## Sobolev inequalities in familiar and unfamiliar settings

Venue: | In S. Sobolev Centenial Volumes, (V. Maz’ja, Ed |

Citations: | 3 - 1 self |

### BibTeX

@INPROCEEDINGS{Saloff-coste_sobolevinequalities,

author = {Laurent Saloff-coste},

title = {Sobolev inequalities in familiar and unfamiliar settings},

booktitle = {In S. Sobolev Centenial Volumes, (V. Maz’ja, Ed},

year = {}

}

### OpenURL

### Abstract

Abstract The classical Sobolev inequalities play a key role in analysis in Euclidean spaces and in the study of solutions of partial differential equations. In fact, they are extremely flexible tools and are useful in many different settings. This paper gives a glimpse of assortments of such applications in a variety of contexts. 1

### Citations

285 | Eigenvalues in Riemannian Geometry - Chavel - 1984 |

268 |
Kernels and Spectral Theory
- Davies
- 1989
(Show Context)
Citation Context ...ed earlier, the generator A is related to the form E and the energy form Γ by ‖(−A) 1/2 f‖ 2 ∫ 2 = E(f, f) = dΓ (f, f), f ∈ Dom((−A) 1/2 ) = D(E). For the following result see [15, 100, 103] and also =-=[33]-=-. M Theorem 4.3. Fix N > 0. Let (M, µ, E, D(E)) be a Dirichlet space with associated semigroup e tA . The following properties are equivalent. • There exists C1 such that ∀ f ∈ L 1 (M, µ), t > 0, ‖e t... |

179 | Groups of polynomial growth and expanding maps
- Gromov
- 1981
(Show Context)
Citation Context ...ent Saloff-Coste V ). This means that a group carrying a recurrent random walk must have a volume growth function satisfying ∀ε > 0, lim inf r→∞ r−(2+ε) V (r) < ∞. By the celebrated theorem of Gromov =-=[54]-=- (and its extension in [99]), the condition ∃ A > 0, lim inf r→∞ r−AV (r) < ∞ (5.5) implies that G contains a nilpotent subgroup of finite index. Since a subgroup of finite index in G has volume growt... |

170 |
Multiple Integrals in the Calculus of Variations
- Morrey
- 1966
(Show Context)
Citation Context ..., s − 2r 2 ) × B and Q+ = (s − r 2 /2, s) × B. Moser’s iteration technique has been adapted and used in hundreds of papers studying various PDE problems. Some early examples are [2, 3, 90]. The books =-=[42, 69, 76]-=- contain many applications of this circle of ideas, as well as further references. The survey paper [83] deals specifically with the heat equation and is most relevant for the purpose of the present p... |

161 |
Finite propagation speed, kernel estimates for functions of the Laplace operator, and the geometry of complete Riemannian manifolds
- Cheeger, Gromov, et al.
(Show Context)
Citation Context ... motivated by our desire to use this approach in other settings such as Riemannian manifolds or more exotic spaces. Early uses of Moser’s iteration technique on manifolds as in the influential papers =-=[22, 23]-=- were actually limited by a misunderstanding of what is really needed to run this technique successfully. Interesting early works that explored the flexibility of Moser’s iteration beyond the classica... |

151 |
On the parabolic kernel of the Schrödinger operator
- Li, Yau
- 1986
(Show Context)
Citation Context ...ction V K/(n−1)(r) of the simply connected space of the same dimension and constant sectional curvature K/(n − 1) (see, for example, [20, p.73], [21, Theorem 3.9] and [41, 3.85; 3.101]). Theorem 2.5 (=-=[14, 22, 74]-=-). A complete Riemannian manifold (M, g) with nonnegative Ricci curvature satisfies the equivalent properties of Theorem 2.2. It is interesting to note that the equivalent properties of Theorem 2.2 wh... |

147 |
Sobolev Spaces
- Maz’ya
- 1985
(Show Context)
Citation Context ...equality must hold uniformly for all geodesic balls B. The exact value of the constants varies from one type of inequality to another. Many results in the spirit of these equivalences can be found in =-=[75]-=- in the context of Euclidean domains. A discussion in a very general setting is in [4] (see also [87, Chapt. 3]). The following theorem describes some of the noteworthy consequences of (2.11). Let ∆M ... |

