GAGLIARDO NIRENBERG INEQUALITIES ON MANIFOLDS

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GAGLIARDO-NIRENBERG INEQUALITIES ON MANIFOLDS NADINE BADR Abstract. We prove Gagliardo-Nirenberg inequalities on some classes of manifolds, Lie groups and graphs. Contents 1. Introduction 1 2. Preliminaries 5 2.1. Besov and Morrey spaces 5 2.2. Sobolev spaces on Riemannian manifolds 6 2.3. Doubling property and Poincare inequalities 7 3. Ledoux's and Sobolev inequalities 8 3.1. The classical Sobolev inequality 9 3.2. Sobolev inequalities for Lorentz spaces 10 4. Proof of Theorem 1.1, 1.6, 1.8 and 1.9 11 References 13 1. Introduction Cohen-Meyer-Oru [5], Cohen-Devore-Petrushev-Xu [4], proved the following Gagliardo- Nirenberg type inequality (1.1) ?f?1? ≤ C? |?f | ? n?1 n 1 ?f? 1 n B?(n?1)∞,∞ for all f ? W 11 (R n) (1? = nn?1). The proof of (1.1) is involved and based on wavelet decompositions, weak type (1,1) estimates and interpolation results. Using a simple method relying on weak type estimates and pseudo-Poincare in- equalities, Ledoux [14] obtained the following extension of (1.1). He proved that for 1 ≤ p < l <∞ and for every f ? W 1p (R n) (1.2) ?f?l ≤ C? |?f | ? ? p?f? 1?? B ? ??1 ∞,∞ where ? = pl and C > 0 only depends on l, p and n.

cohen-devore-petrushev-xu

gagliardo-nirenberg inequalities

haar measure

riemannian manifold

lie group equipped

dµ ≤

ledoux

linear elliptic

lie group

inequality

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Publié par

profil-urra-2012

Nombre de lectures

Langue

English

GAGLIARDO-NIRENBERG INEQUALITIES ON MANIFOLDS

NADINE BADR Abstract. We prove Gagliardo-Nirenberg inequalities on some classes of manifolds, Lie groups and graphs.

Contents

1. Introduction 2. Preliminaries 2.1. Besov and Morrey spaces 2.2. Sobolev spaces on Riemannian manifolds 2.3.DoublingpropertyandPoincare´inequalities 3. Ledoux’s and Sobolev inequalities 3.1. The classical Sobolev inequality 3.2. Sobolev inequalities for Lorentz spaces 4. Proof of Theorem 1.1, 1.6, 1.8 and 1.9 References

1 5 5 6 7 8 9 10 11 13

1. Introduction Cohen-Meyer-Oru [5], Cohen-Devore-Petrushev-Xu [4], proved the following Gagliardo-Nirenberg type inequality n − 1 1 (1.1) k f k 1 ∗ ≤ C k |r f | k 1 n k f k n B −∞ ( ,n ∞− 1) for all f ∈ W 11 ( R n ) (1 ∗ = n − n 1 ). The proof of (1.1) is involved and based on wavelet decompositions, weak type (1,1) estimates and interpolation results. Usingasimplemethodrelyingonweaktypeestimatesandpseudo-Poincare´in-equalities, Ledoux [14] obtained the following extension of (1.1). He proved that for 1 ≤ p < l < ∞ and for every f ∈ W p 1 ( R n ) (1.2 θ k f k 1 − θ θ ) k f k l ≤ C k |r f | k pB θ − 1 ∞ , ∞ where θ = lp and C > 0 only depends on l, p and n . In the same paper, he extended (1.2) to the case of Riemannian manifolds. If p = 2 he observed that (1.2) holds without any assumption on M . If p 6 = 2 he assumed that the Ricci curvature is non-negative and obtained (1.2) with C > 0 only depending on l, p when 1 ≤ p ≤ 2 and on l, p and n when 2 < p < ∞ . 2000 Mathematics Subject Classiﬁcation. 46E30, 26D10, 46B70. Key words and phrases. Gagliardo, Nirenberg, Symmetrization, Sobolev spaces, Interpolation.

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