## Interpolation of Sobolev spaces, Littlewood-Paley inequalities and Riesz transforms on graphs

Venue: | PUBLICACIONS MATEMATIQUES |

Citations: | 5 - 2 self |

### BibTeX

@ARTICLE{Badr_interpolationof,

author = {Nadine Badr and Emmanuel Russ and Université Paris-sud and Université Paul Cézanne},

title = {Interpolation of Sobolev spaces, Littlewood-Paley inequalities and Riesz transforms on graphs},

journal = {PUBLICACIONS MATEMATIQUES},

year = {},

pages = {2009}

}

### OpenURL

### Abstract

Abstract. Let Γ be a graph endowed with a reversible Markov kernel p, and P the associated operator, defined by Pf(x) = ∑ y p(x, y)f(y). Denote by ∇ the discrete gradient. We give necessary and/or sufficient conditions on Γ in order to compare ‖∇f‖ p and ∥ ∥ (I − P)

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Citation Context ...timate follows from the following inequality, valid for any Cn function ϕ on (0, +∞): n∑ C ∣ p n (−1)pϕ(t − pk 2 ) ≤ C sup ∣ t ∣ ∣ (n) ϕ (u) ∣k2n , (3.4) p=0 u≥ n+1 where C > 0 only depends on n (see =-=[30]-=-, problem 16, p. 65). It follows from (3.2) that, for all 0 ≤ m ≤ n, ∑ mk 2 <l≤(m+1)k 2 |αl| 2 l 2 e−c 4j k 2 l ≤ C ∑ ≤ ≤ mk 2 <l≤(m+1)k 2 (m + 1)k 2 l 2 ∫ (m+1)k2 (m + 1)k C 2 mk 2 Ce −c4j where C, c... |

246 |
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Citation Context ...14 and Ω is defined as in the proof of Proposition 1.14. The last inequality follows from the fact that ∑ ⋃ ≤ N and Ω = Bi. Hence ‖∇b‖q ≤ Cα(t)m(Ω) 1 obtain i∈I q . Moreover, since (Mf) ∗ ∼ f ∗∗ (see =-=[11]-=-, Chapter 3, Theorem 3.8), we χBi α(t) = (M(|∇f|) q ) ∗1 q (t) ≤ C (|∇f| q∗∗ ) 1 q (t). Hence, also noting that m(Ω) ≤ t (see [11], Chapter 2, Proposition 1.7), we get K(f, t 1 1 q, W˙ 1,q , W˙ 1,∞ ) ... |

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Citation Context ... entirely similar to the one of Proposition 2.2 in [3] and will therefore be skipped. Let us just mention that it relies on an elliptic Caccioppoli inequality (analogous to the Euclidean version, see =-=[33]-=-), Proposition 1.10 and Gehring’s self-improvement of reverse Hölder inequalities ([32]). 37Appendix We prove Lemma 4.1. For all l ≥ 0, al = (2l)! 4l (l!) 2, and, as already used in Section 7, the St... |

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Citation Context ...ion, which implies the H 1 (Γ) − L 1 (Γ) boundedness of T and therefore its L p (Γ)boundedness for all 1 < p < 2, where H 1 (Γ) denotes the Hardy space on Γ defined in the sense of Coifman and Weiss (=-=[18]-=-). However, the Hörmander integral condition does not yield any information on the L p -boundedness of T for p > 2. The proof of Theorem 1.3 actually relies on a theorem due to Auscher, Coulhon, Duong... |

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Citation Context ...p m(y) ≤ Cr p ∑ y∈B(x,r) |∇f(y)| p m(y). If 1 ≤ p < q < +∞, then (Pp) implies (Pq) (this is a very general statement on spaces of homogeneous type, i.e. on metric measured spaces where (D) holds, see =-=[36]-=-). The converse implication does not hold but an L p Poincaré inequality still has a self-improvement in the following sense: Proposition 1.10 Let (Γ, µ) satisfy (D). Then, for all p ∈ (1, +∞), if (Pp... |

97 |
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Citation Context ... rely on this Calderón-Zygmund decomposition to establish (1.10). The argument also uses the Lp (Γ)-boundedness, for all 2 < p < +∞, of a discrete version of the LittlewoodPaley-Stein g-function (see =-=[46]-=-), which does not seem to have been stated before in this context and is interesting in itself. For all function f on Γ and all x ∈ Γ, define ( ∑ g(f)(x) = l ∣ ) 1/2 ∣ l 2 (I − P)P f(x) . l≥1 Observe ... |

