 < p < 4 + ε, and on L2 (Ω, dx ω ∈ A 3 2 +δ(R 2 ). The ranges of p are known to be sharp =-=[13]-=-. In the case that Ω is a C 1 domain, ∇(L) −1/2 is bounded on L p (Ω) for 1 < p < ∞. We should point out that although our weighted L 2 bounds are new, the boundedness of ∇(−∆) −1/2 on L p (Ω) for Lip... |

89 |
An Lp-estimate for the gradient of solutions of second order elliptic divergence equations, Annali della Scuola Norm
- Meyers
- 1963
(Show Context)
Citation Context ...jk − ajk| 2q′ dx |∇u| 2 dx } 1/2 } 1/2 } 1/(2q ′ ) { 1 r n ∫ D(x0,βr) , |∇u| 2q dx } 1/(2q)16 ZHONGWEI SHEN where q = 1 + δ and δ > 0 is so small that the L 2q estimates hold for solutions of Lu = 0 =-=[15]-=-. Also we have introduced the function φ(r) by ∑ (4.12) φ(r) = C sup x0∈Ω j,k { 1 r n ∫ |ajk − bjk| B(x0,αr) 2q′ dx } 1/(2q ′ ) Finally we note that by the John-Nirenberg inequality, if ajk ∈ V MO(R n... |

36 |
Riesz transforms on manifolds and heat kernel regularity
- Auscher, Coulhon, et al.
(Show Context)
Citation Context ...oved in section 4. Finally in section 5 we give the proof of Theorems 3.1 and 3.2. Acknowledgment. After this paper was submitted, the author was informed kindly by S. Hofmann of two recent preprints =-=[1,2]-=- on the study of Riesz transforms. In these two ωBOUNDS OF RIESZ TRANSFORMS ON L p SPACES 5 papers necessary and sufficient conditions are obtained for the L p boundedness of Riesz transforms on mani... |

36 |
Square root problem for divergence operators and related topics, Astérisque 249
- Auscher, Tchamitchian
- 1998
(Show Context)
Citation Context ... ξ, x ∈ R n and some µ > 0. Under these assumptions, it is known that the Riesz transform ∇(L) −1/2 is bounded on L p (Ω) for 1 < p < 2 + ε where ε = ε(n, µ) > 0, and is of weak type (1, 1) (see e.g. =-=[4,7]-=-). Moreover, the range of p is sharp. The main purpose of this paper is to investigate the L p boundedness of the Riesz transform for p > 2, as well as the closely related boundedness on weighted L 2 ... |

35 |
Hardy spaces and the Neumann problem in L p for Laplace’s equation in Lipschitz domains
- Dahlberg, Kenig
- 1987
(Show Context)
Citation Context ...4 if n = 2. Finally we point out that since u = 0 on ∆α2r, (4.4) follows from the L2 solvability of the regularity problem for Laplace’s equation on Lipschitz domains, by an integration argument (see =-=[8]-=-). Remark 4.5. If Ω is a C 1 domain, then Hölder estimate (4.3) holds for any 0 < η < 1. It follows that operators with constant coefficients satisfy condition (i) in Theorem B for any p > 2. Conseque... |

30 |
On necessary and sufficient conditions for L p estimates of Riesz transforms associated to elliptic operators on R n and related estimates
- Auscher
(Show Context)
Citation Context ...oved in section 4. Finally in section 5 we give the proof of Theorems 3.1 and 3.2. Acknowledgment. After this paper was submitted, the author was informed kindly by S. Hofmann of two recent preprints =-=[1,2]-=- on the study of Riesz transforms. In these two ωBOUNDS OF RIESZ TRANSFORMS ON L p SPACES 5 papers necessary and sufficient conditions are obtained for the L p boundedness of Riesz transforms on mani... |

23 |
On W 1,p estimates for elliptic equations in divergence form
- CAFFARELLI, PERAL
- 1998
(Show Context)
Citation Context ...plies (ii), we use a new and refined version of the celebrated Calderón-Zygmund Lemma. See Theorem 3.1. This theorem, formulated by the author in [17], was inspired by a paper of Caffarelli and Peral =-=[6]-=- as well as a recent work of L. Wang [19]. For any fixed p > 2, it gives a sufficient condition for an L 2 bounded sublinear operator to be bounded on L q for all 2 < q < p. It enables us to show that... |