118 |
Analysis and geometry on groups
- Coulhon, Saloff-Coste, et al.
- 1993
(Show Context)
Citation Context ...he triviality of positive harmonic functions. This, in turns, implies polynomial volume growth by [13, Theorem 1.4 or 1.6]. The stated theorem follows. ⊓⊔ For more general results in this setting see =-=[103]-=-. Harmonic functions of polynomial growth on Lie groups of polynomial growth are studied in [1]. Example 2.4 (coverings of compact manifolds). Let (M, g) be a complete Riemannian manifold such that th... |

113 |
Continuity of solutions of parabolic and elliptic equations
- Nash
- 1958
(Show Context)
Citation Context ... basic question in this context is that of the boundedness and continuity of solutions of the equation Lau = 0 in the interior of an open set Ω. This was solved earlier by De Giorgi [34] (and by Nash =-=[81]-=- in the parabolic case), but Moser proposed an alternative method, squarely based on the use of Sobolev inequality (2.5). To understand why one might hope this is possible, observe that the argument g... |

103 | Riemannian geometry - Gallot, Hulin, et al. - 1990 |

101 |
Lectures on Analysis on Metric Spaces. Universitext
- Heinonen
- 2001
(Show Context)
Citation Context ...setting. Indeed, the theory of Sobolev spaces on metric measure spaces is also of interest because of the many similar, but different setting it unifies. We refer the reader to the entertaining books =-=[59, 62, 89]-=- and the review paper [63] for glimpses of the general viewpoint on “first order calculus.” There are many interesting natural metric spaces (of finite dimension type) on which one wants to do some an... |

97 |
Sobolev met Poincaré
- Hajlasz, Koskela
(Show Context)
Citation Context ...setting. Indeed, the theory of Sobolev spaces on metric measure spaces is also of interest because of the many similar, but different setting it unifies. We refer the reader to the entertaining books =-=[59, 62, 89]-=- and the review paper [63] for glimpses of the general viewpoint on “first order calculus.” There are many interesting natural metric spaces (of finite dimension type) on which one wants to do some an... |

85 |
On the Schrödinger equation and the eigenvalue problem
- Li, Yau
- 1983
(Show Context)
Citation Context ...s known as the Rozenblum–Cwikel–Lieb inequality. We refer the reader to the review of the literature in [70, 84]. The following elegant result is taken from [70] and is based on the technique used in =-=[73]-=- in Euclidean space. Theorem 4.5 ([70]). Let (M, µ, E, D(E)) be a Dirichlet space with associated semigroup e tA . Assume that the Sobolev inequality R N ∀ f ∈ D(E), ‖f‖ 2 2N/(N−2) � S2 E(f, f) holds ... |

83 |
On Harnack’s theorem for elliptic differential equations
- Moser
- 1961
(Show Context)
Citation Context ...‖|u|2‖p = ‖u2‖∞. The desired conclusion 2.2 Harnack inequalities The technique illustrated above is the simplest instance of what is widely known as Moser’s iteration technique. In a series of papers =-=[77]-=-–[80], Moser developed this technique as the basis for the study of divergence form uniformly elliptic operators in Rn , i.e., operators of the form (we use ∂i = ∂/∂xi) La = ∑ ∂i(ai,j(x)∂j) i,j with r... |

83 | Random walks on infinite graphs and groups
- Woess
- 2000
(Show Context)
Citation Context ...rst applications of Sobolev inequalities on graphs, namely, Varopoulos’ solution of Kesten’s conjecture regarding random walks on finitely generated groups. We refer to [47] for a short survey and to =-=[98, 105]-=- for a detailed treatment of some aspects.Sobolev Inequalities in Familiar and Unfamiliar Settings 331 5.1 Graphs of bounded degree In what follows, a graph is a pair (V, E), where E is a symmetric s... |

79 |
Trudinger, N.S.: Elliptic Partial Differential Equations of Second Order
- Gilbarg
- 1983
(Show Context)
Citation Context ..., s − 2r 2 ) × B and Q+ = (s − r 2 /2, s) × B. Moser’s iteration technique has been adapted and used in hundreds of papers studying various PDE problems. Some early examples are [2, 3, 90]. The books =-=[42, 69, 76]-=- contain many applications of this circle of ideas, as well as further references. The survey paper [83] deals specifically with the heat equation and is most relevant for the purpose of the present p... |