72 |
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Citation Context ...erse Hölder inequality. Remember that such an inequality always holds for solutions of div(A∇u) = 0 on any ball B ⊂ R n , if u is assumed to be in H 1 (B) and A is bounded and uniformly elliptic (see =-=[41]-=-). In the present context, a similar self-improvement result can be shown: Proposition 1.8 Assume that (D), (∆(α)) and (P2) hold. Then there exists p0 > 2 such that (RHp) holds for any p ∈ (2, p0). As... |

64 |
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Citation Context ...rential and go through a property analogous to (Πp) in [3], see Section 8 for detailed definitions. Proposition 1.8 follows essentially from Gehring’s self-improvement of reverse Hölder inequalities (=-=[32]-=-). The plan of the paper is as follows. After recalling some well-known estimates for the iterates of p and deriving some consequences (Section 2), we first prove Theorem 1.15, which is of independent... |

59 | Upper bounds for symmetric Markov transition functions - Carlen, Kusuoka, et al. - 1987 |

42 |
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Citation Context ... 1 2 ν − ‖f‖q ≤ CrV (B) 2 ‖∇f‖2 (8.7) with q = 2 . This inequality is actually equivalent to a relative Faber-Krahn inequality, 1−ν which is itself equivalent to the conjunction of (D) and (DUE), see =-=[23, 35, 15, 20, 10, 26]-=-. Let B and f as in the statement of Lemma 8.1. Since I − P = −δd, (8.6) is equivalent to for all v ∈ W 1,2 0 (B). For all u, v ∈ W 1,2 is a continuous bilinear form on W 1,2 0 〈dh, dv〉 L 2 (E) = 〈f, ... |

37 | Gaussian estimates for Markov chains and random walks on groups. Annals of Probability 21
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- 1993
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Citation Context ...for all λ > 0 and all function f ∈ L 1 (Γ), m ({ x ∈ Γ; ∇(I − P) −1/2 f(x) > λ }) ≤ C λ ‖f‖ 1 . As a consequence, under the same assumptions, (RRp) holds for all 2 ≤ p < +∞. Notice that, according to =-=[37]-=-, the assumptions of Theorem 1.2 hold, for instance, when Γ is the Cayley graph of a group with polynomial volume growth and p(x, y) = h(y −1 x), where h is a symmetric bounded probability density sup... |

34 |
Random walks on graphs with regular volume growth
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Citation Context ....3) in the second one. If we define now the operator “length of the gradient” by ∇f(x) = ( 1 2 ∑ p(x, y) |f(y) − f(x)| 2 ) 1/2 y∈Γ for all function f on Γ and all x ∈ Γ (this definition is taken from =-=[23]-=-), (1.5) shows that 〈(I − P)f, f〉 L2 (Γ) = ‖∇f‖ 2 L2 (Γ) . (1.6) Because of (1.3), the operator P is self-adjoint on L 2 (Γ) and I − P, which, by (1.6) , can be considered as a discrete “Laplace” oper... |

33 |
Riesz transform on manifolds and heat kernel regularity
- Auscher, Coulhon, et al.
(Show Context)
Citation Context ...ained in (1.1) holds for a range of p’s (which is, in general, different for the two inequalities). The second inequality in (1.1) means that the Riesz transform d∆ −1/2 is L p -bounded. We refer to (=-=[3, 5, 9, 22]-=-) and the references therein. In the present paper, we consider a graph equipped with a discrete gradient and a discrete Laplacian and investigate the corresponding counterpart of (1.1). To that purpo... |

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 ...ions for an L 2 -bounded sublinear operator to be L p -bounded for 2 < p < p0. Let us recall this theorem here in the form to be used in the sequel for the sake of completeness (see [5], Theorem 2.1, =-=[2]-=-, Theorem 2.2): Theorem 1.13 Let p0 ∈ (2, +∞]. Assume that Γ satisfies the doubling property (D) and let T be a sublinear operator acting on L 2 (Γ). For any ball B, let AB be a linear operator acting... |

