REAL INTERPOLATION OF SOBOLEV SPACES
25
pages
English
Documents
Obtenez un accès à la bibliothèque pour le consulter en ligne En savoir plus
Découvre YouScribe en t'inscrivant gratuitement
Découvre YouScribe en t'inscrivant gratuitement
25
pages
English
Documents
Obtenez un accès à la bibliothèque pour le consulter en ligne En savoir plus
REAL INTERPOLATION OF SOBOLEV SPACES NADINE BADR Abstract. We prove that W 1p is a real interpolation space between W 1 p1 and W 1 p2 for p > q0 and 1 ≤ p1 < p < p2 ≤ ∞ on some classes of manifolds and general metric spaces, where q0 depends on our hypotheses. Contents 1. Introduction 1 2. Preliminaries 3 2.1. The doubling property 3 2.2. The K-method of real interpolation 4 3. Non-homogeneous Sobolev spaces on Riemannian manifolds 5 3.1. Non-homogeneous Sobolev spaces 5 3.2. Estimation of the K-functional of interpolation 5 4. Interpolation Theorems 13 5. Homogeneous Sobolev spaces on Riemannian manifolds 15 6. Sobolev spaces on compact manifolds 18 7. Metric-measure spaces 18 7.1. Upper gradients and Poincare inequality 18 7.2. Interpolation of the Sobolev spaces H1p 19 8. Applications 21 8.1. Carnot-Caratheodory spaces 21 8.2. Weighted Sobolev spaces 22 8.3. Lie Groups 22 9. Appendix 23 References 24 1. Introduction Do the Sobolev spaces W 1p form a real interpolation scale for 1 < p <∞? The aim of the present work is to provide a positive answer for Sobolev spaces on some metric spaces. Let us state here our main theorems for non-homogeneous Sobolev spaces (resp. homogeneous Sobolev spaces) on Riemannian manifolds.
Voir
- measure space
- riemannian manifold
- interpolation
- interpolation space between
- can take
- p2 ≤
- general spaces
- calderon- zygmund decomposition
REAL INTERPOLATION OF SOBOLEV SPACES
NADINE BADR
Abstract.We prove thatWp1is a real interpolation space betweenWp11andWp12 forp > q0and 1≤p1< p < p2≤ ∞on some classes of manifolds and general metric spaces, whereq0depends on our hypotheses.
Contents
1. Introduction 2. Preliminaries 2.1. The doubling property 2.2. TheK-method of real interpolation 3. Non-homogeneous Sobolev spaces on Riemannian manifolds 3.1. Non-homogeneous Sobolev spaces 3.2. Estimation of theK-functional of interpolation 4. Interpolation Theorems 5. Homogeneous Sobolev spaces on Riemannian manifolds 6. Sobolev spaces on compact manifolds 7. Metric-measure spaces 7.1.UppergradientsandPoincare´inequality 7.2. Interpolation of the Sobolev spacesHp1 8. Applications 8.1. Carnot-Caratheodory spaces ´ 8.2. Weighted Sobolev spaces 8.3. Lie Groups 9. Appendix References
1 3 3 4 5 5 5 13 15 18 18 18 19 21 21 22 22 23 24
1.ucodtrInonti Do the Sobolev spacesWp1form a real interpolation scale for 1< p <∞ aim? The of the present work is to provide a positive answer for Sobolev spaces on some metric spaces. Let us state here our main theorems for non-homogeneous Sobolev spaces (resp. homogeneous Sobolev spaces) on Riemannian manifolds.
Theorem 1.1.LetMbe a complete non-compact Riemannian manifold satisfying the local doubling property(Dloc)ilytcalodaanauqenie´racnioPl(Pqloc), for some 2000Mathematics Subject Classification.46B70, 46M35. Key words and phrases.InolatterpoboS;noiecapsvelcainPos;quneeir´repo;ytliubprngitalDoy; Riemannian manifolds; Metric-measure spaces. 1
1≤q <∞ for. Then1≤r≤q < p <∞,Wp1is a real interpolation space between Wr1andW1∞. To prove Theorem 1.1, we characterize theK-functional of real interpolation for non-homogeneous Sobolev spaces:
Theorem 1.2.LetMbe as in Theorem 1.1. Then 1.there existsC1>0such that for allf∈Wr1+W∞1andt >0 K(f, tr1, Wr1, W1∞)≥C1t1r|f|r∗∗1r(t) +|rf|r∗∗r1(t); 2.forr≤q≤p <∞, there isC2>0such that for allf∈Wp1andt >0 K(f, t1r, Wr1, W1)≤C2tr1|f|q∗∗1q(t) +|rf|q∗∗q1(t). ∞ In the special caser=q, we obtain the upper bound ofK every forin point 2. f∈Wq1+W1∞and hence get a true characterization ofK. TheproofofthistheoremreliesonaCalder´on-ZygmunddecompositionforSobolev functions (Proposition 3.5). Above and from now on,|g|q∗∗q1means (|g|q∗∗)1q–see section 2 for the definition of ∗∗ g–. The reiteration theorem ([6], Chapter 5, Theorem 2.4 p.311) and an improvement resultfortheexponentofaPoincare´inequalityduetoKeith-Zhongyieldamore general version of Theorem 1.1. Defineq0= inf{q∈[1,∞[: (Pqloc) holds}. Corollary 1.3.For1≤p1< p < p2≤ ∞withp > q0,Wp1is a real interpolation space betweenWp11andWp12 precisely. More Wp1= (Wp11, Wp12)θ,p 1−θ . where0< θ <1such that1p=p1+θp2 However, ifp≤q0, we only know that (Wp11, Wp12)θ,p⊂Wp1. For the homogeneous Sobolev spaces, a weak form of Theorem 1.2 is available. This result is presented in section 5. The consequence for the interpolation problem is stated as follows.
Theorem 1.4.LetMbe a complete non-compact Riemannian manifold satisfying the operty(D)and a global Poinc ´ i equality(Pq)for some1≤q <∞. global doubling pr are n ˙ ˙ ˙ Then, for1≤r≤q < p <∞,Wp1is a real interpolation space betweenWr1andW∞1. Again, the reiteration theorem implies another version of Theorem 1.4; see section 5 below. ForRnand the non-homogeneous Sobolev spaces, our interpolation result follows from the leading work of Devore-Scherer [14]. The method of [14] is based on spline functions.Later,simplerproofsweregivenbyCalder´on-Milman[9]andBennett-Sharpley [6], based on the Whitney extension and covering theorems. SinceRnadmits (D) and (P1 applying Theorem 1.4, Moreover,), we recover this result by our method. we obtain the interpolation of the homogeneous Sobolev spaces onRn that. Notice this result is not covered by the existing references. 2