72 |
On a theorem of functional analysis
- Sobolev
- 1938
(Show Context)
Citation Context ... glimpse of assortments of such applications in a variety of contexts. 1 Introduction There are few articles that have turned out to be as influential and truly important as S.L. Sobolev 1938 article =-=[93]-=- (the American translation appeared in 1963), where he introduces his famed inequalities. It is the idea of a functional inequality itself that Sobolev brings to life in his paper, as well as the now ... |

71 |
Sulla differenziabilità e l’analiticità delle estremali degli integrali multipli regolari
- Giorgi
- 1957
(Show Context)
Citation Context ...ficients, the most basic question in this context is that of the boundedness and continuity of solutions of the equation Lau = 0 in the interior of an open set Ω. This was solved earlier by De Giorgi =-=[34]-=- (and by Nash [81] in the parabolic case), but Moser proposed an alternative method, squarely based on the use of Sobolev inequality (2.5). To understand why one might hope this is possible, observe t... |

65 |
A Harnack inequality for parabolic differential equations
- Moser
- 1964
(Show Context)
Citation Context ...m. The ball 2B is concentric with B with twice the radius of B). He then proceeded to prove this Harnack inequality by variations on the argument outlined in the previous section. In his later papers =-=[78]-=-–[80], Moser obtained a parabolic version of the above Harnack inequality. Namely, he proved that there exists a constant C(n, ε) such that any nonnegative solution u of the heat equation (∂t−La)u = 0... |

58 |
Local behavior of solutions of quasilinear elliptic equations
- Serrin
- 1964
(Show Context)
Citation Context ... where Q− = (s − 3r 2 , s − 2r 2 ) × B and Q+ = (s − r 2 /2, s) × B. Moser’s iteration technique has been adapted and used in hundreds of papers studying various PDE problems. Some early examples are =-=[2, 3, 90]-=-. The books [42, 69, 76] contain many applications of this circle of ideas, as well as further references. The survey paper [83] deals specifically with the heat equation and is most relevant for the ... |

57 |
The heat equation on non compact Riemannian manifold
- Grigory’an
- 1992
(Show Context)
Citation Context ... and let h(t, x, y) be the (minimal) fundamental solution of the heat equation (∂t − ∆M )u = 0 on M, i.e., the kernel of the heat semigroup et∆M . For complete discussions, surveys, and variants, see =-=[43, 44, 46, 45, 48, 85, 86, 87]-=-. Theorem 2.1. Assume that (M, g) is a complete Riemannian manifolds which satisfies the scale invariant family of Sobolev inequalities (2.11) (with some parameter ν > 2). Then the following propertie... |

56 |
Analytic and geometric background of recurrence and non-explosion of the Brownian motion on Riemannian manifolds
- Grigor’yan
- 1999
(Show Context)
Citation Context ... and let h(t, x, y) be the (minimal) fundamental solution of the heat equation (∂t − ∆M )u = 0 on M, i.e., the kernel of the heat semigroup et∆M . For complete discussions, surveys, and variants, see =-=[43, 44, 46, 45, 48, 85, 86, 87]-=-. Theorem 2.1. Assume that (M, g) is a complete Riemannian manifolds which satisfies the scale invariant family of Sobolev inequalities (2.11) (with some parameter ν > 2). Then the following propertie... |

52 |
A note on the isoperimetric constant
- Buser
- 1982
(Show Context)
Citation Context ...ction V K/(n−1)(r) of the simply connected space of the same dimension and constant sectional curvature K/(n − 1) (see, for example, [20, p.73], [21, Theorem 3.9] and [41, 3.85; 3.101]). Theorem 2.5 (=-=[14, 22, 74]-=-). A complete Riemannian manifold (M, g) with nonnegative Ricci curvature satisfies the equivalent properties of Theorem 2.2. It is interesting to note that the equivalent properties of Theorem 2.2 wh... |

49 | Nonlinear analysis on manifolds: Sobolev spaces and inequalities - Hebey - 1999 |