29 |
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Citation Context ... 1 2 ν − ‖f‖q ≤ CrV (B) 2 ‖∇f‖2 (8.7) with q = 2 . This inequality is actually equivalent to a relative Faber-Krahn inequality, 1−ν which is itself equivalent to the conjunction of (D) and (DUE), see =-=[23, 35, 15, 20, 10, 26]-=-. Let B and f as in the statement of Lemma 8.1. Since I − P = −δd, (8.6) is equivalent to for all v ∈ W 1,2 0 (B). For all u, v ∈ W 1,2 is a continuous bilinear form on W 1,2 0 〈dh, dv〉 L 2 (E) = 〈f, ... |

29 |
Analysis of the Laplacian on the complete Riemannian manifold
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(Show Context)
Citation Context ...esting in particular because ∇ is a local operator, while (−∆) 1/2 is not. Generalizations of this result to geometric contexts can be given. On a Riemannian manifold M, it was asked by Strichartz in =-=[47]-=- whether, if 1 < p < +∞, there exists Cp > 0 such that, for all function f ∈ C∞ 0 (M), C −1 p ∥ ∥ ∆ 1/2 f ∥ ∥p ≤ ‖|df|‖ p ≤ Cp ∥ ∥ 1/2 ∆ f∥p , (1.1) where ∆ stands for the Laplace-Beltrami operator on... |

24 |
Etude des transformations de Riesz dans les variétés riemanniennes à courbure de Ricci minorée
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Citation Context ...ained in (1.1) holds for a range of p’s (which is, in general, different for the two inequalities). The second inequality in (1.1) means that the Riesz transform d∆ −1/2 is L p -bounded. We refer to (=-=[3, 5, 9, 22]-=-) and the references therein. In the present paper, we consider a graph equipped with a discrete gradient and a discrete Laplacian and investigate the corresponding counterpart of (1.1). To that purpo... |

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

18 |
Riesz transforms for 1 ≤ p ≤ 2
- Coulhon, Duong
- 1999
(Show Context)
Citation Context ...ained in (1.1) holds for a range of p’s (which is, in general, different for the two inequalities). The second inequality in (1.1) means that the Riesz transform d∆ −1/2 is L p -bounded. We refer to (=-=[3, 5, 9, 22]-=-) and the references therein. In the present paper, we consider a graph equipped with a discrete gradient and a discrete Laplacian and investigate the corresponding counterpart of (1.1). To that purpo... |

17 |
The conservation property of the heat equation on Riemannian manifolds
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(Show Context)
Citation Context ...p0), all f supported in Ci(B) and all l ≥ 1, ∥ ∥ l ∇P f∥L p ≤ (B) Cp √le −c 4ik 2 l ‖f‖Lp (Ci(B)) . (2.5) Inequalities (2.4) and (2.5) may be regarded as “Gaffney” type inequalities, in the spirit of =-=[31]-=-. 3 Littlewood-Paley inequalities In this section, we establish Theorem 1.15. The case 1 < p < 2: We apply the vector-valued version of Theorem 1.16 with T = g and p0 = 1 and, for all ball B with radi... |

15 |
On L p -estimates for square roots of second order elliptic operators
- Auscher
- 2004
(Show Context)
Citation Context ...sumptions. Following [3], we first prove (1.10). The proof relies on a Calderón-Zygmund decomposition for Sobolev functions, which is the adaptation to our context of Proposition 1.1 in [3] (see also =-=[1]-=- in the Euclidean case and [6] for the extension to a weighted Lebesgue measure): Proposition 1.14 Assume that (D) and (Pq) hold for some q ∈ [1, ∞) and let p ∈ [q, +∞). Let f ∈ ˙ E1,p (Γ) and α > 0. ... |

12 | Random walks and geometry on infinite graphs Lecture notes on analysis on metric spaces - Coulhon - 2000 |

12 |
L p bounds for Riesz transforms and square roots associated to second order elliptic operators
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Citation Context ...(7.4) As far as the second term in the right-hand side of (7.4) is concerned, it can be estimated by ({ m x /∈ ⋃ }) ∑ 4Bi; Tibi(x) > λ ≤ ∣ ∣ 1 λ2 ∑ ∑ χΓ\4Bi ∣ (x)Tibi(x) 2 m(x). ∣ i i∈I Arguing as in =-=[3, 13, 38]-=-, we estimate this last quantity by duality. Fix a function u ∈ L 2 (Γ, m) with ‖u‖ 2 = 1. One has x∈Γ i∈I ∑∑ χΓ\4Bi ∣ (x)Tibi(x)u(x)m(x) ∑ ≤ ∣ x∈Γ i∈I 28 i∈I +∞∑ Ai,j j=2where, for all i ∈ I and all... |