18 |
Lp estimates for divergence form elliptic equations with discontinuous coefficients
- DiFazio
- 1996
(Show Context)
Citation Context ... L p (Ω) for any 1 < p < ∞, and on L 2 (Ω, ωdx) for any ω ∈ A2(R n ). Remark 1.11. For divergence form elliptic equations on C 1,1 domains with V MO coefficients, the W 1,p estimates were obtained in =-=[11]-=- for any 1 < p < ∞. The result was extended in [3] to the case of C 1 domains, for operators with complex coefficients. Our approach to Theorem C, which is very different from that in [11,3], is based... |

17 |
Fourier analysis, Graduate Studies
- Duoandikoetxea
- 2001
(Show Context)
Citation Context ...implies that ω ∈ Aq for some q < 2, and 1 ω ∈ A2. For the classical Riesz transform ∇(−∆) −1/2 , the boundedness on Lp (Rn) for 1 < p < ∞, and on L2 (Rn, ωdx) with ω ∈ A2(Rn ) is well known (see e.g. =-=[18,10]-=-). Remark 1.8. For a general second order elliptic operator L with real, symmetric, bounded measurable coefficients, ∇(L) −1/2 is bounded on L p for 1 < p < 2 + ε. The range of p was shown to be optim... |

14 |
Square roots of elliptic second order divergence operators on strongly Lipschitz domains: L2 theory
- Auscher, Tchamitchian
(Show Context)
Citation Context ...t is worth mentioning that the boundedness of ∇(L) −1/2 on L p is equivalent to the inequality ‖∇f‖p ≤ C ‖L 1/2 f‖p. The reverse inequality ‖L 1/2 f‖p ≤ C ‖∇f‖p, nevertheless, holds for all 1 < p < ∞ =-=[4,5]-=-. The proof of Theorems A and B depends on this fact. Remark 1.9. By a simple geometric observation, one may see that condition (i) in Theorem B is equivalent to the following. There exist C1 > 0, α4 ... |

14 | Riesz transforms for 1 - Coulhon, Duong |

7 |
Observations on W 1,p estimates for divergence elliptic equations with VMO coefficients
- Auscher, Qafsaoui
(Show Context)
Citation Context ... any ω ∈ A2(R n ). Remark 1.11. For divergence form elliptic equations on C 1,1 domains with V MO coefficients, the W 1,p estimates were obtained in [11] for any 1 < p < ∞. The result was extended in =-=[3]-=- to the case of C 1 domains, for operators with complex coefficients. Our approach to Theorem C, which is very different from that in [11,3], is based on Theorem B and a perturbation argument found in... |

4 | The L p Dirichlet problem for elliptic systems on Lipschitz domains, preprint
- Shen
- 2004
(Show Context)
Citation Context ... is much more involved. To prove that condition (i) implies (ii), we use a new and refined version of the celebrated Calderón-Zygmund Lemma. See Theorem 3.1. This theorem, formulated by the author in =-=[17]-=-, was inspired by a paper of Caffarelli and Peral [6] as well as a recent work of L. Wang [19]. For any fixed p > 2, it gives a sufficient condition for an L 2 bounded sublinear operator to be bounded... |

2 |
Factorization theory and Ap weights
- Francia, L
- 1984
(Show Context)
Citation Context ...ition (i). We will also show that condition (i) leads to an L p estimate on the kernel function of the resolvent (L + λ) −1 for λ > 0. The following proposition is essentially due to Rubio de Francia =-=[16]-=-. Proposition 2.1. Let T be a bounded operator on L2 (E) where E is a measurable subset of Rn . Let 0 < δ ≤ 1. Suppose that ∫ ∫ 2 dx 2 dx (2.2) |Tf| ≤ C |f| for any ω ∈ A1+δ(R ω ω n ), E E where C dep... |

2 | A geometric approach to the Calderon-Zygmund estimates, Preprint. Mei-Chi
- Wang
(Show Context)
Citation Context ...sion of the celebrated Calderón-Zygmund Lemma. See Theorem 3.1. This theorem, formulated by the author in [17], was inspired by a paper of Caffarelli and Peral [6] as well as a recent work of L. Wang =-=[19]-=-. For any fixed p > 2, it gives a sufficient condition for an L 2 bounded sublinear operator to be bounded on L q for all 2 < q < p. It enables us to show that the operator ∇L −1 div is bounded on L p... |