49 |
Aspects of Sobolev-type inequalities
- Saloff-Coste
(Show Context)
Citation Context ... and let h(t, x, y) be the (minimal) fundamental solution of the heat equation (∂t − ∆M )u = 0 on M, i.e., the kernel of the heat semigroup et∆M . For complete discussions, surveys, and variants, see =-=[43, 44, 46, 45, 48, 85, 86, 87]-=-. Theorem 2.1. Assume that (M, g) is a complete Riemannian manifolds which satisfies the scale invariant family of Sobolev inequalities (2.11) (with some parameter ν > 2). Then the following propertie... |

42 |
Saloff-Coste L., Isopérimétrie pour les groupes et les variétés
- Coulhon
- 1993
(Show Context)
Citation Context ...≃ rn . Example 4.2. Let (M, g) be a connected unimodular Lie group equipped with a left-invariant Riemannian metric. Then the pseudo-Poincaré inequality of Definition 4.1 holds for any 1 � p � ∞ (see =-=[31]-=- or [87, 3.3.4]). The inequality ∀ f ∈ C ∞ c (M), ‖f‖ pN/(N−p) � S(M, p)‖∇f‖p holds if and only if V (r) � cr N for all r > 0. For instance, if M is the group of upper-triangular 3 by 3 matrices with ... |

42 |
Heat kernel upper bounds on a complete non-compact manifold
- Grigor’yan
- 1994
(Show Context)
Citation Context ...aurent Saloff-Coste ∫ |f| 2(1+2/ν) dµ � CM r2 µ(B) 2/ν ∫ [ 2 −2 2 |∇f| + r |f| ] ⎛ ⎞ ∫ dµ ⎝ |f|dµ ⎠ B B (see [81] and [75, Sect. 2.3]). The third is often referred to as a Faber–Krahn inequality (see =-=[44]-=-) and reads λD(Ω) � cM r 2 ( ) 2/ν µ(Ω) , µ(B) where λD(Ω) is the lowest Dirichlet eigenvalue in Ω, an arbitrary subset of the ball B of radius r. In each case, r is the radius of B and the inequality... |

42 |
Croissance polynomiale et période des fonctions harmoniques
- Guivarc’h
(Show Context)
Citation Context ..., a ∈ (0, ∞) such that V (e, r) � Cr a for all r � 1. A group G with polynomial volume growth must be unimodular (left-invariant Haar measures are also right-invariant) and, by a theorem of Guivarc’h =-=[55]-=-, there exists an integer N such that c0 � r −N V (e, r) � C0 for all r � 1. It follows that (G, g) satisfies the volume doubling property. By a simple direct argument (see, for example, [87, Theorem ... |

38 | Sobolev inequalities in disguise
- Coulhon, Saloff-Coste
- 1995
(Show Context)
Citation Context ... varies from one type of inequality to another. Many results in the spirit of these equivalences can be found in [75] in the context of Euclidean domains. A discussion in a very general setting is in =-=[4]-=- (see also [87, Chapt. 3]). The following theorem describes some of the noteworthy consequences of (2.11). Let ∆M be the Laplace operator on M, and let h(t, x, y) be the (minimal) fundamental solution... |

38 |
On the upper estimate of the heat kernel of a complete Riemannian
- Cheng, Li, et al.
- 1981
(Show Context)
Citation Context ... motivated by our desire to use this approach in other settings such as Riemannian manifolds or more exotic spaces. Early uses of Moser’s iteration technique on manifolds as in the influential papers =-=[22, 23]-=- were actually limited by a misunderstanding of what is really needed to run this technique successfully. Interesting early works that explored the flexibility of Moser’s iteration beyond the classica... |

36 | Riemannian Geometry–A Modern Introduction, Cambridge Tracts - Chavel - 1993 |

34 |
Rough isometries and combinatorial approximations of geometries of noncompact riemannian manifolds
- Kanai
- 1985
(Show Context)
Citation Context ... p < ∞. 3.6 Rough isometries One of the strengths of the techniques and results discussed in this paper is their robustness. In the present context, the idea of rough isometry was introduced by Kanai =-=[64, 66, 65]-=- and developed further in [32]. It has also been made very popular by the work of M. Gromov. Note that rough isometries as defined below do not preserve the small scale structure of the space. Definit... |