11 | décroissance du noyau de la chaleur et transformations de Riesz: un contre-exemple, Ark. för Mat - Coulhon, Ledoux, et al. - 1994 |

11 | Bounds of Riesz transforms on L p spaces for second order elliptic operators
- Shen
(Show Context)
Citation Context ...forms and harmonic functions We also obtain another characterization of the validity of (Rp) for p > 2 in terms of reverse Hölder inequalities for the gradient of harmonic functions, in the spirit of =-=[45]-=- (in the Euclidean context for second order elliptic operators in divergence form) and [3] (on Riemannian manifolds). If B is any ball in Γ and u a function on B, say that u is harmonic on B if, for a... |

10 |
Calderón-Zygmund theory for non-integral operators and the H ∞ functional calculus
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(Show Context)
Citation Context ...phs. The proof of Theorem 1.15 for p > 2 relies on the vector-valued version of Theorem 1.13, while, for p < 2, we use the vector-valued version of the following result (see [2], Theorem 2.1 and also =-=[13]-=- for an earlier version): Theorem 1.16 Let p0 ∈ [1, 2). Assume that Γ satisfies the doubling property (D) and let T be a sublinear operator of strong type (2, 2). For any ball B, let AB be a linear op... |

10 |
Espaces de Lipschitz et inégalités de Poincaré
- Coulhon
- 1996
(Show Context)
Citation Context ... 1 2 ν − ‖f‖q ≤ CrV (B) 2 ‖∇f‖2 (8.7) with q = 2 . This inequality is actually equivalent to a relative Faber-Krahn inequality, 1−ν which is itself equivalent to the conjunction of (D) and (DUE), see =-=[23, 35, 15, 20, 10, 26]-=-. Let B and f as in the statement of Lemma 8.1. Since I − P = −δd, (8.6) is equivalent to for all v ∈ W 1,2 0 (B). For all u, v ∈ W 1,2 is a continuous bilinear form on W 1,2 0 〈dh, dv〉 L 2 (E) = 〈f, ... |

10 |
Axiomatic Theory of Sobolev Spaces
- Gol’dshtein, Troyanov
(Show Context)
Citation Context ...ity has an interpretation in terms of Sobolev spaces defined through ∇. Let 1 ≤ p ≤ +∞. Say that a scalar-valued function f on Γ belongs to the (inhomogeneous) Sobolev space W 1,p (Γ) (see also [42], =-=[34]-=-) if and only if ‖f‖ W 1,p (Γ) := ‖f‖ L p (Γ) + ‖∇f‖ L p (Γ) < +∞. If B is any ball in Γ and 1 ≤ p ≤ +∞, denote by W 1,p 0 (B) the subspace of W 1,p (Γ) made of functions supported in B. 4We will als... |

9 |
Gaussian lower bounds for random walks from elliptic regularity
- Auscher, Coulhon
- 1999
(Show Context)
Citation Context ... y f(x)G(x, y)µxy 〈df, G〉 L 2 (E) = −〈f, δG〉 L 2 (Γ) 32whenever f ∈ L 2 (Γ), dF ∈ L 2 (E), G ∈ L 2 (E) and δG ∈ L 2 (Γ). Notice also that I−P = −δd. The following lemma, very similar to Lemma 4.2 in =-=[4]-=-, holds: Lemma 8.1 Assume that (D), (∆(α)) and (DUE) hold. There exists C > 0 such that, for all ball B and all function f ∈ L2 (Γ) supported in B, there exists a unique function h ∈ W 1,2 0 (B) such ... |

6 | Coulhon T., Riesz transforms on manifolds and Poincaré inequalities, preprint 2004
- Auscher
(Show Context)
Citation Context |

