## Contents (2007)

### BibTeX

@MISC{Hytönen07contents,

author = {Tuomas Hytönen and Alan Mcintosh and Pierre Portal},

title = {Contents},

year = {2007}

}

### OpenURL

### Abstract

Abstract. Let L be an elliptic differential operator with bounded measurable coefficients, acting in Bochner spaces L p (R n; X) of X-valued functions on R n. We characterize Kato’s square root estimates ‖ √ Lu‖p � ‖∇u‖p and the H ∞-functional calculus of L in terms of R-boundedness properties of the resolvent of L, when X is a Banach function lattice with the UMD property, or a noncommutative L p space. To do so, we develop various vector-valued analogues of classical objects in Harmonic Analysis, including a maximal function for Bochner spaces. In the special case X = C, we get a new approach to the L p theory of square roots of elliptic operators, as well as an L p version of Carleson’s inequality.

### Citations

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Positive Operators
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(Show Context)
Citation Context ... unconditional basis may be viewed as a Banach lattice of functions defined on Z+. One can also give an abstract definition of a Banach lattice without a postulated function space structure (see e.g. =-=[2]-=-), but we restrict ourselves to the concrete situation, which is the context where Banach lattices with the UMD property have been studied by Rubio de Francia [39]. In this situation, the harmonic ana... |

35 |
Some remarks on Banach spaces in which martingale difference sequences are unconditional,” Arkiv für
- Bourgain
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(Show Context)
Citation Context ...n L p (Ω; X) for 1 < p < ∞ and Ω any probability space. This class of spaces is the most important one for vector-valued Harmonic Analysis. All UMD spaces are reflexive (and even super-reflexive; cf. =-=[11]-=-). The principal examples include the reflexive Lebesgue, Lorentz, Sobolev, and Orlicz spaces, as well as the reflexive noncommutative L p spaces. A recent survey paper on UMD spaces is [14]. The abov... |

32 |
On necessary and sufficient conditions for L p -estimates of Riesz transforms associated to elliptic operators on R n and related estimates, preprint 2004-04, Université de Paris-Sud, Mathématiques
- Auscher
(Show Context)
Citation Context ...o extend L 2 results to the L p setting for p in an open interval (p−, p+), which may be strictly smaller than the whole reflexive range (1, ∞) admissible for classical operators. P. Auscher’s memoir =-=[3]-=- presents the large range of applications of this method and demonstrates that the L p behavior of objects associated with an elliptic operator L (its functional calculus, Riesz transforms, square fun... |

31 |
Square root problem for divergence operators and related topics
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(Show Context)
Citation Context ..., equivalent to conditional expectations with respect to the dyadic filtration of R n (Corollary 5.6). This result, which is handled in the classical case by a T(1) Theorem for Carleson measures (see =-=[7]-=-), is obtained in our context by extending ideas from [9]. To do so, we develop Banach space valued analogues of classical estimates such as Poincaré’s inequality, and Schur’s Lemma. Finally we establ... |

27 |
Vector-valued singular integrals and the H 1 − BMO duality. Probability theory and harmonic
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Citation Context ...ent in the scalar-valued setting. During the 1980’s, this theory advanced from D. L. Burkholder’s [13] extension of M. Riesz’ classical theorem on the Hilbert transform boundedness, via J. Bourgain’s =-=[12]-=-, T. R. McConnell’s [32] and F. Zimmermann’s [42] results on Calderón–Zygmund principal value convolutions and Marcinkiewicz–Mihlin multipliers, to T. Figiel’s [19] vector-valued Date: 26 February 200... |

26 |
Martingales and singular integrals in Banach spaces
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(Show Context)
Citation Context ...xive; cf. [11]). The principal examples include the reflexive Lebesgue, Lorentz, Sobolev, and Orlicz spaces, as well as the reflexive noncommutative L p spaces. A recent survey paper on UMD spaces is =-=[14]-=-. The abovementioned equivalence with the Hilbert transform boundedness, due in one direction to Burkholder [13] and in the other to Bourgain [11], lies at the heart of the theory, and is characterist... |

26 |
Banach space operators with a bounded H ∞ functional calculus
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(Show Context)
Citation Context ...u = f(0)P 0 u + lim n→∞ ψn(A)u. We also often omit the angle and just say that A has an H ∞ functional calculus. The detailed construction of this calculus, and much more information, can be found in =-=[15, 21, 30]-=-. A crucial aspect of the functional calculus is its harmonic analytic characterization. If Y is a Hilbert space, it is shown in [34] that A has an H ∞ functional calculus with angle ν if and only if ... |

25 |
A geometric condition that implies the existence of certain singular integrals of Banach-space-valued functions
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(Show Context)
Citation Context ...celerated replay — with new actors, insight, and considerable improvisation — of the original development in the scalar-valued setting. During the 1980’s, this theory advanced from D. L. Burkholder’s =-=[13]-=- extension of M. Riesz’ classical theorem on the Hilbert transform boundedness, via J. Bourgain’s [12], T. R. McConnell’s [32] and F. Zimmermann’s [42] results on Calderón–Zygmund principal value conv... |

22 |
Topics in Banach Space Theory
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(Show Context)
Citation Context ...in a circle of ideas which can be roughly referred to as “averaging in Banach spaces”, and which forms the core of vector-valued harmonic analysis. A gentle introduction to this topic can be found in =-=[1]-=-. In addition to the above conditions, which are well known in the theory of Banach spaces, we need to introduce a new class of spaces, the defining property of which is the boundedness of the followi... |