33 |
Hardy-Littlewood Theory for Semigroups
- Varopoulos
- 1985
(Show Context)
Citation Context ...erator etA jointly defined on the spaces Lp (M, µ), 1 � p < ∞. One of the most straightforward results in this context is the following theorem from [26] which extends an earlier result of Varopoulos =-=[100]-=- (see also [103]). For α > 0 we set ∫∞ (−A) −α/2 = Γ (α/2) −1 0 t −1+α/2 e tA dt. Theorem 4.2. Fix p ∈ (1, ∞). Assume that e tA is a bounded holomorphic semigroup of operator on L p (M, µ) which exten... |

31 |
Estimates of heat kernels on Riemannian manifolds
- Grigor’yan
- 1999
(Show Context)
Citation Context |

30 | Telcs A., Sub-Gaussian estimates of heat kernels on infinite graphs
- Grigor’yan
(Show Context)
Citation Context ...olev inequalities because this case is quite interesting and important and does avoid most technical difficulties. For developments paralleling the ideas and results of Sect. 2 we refer the reader to =-=[28, 29, 30, 52, 53, 98]-=- and the references therein.332 Laurent Saloff-Coste 5.2 Sobolev inequalities and volume growth We start with the following two theorems. Theorem 5.1. Fix ν > 0. For a graph (V, E) as above, the foll... |

29 | G.,Inégalités isopérimétriques de Faber-Krahn et conséequences, Actes de la table ronde de géométrie différentielle (Luminy - Carron - 1992 |

29 | Telcs A., Harnack inequalities and sub-Gaussian estimates for random walks
- Grigor’yan
- 2002
(Show Context)
Citation Context ...olev inequalities because this case is quite interesting and important and does avoid most technical difficulties. For developments paralleling the ideas and results of Sect. 2 we refer the reader to =-=[28, 29, 30, 52, 53, 98]-=- and the references therein.332 Laurent Saloff-Coste 5.2 Sobolev inequalities and volume growth We start with the following two theorems. Theorem 5.1. Fix ν > 0. For a graph (V, E) as above, the foll... |

28 |
Heat kernels on weighted manifolds and applications
- Grigor’yan
- 2006
(Show Context)
Citation Context |

27 |
Isoperimetric inequalities and Markov chains
- Varopoulos
- 1985
(Show Context)
Citation Context ...(y) is the probability that our particle is at y at (discrete) time t given that it started at x at time 0. The idea of applying Sobolev type inequalities in this context was introduced by Varopoulos =-=[101]-=- and produced a remarkable breakthrough in the study of random walks on graphs and finitely generated groups. The book [105] gives a detailed treatment of many aspects of the resulting developments. T... |

26 | Variétés Riemanniennes isométriques à l’infini - Coulhon, Saloff-Coste - 1995 |

26 | On the relation between elliptic and parabolic Harnack inequalities
- Hebisch, Saloff-Coste
(Show Context)
Citation Context ...ies are related to the elliptic version of Harnack inequality. This is not entirely understood, but the following result involving the Sobolev inequality (2.11) sheds some light on this question (see =-=[61]-=-). Theorem 2.3. Let M be a complete manifold satisfying the Sobolev inequality (2.11) for some q > 1. Then the following properties are equivalent. • The scale invariant L 2 Poincaré inequality. • The... |

25 |
Ultracontractivity and Nash type inequalities
- Coulhon
- 1996
(Show Context)
Citation Context ... SM being the same constant as in the Euclidean n-space. The first property in Theorem 4.3 obviously calls for a more general formulation. The following general elegant result was obtained by Coulhon =-=[27]-=- (after many attempts by different authors). A smooth positive function Φ defined on [0, ∞) satisfies condition (D) if there exists ε ∈ (0, 1) such that ϕ ′ (s) � εϕ ′ (t) for all t > 0 and s ∈ [t, 2t... |

24 |
Local behavior of solutions of quasi-linear parabolic equations. Archive for Rational Mechanics and Analysis 25
- Aronson, Serrin
- 1967
(Show Context)
Citation Context ... where Q− = (s − 3r 2 , s − 2r 2 ) × B and Q+ = (s − r 2 /2, s) × B. Moser’s iteration technique has been adapted and used in hundreds of papers studying various PDE problems. Some early examples are =-=[2, 3, 90]-=-. The books [42, 69, 76] contain many applications of this circle of ideas, as well as further references. The survey paper [83] deals specifically with the heat equation and is most relevant for the ... |