6 |
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Citation Context ...P n f ∥ ∥p ≤ C′ p √n ‖f‖ p . More precisely, as was explained in [43], assumption ∆(α) implies that −1 does not belong to the spectrum of P on L 2 (Γ). As a consequence, P is analytic on L 2 (Γ) (see =-=[25]-=-, Proposition 3), and since P is submarkovian, P is also analytic on L p (Γ) (see [25], p. 426). Proposition 2 in [25] therefore yields the second inequality in (Gp). Thus, condition (Gp) is necessary... |

6 |
Desigualdades con pesos en el Análisis de Fourier: de los espacios de tipo homogéneo a las medidas no doblantes, Ph. D., Universidad Autónoma de Madrid
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- 2001
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Citation Context ... θ > 1, V (x, θr) ≤ Cθ D V (x, r). (1.8) Remark 1.1 Observe also that, since (Γ, µ) is infinite, it is also unbounded (since it is locally uniformly finite) so that, if (D) holds, then m(Γ) = +∞ (see =-=[40]-=-). The second assumption on (Γ, µ) is a uniform lower bound for p(x, y) when x ∼ y, i.e. when p(x, y) > 0. For α > 0, say that (Γ, µ) satisfies the condition ∆(α) if, for all x, y ∈ Γ, (x ∼ y ⇔ µxy ≥ ... |

5 |
Real interpolation of Sobolev spaces
- Badr
(Show Context)
Citation Context ...nt interest, in Section 3. In Section 4, we prove Theorem 1.3 using Theorem 1.13. Section 5 is devoted to the proof of Proposition 1.14. Theorem 1.17 is established in Section 6 by methods similar to =-=[8]-=- and, in Section 7, we prove Theorem 1.11. Finally, Section 8 contains the proof of Theorem 1.7 and of Proposition 1.8. 2 Kernel bounds In this section, we gather some estimates for the iterates of p ... |

5 |
The Poincaré inequality is an open ended condition, preprint 2003
- Keith, Zhong
(Show Context)
Citation Context ...D). Then, for all p ∈ (1, +∞), if (Pp) holds, there exists ε > 0 such that (Pp−ε) holds. This deep result actually holds in the general context of spaces of homogeneous type, i.e. when (D) holds, see =-=[39]-=-. Assuming that (Pq) holds for some q < 2, we establish (RRp) for q < p < 2: Theorem 1.11 Let 1 ≤ q < 2. Assume that (D), (∆(α)) and (Pq) hold. Then, for all q < p < 2, (RRp) holds. Moreover, there ex... |

4 |
Tchamitchian P., Square root problem for divergence operators and related topics, Astérisque
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Citation Context ...d ‖∇ϕ‖∞ ≤ C . Up to an additive constant, k one may assume that the mean value of u in 16B is 0. In order to control the left-hand side of (RHp), it suffices to estimate ∑ x∈B |∇(uϕ)(x)|p m(x). As in =-=[7]-=- p. 35 and [3], Section 2.4, write so that uϕ = P k2 k (uϕ) + 2 ∑−1 P l (I − P)(uϕ), l=0 ( ∇(uϕ) ≤ ∇ P k2 ) k (uϕ) + 2 ∑−1 ∇ ( P l (I − P)(uϕ) ) . (8.10) To treat the first term in the right-hand side... |

4 |
1 − L 1 boundedness of Riesz transforms on Riemannian manifolds and on graphs
- Russ, H
- 2001
(Show Context)
Citation Context ... = ∇ akP k ) , where the ak’s are defined by the expansion k=0 (1 − x) −1/2 = 9 +∞∑ k=0 akx k (1.11)for −1 < x < 1. The kernel of T is therefore given by ( +∞∑ ) akpk(x, y) . ∇x k=0 It was proved in =-=[44]-=- that, under (D) and (P2), this kernel satisfies the Hörmander integral condition, which implies the H 1 (Γ) − L 1 (Γ) boundedness of T and therefore its L p (Γ)boundedness for all 1 < p < 2, where H ... |

2 |
Perturbation of analytic operators and temporal regularity of discrete heat kernels
- Blunck
- 2000
(Show Context)
Citation Context .... (2.1) This “time regularity” estimate, which is a consequence of the L 2 analyticity of P, was first proved by Christ ([16]) by a quite difficult argument. Simpler proofs have been given by Blunck (=-=[12]-=-) and, more recently, by Dungey ([28]). Thus, if B is a ball in Γ with radius k, f any function supported in B and i ≥ 2, one has, for all x ∈ Ci(B) and all l ≥ 1, ∣ P l f(x) ∣ ∣ + l ∣ ∣(I − P)P l f(x... |

