Sobolev spaces in the presence of symmetries

Hebey, Emmanuel; Vaugon, Michel

doi:10.1016/s0021-7824(97)89975-8

Cited by 66 publications

(36 citation statements)

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“…Assume that Gx 0 is an orbit of concentration of dimension k > k. Then there exists δ > 0 such that dim Gx ≥ k > k for any x ∈ B Gx0 (δ) (see Faget [3] lemma 2). It thus follows from Hebey-Vaugon (corollary 2 of [6]) and the inequality (n−k )p …”

Section: Vol 15 2008mentioning

confidence: 83%

“…The set of smooth G-invariant functions on M being dense in H p 1,G (M ) (see HebeyVaugon [6]), we can assume that the v α 's are smooth. Independently, since the v α 's don't converge strongly to 0, the definition of a (P-S) sequence implies that…”

Section: Vol 15 2008mentioning

confidence: 99%

“…∇u is the p-Laplacian of u, p * = (n−k)p n−k−p is the critical exponent for the injection from the Sobolev space H p 1,G (M ) of G-invariant functions in L p (M ) whose gradient is also in L p (M ), into the Lebesgue spaces L q G (M ) of G-invariant functions in L q (M ) (cf HebeyVaugon [6]), f is a C 1 G-invariant function, and (h α ) is a sequence of continuous G-invariant functions converging uniformly to some continous G-invariant function h ∞ . The solutions we consider are in H p 1 ; therefore, a solution to (E α ) has to be taken in the distribution sense.…”

mentioning

confidence: 99%

“…We refer to Bredon [1] for more details (see also Hebey-Vaugon [6] and Faget [2]). Let G be a closed subgroup of Isom g (M ).…”

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Blow-up theory for symmetric critical equations involving the p-Laplacian

Saintier

2008

Nonlinear differ. equ. appl.

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Section: Vol 15 2008mentioning

confidence: 83%

Section: Vol 15 2008mentioning

confidence: 99%

mentioning

confidence: 99%

“…We refer to Bredon [1] for more details (see also Hebey-Vaugon [6] and Faget [2]). Let G be a closed subgroup of Isom g (M ).…”

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confidence: 99%

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Blow-up theory for symmetric critical equations involving the p-Laplacian

Saintier

2008

Nonlinear differ. equ. appl.

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“…[2,8,9,11,12]. In particular, the Sobolev spaces on Riemannian manifolds were studied by Hebey, Vaugon [5,6], and one of the authors [13]. A more abstract approach was developed by the second author and his collaborators [3,10,17] The purpose of this paper is to identify a general geometric condition that is necessary and sufficient for such a compactness phenomenon to occur in a setting when the original embedding is inherently non-compact.…”

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confidence: 99%

A geometric criterion for compactness of invariant subspaces

Skrzypczak

Tintarev

2013

Arch. Math.

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Abstract. Let M be a non-compact homogeneous Riemannian manifold, and let Ω be a compact subgroup of isometries of M . We show, under general conditions, that the Ω-invariant subspace AΩ of a normed vector space A → L q (M ) is compactly embedded into L q (M ) if and only if the group Ω has no orbits with a uniformly bounded diameter in a neighborhood of infinity. Mathematics Subject Classification (2010). Primary 46B50; Secondary 46E35, 46N20.

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Entropy of Sobolev embeddings of radial functions and radial eigenvalues of Schrödinger operators on isotropic manifolds

Skrzypczak

Tomasz

2007

Mathematische Nachrichten

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Key words Sobolev embeddings, entropy numbers, hyperbolic spaces, Schrödinger operator MSC (2000) 46E35, 41A46, 47B06We consider Sobolev embeddings between Sobolev and Besov spaces of radial functions on noncompact symmetric spaces of rank one. An asymptotic behaviour of entropy numbers of the compact embeddings is described. The estimates are used for investigation of the negative spectrum of Schrödinger type operators.Let X be a connected Riemannian manifold and let o be a fixed point of X. We call a function f (x) radial if its value at x depends only on a distance of x to the point o. It was noticed in late seventies of the last century that radiality implies compactness of Sobolev embeddings. First, it was proved for the first order Sobolev spaces of radial functions defined on R d by Strauss [27] and Coleman, Glazer, and Martin [3], cf. also Berestycki and Lions [1]. Later the corresponding theory was developed for other type of function spaces, cf. [22], more general symmetry conditions, cf. [19] and [25], as well as on function spaces defined on Riemannian manifolds, cf. [11], [13], and [24]. On manifolds the symmetry conditions are expressed in terms of invariance with respect to the action of a compact group of isometries.Isotropic Riemannian manifolds seem to be of special interest here since on such manifolds radial means invariant with respect to the action of isotropy group of the point o. A Riemannian manifold X is called isotropic if for each y ∈ X the linear isotropy group acts transitively on the unit sphere in the tangent space T y X. It is well-known that a noncompact Riemannian manifold is isotropic if and only if it is either an Euclidean space or a noncompact Riemannian globally symmetric space of rank one.The quantitative approach to the compactness of Sobolev embeddings of radial Sobolev and Besov spaces in terms of entropy numbers was developed in [17]. The n-th (dyadic) entropy number of a bounded linear operator T : X → Y between two Banach spaces is defined aswhere B X and B Y stand for the closed unit balls in X and Y , respectively. For basic properties of entropy numbers and more background we refer to the literature, see e.g. [2], [5], [16] or [20]. The operator T is compact if and only if lim n→∞ e n (T ) = 0. The quantitative approach to compactness means that we describe an asymptotic behaviour of e n (T ) as n tends to infinity. Let RH s p R d , s ∈ R, 1 < p < ∞, denote a subspace of the fractional Sobolev space H s p R d consisted of radial distributions. In a similar way we define a radial Besov space RB s p,q R d , 1 ≤ p, q ≤ ∞. The Sobolev embeddings RH s0 p0 R d → RH s1 p1 R d and RB s0 p0,q0 R d → RB s1 p1,q1 R

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Sobolev spaces in the presence of symmetries

Cited by 66 publications

References 7 publications

Blow-up theory for symmetric critical equations involving the p-Laplacian

Blow-up theory for symmetric critical equations involving the p-Laplacian

A geometric criterion for compactness of invariant subspaces

Entropy of Sobolev embeddings of radial functions and radial eigenvalues of Schrödinger operators on isotropic manifolds

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