24 | Manifolds and graphs with slow heat kernel decay
- Barlow, Coulhon, et al.
(Show Context)
Citation Context ...e depends on the validity of a pseudoPoincaré inequality. Under that extra hypothesis, the Nash inequality one obtains is, in fact, equivalent to the volume lower bound. Both results are optimal (see =-=[6]-=-). Theorem 5.2 ([6, 31]). Fix ν > 0 and assume that a graph (V, E) has volume growth bounded from below: ∀ x ∈ V, r > 0, V (x, r) � cr ν . • In all the cases, ∀ f ∈ Cc(V ), ‖f‖ (1+1/γ) 2 � N‖∇f‖2‖f‖ 1... |

24 |
On a pointwise estimate for parabolic differential equations
- MOSER
- 1971
(Show Context)
Citation Context ...‖p = ‖u2‖∞. The desired conclusion 2.2 Harnack inequalities The technique illustrated above is the simplest instance of what is widely known as Moser’s iteration technique. In a series of papers [77]–=-=[80]-=-, Moser developed this technique as the basis for the study of divergence form uniformly elliptic operators in Rn , i.e., operators of the form (we use ∂i = ∂/∂xi) La = ∑ ∂i(ai,j(x)∂j) i,j with real m... |

24 |
Gromov’s theorem on groups of polynomial growth and elementary logic
- Dries, Wilkie
- 1984
(Show Context)
Citation Context ...means that a group carrying a recurrent random walk must have a volume growth function satisfying ∀ε > 0, lim inf r→∞ r−(2+ε) V (r) < ∞. By the celebrated theorem of Gromov [54] (and its extension in =-=[99]-=-), the condition ∃ A > 0, lim inf r→∞ r−AV (r) < ∞ (5.5) implies that G contains a nilpotent subgroup of finite index. Since a subgroup of finite index in G has volume growth comparable to that of G a... |

23 |
A note on Poincaré
- Saloff-Coste
- 1992
(Show Context)
Citation Context |

23 |
Analysis on local Dirichlet spaces. I: Recurrence, conservativeness and L p -Liouville properties
- Sturm
(Show Context)
Citation Context ...ompact closure given by the associated closed ball. Set V (x, r) = µ(B(x, r)). For each fixed x ∈ M, r > 0 the function δ(y) = max{0, r − ρ(x, y)} is in D(E) ∩ Cc(M) and satisfies dΓ (δ, δ) � dµ (see =-=[10, 11, 12, 94, 95, 96, 97]-=- for details). 3.3 Local weak solutions of the Laplace and heat equations Recall that A is the infinitesimal generator of the semigroup of operators associated to our Dirichlet form. Identify L 2 (M, ... |

22 |
Analysis on local Dirichlet spaces. III. The parabolic Harnack inequality
- Sturm
- 1996
(Show Context)
Citation Context ...ompact closure given by the associated closed ball. Set V (x, r) = µ(B(x, r)). For each fixed x ∈ M, r > 0 the function δ(y) = max{0, r − ρ(x, y)} is in D(E) ∩ Cc(M) and satisfies dΓ (δ, δ) � dµ (see =-=[10, 11, 12, 94, 95, 96, 97]-=- for details). 3.3 Local weak solutions of the Laplace and heat equations Recall that A is the infinitesimal generator of the semigroup of operators associated to our Dirichlet form. Identify L 2 (M, ... |

20 |
A Saint-Venant type principle for Dirichlet forms on discontinuous
- Biroli, Mosco
- 1995
(Show Context)
Citation Context ...hain rule and Leibnitz rule apply. In what follows, we work under314 Laurent Saloff-Coste additional assumptions that imply that the set of those u in D(E) such that dΓ/dµ exists is rich enough (see =-=[10, 97]-=- for further details). We now introduce a key ingredient to our discussion: the intrinsic distance. Definition 3.1. Let (M, µ, E, D(E)) be a strictly local regular Dirichlet space. For x, y in M we se... |