2 |
Parabolic Harnack inequality, Rev
- Delmotte
- 1999
(Show Context)
Citation Context ... well as L p bounds for Littlewood-Paley functionals. 21.1 Presentation of the discrete framework Let us give precise definitions of our framework. The following presentation is partly borrowed from =-=[27]-=-. Let Γ be an infinite set and µxy = µyx ≥ 0 a symmetric weight on Γ × Γ. We call (Γ, µ) a weighted graph. In the sequel, we write most of the time Γ instead of (Γ, µ), somewhat abusively. If x, y ∈ Γ... |

2 |
A note on time regularity for discrete time heat kernels, Semigroup Forum
- Dungey
- 2006
(Show Context)
Citation Context ...te, which is a consequence of the L 2 analyticity of P, was first proved by Christ ([16]) by a quite difficult argument. Simpler proofs have been given by Blunck ([12]) and, more recently, by Dungey (=-=[28]-=-). Thus, if B is a ball in Γ with radius k, f any function supported in B and i ≥ 2, one has, for all x ∈ Ci(B) and all l ≥ 1, ∣ P l f(x) ∣ ∣ + l ∣ ∣(I − P)P l f(x) ∣ ∣ ≤ 13 C V (B) e−c 4i k 2 l ‖f‖ L... |

1 |
Temporal regularity for random walk on discrete nilpotent groups
- Christ
- 1995
(Show Context)
Citation Context ... Γ and all l ∈ N∗ , |pl(x, y) − pl+1(x, y)| ≤ Cm(y) lV (x, √ l) e−c d2 (x,y) l . (2.1) This “time regularity” estimate, which is a consequence of the L 2 analyticity of P, was first proved by Christ (=-=[16]-=-) by a quite difficult argument. Simpler proofs have been given by Blunck ([12]) and, more recently, by Dungey ([28]). Thus, if B is a ball in Γ with radius k, f any function supported in B and i ≥ 2,... |

1 |
Suite d’opérateurs à puissances bornées dans les espaces ayant la propriété de Dunford-Pettis, Séminaire d’analyse fonctionnelle 84/85, Publications mathématiques de l’Université Paris VII
- Coulhon
- 1986
(Show Context)
Citation Context ...∥ ∥ 1/2 (I − P) v1 (8.17) 1 V (B) 1/p ‖∇v2‖ L p (B) . (8.18) ∥ L p (Γ) ≤ C ‖v1‖ L p (Γ) , 1 2 . 36where the last inequality follows from the L p -boundedness of (I −P) 1/2 (see [25], p. 423 and also =-=[19]-=-). But v1 is supported in 4B and, for all x ∈ 4B, As a consequence, |v1(x)| ≤ C k |u(x)|. ‖v1‖Lp (Γ) ≤ C k ‖u‖Lp C (4B) ≤ k ‖uψ‖Lp (8B) , where ψ is a nonnegative function equal to 1 on 4B, supported ... |

1 |
A Littewood-Paley-Stein estimate on graphs and groups
- Dungey
(Show Context)
Citation Context ...ies (D), (P2) and (∆(α)). Let p0 ∈ (2, +∞]. Then, the following two assertions are equivalent: (i) for all p ∈ (2, p0), (Gp) holds, (ii) for all p ∈ (2, p0), (Rp) holds. Remark 1.5 In the recent work =-=[29]-=-, property (Gp) is shown to be true for all p ∈ (1, 2] under the sole assumption that Γ satisfies a local doubling property for the volume of balls. Remark 1.6 On Riemannian manifolds, the L 2 Poincar... |

1 |
Sobolev spaces on graphs, Quaest
- Ostrovskii
- 2005
(Show Context)
Citation Context ... the corresponding counterpart of (1.1). To that purpose, we prove, among other things, an interpolation result for Sobolev spaces defined via the differential, similar to those already considered in =-=[42]-=-, as well as L p bounds for Littlewood-Paley functionals. 21.1 Presentation of the discrete framework Let us give precise definitions of our framework. The following presentation is partly borrowed f... |