22 | Weighted norm inequalities, off diagonal estimates and elliptic operators. Part I: General operator theory and weights - Auscher, Martell |

18 |
boundedness, Fourier multipliers and problems of elliptic and parabolic type
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- 2003
(Show Context)
Citation Context ... of applications especially in the field of Partial Differential Equations. An important opening move into this direction was made by L. Weis [41]; further developments and references are recorded in =-=[16, 30]-=-. The aim of the present paper is to continue the vector-valued program so as to catch up with some of the latest achievements in scalar-valued Harmonic Analysis. More precisely, we are going to devel... |

13 |
Quadratic estimates and functional calculi of perturbed Dirac operators, Inventiones Mathematicae 163
- Axelsson, Keith, et al.
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(Show Context)
Citation Context ...d by P. Auscher, S. Hofmann, M. Lacey, A. McIntosh and Ph. Tchamitchian [5], and more generally for the perturbed Dirac operators treated in a subsequent work by A. Axelsson, S. Keith and A. McIntosh =-=[9]-=-. These objects are no longer Calderón–Zygmund operators, and may even fail to have a pointwise defined kernel. For this reason, their study is considered a move beyond Calderón–Zygmund theory. In the... |

10 |
The solution of the Kato square root problem for second order elliptic operators on R n
- Tchamitchian
(Show Context)
Citation Context ...ach space theory for the square roots of elliptic operators appearing in the famous problem of T. Kato, which was recently solved by P. Auscher, S. Hofmann, M. Lacey, A. McIntosh and Ph. Tchamitchian =-=[5]-=-, and more generally for the perturbed Dirac operators treated in a subsequent work by A. Axelsson, S. Keith and A. McIntosh [9]. These objects are no longer Calderón–Zygmund operators, and may even f... |

10 | Calderón-Zygmund theory for non-integral operators and the H ∞ functional calculus - Blunck, Kunstmann |

7 | Functional calculus of Dirac operators and complex perturbations of Neumann and Dirichlet problems
- Auscher, Axelsson, et al.
(Show Context)
Citation Context ...rn to the proof of Proposition 9.2. Let us fix ν ∈ OΓ ⊆ L (C N ) of norm 1, and let w, ˆw ∈ C N also be of norm 1, and such that w = ν ∗ ( ˆw) = PΓν ∗ ( ˆw). Hence w ∈ WΓ. We can now construct (as in =-=[4]-=-, Lemma 4.10) the following kind of auxiliary functions for each Q ∈ △: wQ ∈ R(Γ), suppwQ ⊆ 3Q, wQ(x) ≡ w ∀x ∈ 2Q, ‖wQ‖ ∞ � 1. To do so, we take an affine function uQ such that ΓuQ ≡ w and ‖1QuQ‖∞ � ℓ... |

7 |
On equivalence of some bases to the Haar system in spaces of vector-valued functions
- Figiel
- 1989
(Show Context)
Citation Context ... 2.5 E∥ ∑ ∑ 1Q(y)uQ(y) ∥ ≃ E∥ X ∑ ∑ hQ(y)uQ(y) ∥ . X k εk Q∈△ 2 k k εk Q∈△ 2 k ) 1/p This gives the first equivalence. A similar argument applies to the second. We next recall a result of Figiel from =-=[18]-=-. Our need for it is no surprise, since it is also a fundamental ingredient in Figiel’s vector-valued T(1) theorem [19]. Proposition 4.4 (Figiel). Let X be a UMD Banach space, and 1 < p < ∞. Then for ... |

7 |
Singular integral operators: a martingale approach
- Figiel
- 1989
(Show Context)
Citation Context ...form boundedness, via J. Bourgain’s [12], T. R. McConnell’s [32] and F. Zimmermann’s [42] results on Calderón–Zygmund principal value convolutions and Marcinkiewicz–Mihlin multipliers, to T. Figiel’s =-=[19]-=- vector-valued Date: 26 February 2007. 2000 Mathematics Subject Classification. 46B09, 46E40, 47A60, 47F05, 60G46. 12 T. HYTÖNEN, A. MCINTOSH, AND P. PORTAL generalization of the T(1) theorem of G. D... |

4 |
Absence de principe du maximum pour certaines équations paraboliques complexes
- Tchamitchian
- 1996
(Show Context)
Citation Context ... situation, this is no longer the case; in fact, already when X = C but p ̸= 2, there exist elliptic second order differential operators which are not sectorial in Lp (Rn;C) for some values of p (see =-=[8]-=-). Thus we need to redefine the problem slightly, so as to adopt the analogues of some of the operator-theoretic conclusions in [9] as the assumptions for our Harmonic Analysis. In particular, we assu... |

1 |
Absolutely Summing Operators. Cambridge stud. adv. math. 43, Cambridge Univ
- Diestel, Jarchow, et al.
- 1995
(Show Context)
Citation Context ... and cotype t = s = 2. For the present purposes, the most important thing is to know that every UMD space has both nontrivial type and cotype. The property of finite cotype is also characterized (see =-=[17]-=-, 12.27) by the comparability of Rademacher and Gaussian random sums, E∥ ∑ ∥ � E∥ X ∑ ∥ ⇔ X has finite cotype, (2) X k∈Z εkxk k∈Z γkxk8 T. HYTÖNEN, A. MCINTOSH, AND P. PORTAL where the γk are indepen... |