MATH. SCAND. 105 (2009), 235–264
REAL INTERPOLATION OF SOBOLEV SPACES
NADINE BADR
Abstract We prove that W 1 is a real interpolation space between W 1 and W 1 for p>q and 1 ≤ p < p p1 p2 0 1 p 1. Introduction 1 ∞ Do the Sobolev spaces Wp form a real interpolation scale for 1 Received January 9, 2007; in revised form June 11, 2008. 236 nadine badr The proof of this theorem relies on a Calderón-Zygmund decomposition for Sobolev functions (Proposition 3.5). q∗∗ 1 ∗∗ 1 Above and from now on, |g| q means (|g|q ) q – see Section 2 for the definition of g∗∗. The reiteration theorem ([6], Chapter 5, Theorem 2.4, p. 311) and an im- provement result for the exponent of a Poincaré inequality due to Keith-Zhong yield a more general version of Theorem 1.1. Define q0 = inf{q ∈ [1, ∞[: (Pqloc) holds}. ≤ ≤∞ 1 Corollary 1.3. For 1 p1 2. Preliminaries Throughout this paper we will denote by 11E the characteristic function of a set E and Ec the complement of E.IfX is a metric space, Lip will be the set of real Lipschitz functions on X and Lip0 the set of real, compactly supported Lipschitz functions on X. For a ball B in a metric space, λB denotes the ball co-centered with B and with radius λ times that of B. Finally, C will be a constant that may change from an inequality to another and we will use u ∼ v to say that there exists two constants C1, C2 > 0 such that C1u ≤ v ≤ C2u. 2.1. The doubling property By a metric-measure space, we mean a triple (X,d,μ)where (X, d) is a metric space and μ a non negative Borel measure. Denote by B(x,r) the open ball of center x ∈ X and radius r>0. Definition 2.1. Let (X,d,μ) be a metric-measure space. One says that X satisfies the local doubling property (Dloc) if there exist constants r0 > 0, 238 nadine badr 0 (Dloc) μ(B(x, 2r)) ≤ Cμ(B(x, r)). Furthermore X satisfies a global doubling property or simply doubling property (D) if one can take r0 =∞. We also say that μ is a locally (resp. globally) doubling Borel measure. Observe that if X is a metric-measure space satisfying (D) then diam(X) < ∞⇔μ(X) < ∞ ([1]). Theorem 2.2 (Maximal theorem ([11])). Let (X,d,μ)be a metric-measure space satisfying (D). Denote by M the uncentered Hardy-Littlewood maximal function over open balls of X defined by Mf(x)= sup |f |B B:x∈B 1 where fE := − fdμ:= fdμ. Then E μ(E) E { M } ≤ C | | 1. μ( x : f(x)>λ) λ X f dμ for every λ>0; M ≤ ≤∞ 2. f Lp Cp f Lp , for 1 2.2. The K-method of real interpolation The reader can refer to [6], [7] for details on the development of this theory. Here we only recall the essentials to be used in the sequel. Let A0, A1 be two normed vector spaces embedded in a topological Haus- dorff vector space V . For each a ∈ A0 + A1 and t>0, we define the K- functional of interpolation by = + K(a,t,A0,A1) inf ( a0 A0 t a1 A1 ). a=a0+a1 For 0 <θ<1, 1 ≤ q ≤∞, we denote by (A0,A1)θ,q the interpolation space between A0 and A1: (A0,A1)θ,q = a ∈ A0 + A1 ∞ 1 q −θ q dt : a θ,q = t K(a,t,A0,A1) < ∞ . 0 t It is an exact interpolation space of exponent θ between A0 and A1, see [7], Chapter II. real interpolation of sobolev spaces 239 Definition 2.3. Let f be a measurable function on a measure space (X, μ). The decreasing rearrangement of f is the function f ∗ defined for every t ≥ 0 by f ∗(t) = inf{λ : μ({x : |f(x)| >λ}) ≤ t}. The maximal decreasing rearrangement of f is the function f ∗∗ defined for every t>0by 1 t f ∗∗(t) = f ∗(s)ds. t 0 It is known that (Mf)∗ ∼ f ∗∗ and μ({x : |f(x)| >f∗(t)}) ≤ t for all t>0. We refer to [6], [7], [8] for other properties of f ∗ and f ∗∗. We conclude the preliminaries by quoting the following classical result ([7] p. 109): Theorem 2.4. Let (X, μ) be a measure space where μ is a totally σ-finite positive measure. Let f ∈ Lp + L∞, 0 3.1. Non-homogeneous Sobolev spaces Definition 3.1 ([2]). Let M be a C∞ Riemannian manifold of dimension 1 ∞ |∇ |∈ n. Write Ep for the vector space of C functions ϕ such that ϕ and ϕ ≤ ∞ 1 1 Lp, 1 p< . We define the Sobolev space Wp as the completion of Ep for the norm ϕ 1 = ϕ + |∇ϕ| . Wp p p 1 We denote W∞ for the set of all bounded Lipschitz functions on M. Proposition 3.2 ([2], [20]). Let M be a complete Riemannian manifold. ∞ 1 ≤ ∞ Then C0 and in particular Lip0 is dense in Wp for 1 p< . Definition 3.3 (Poincaré inequality on M). We say that a complete Riemannian manifold M admits a local Poincaré inequality (Pqloc) for some 240 nadine badr 1 ≤ q<∞ if there exist constants r1 > 0,C= C(q,r1)>0 such that, for ∈ every function f Lip0 and every ball B of M of radius 0 M admits a global Poincaré inequality (Pq ) if we can take r1 =∞in this definition. ∞ 1 ∞ Remark 3.4. By density of C0 in Wp , we can replace Lip0 by C0 . 3.2. Estimation of the K-functional of interpolation In the first step, we prove Theorem 1.2 in the global case. This will help us to understand the proof of the more general local case. 3.2.1. The global case . Let M be a complete Riemannian manifold satis- fying (D) and (Pq ), for some 1 ≤ q<∞. Before we prove Theorem 1.2, we make a Calderón-Zygmund decomposition for Sobolev functions inspired by the one done in [3]. To achieve our aims, we state it for more general spaces ∞ (in [3], the authors only needed the decomposition for the functions f in C0 ). This will be the principal tool in the estimation of the functional K. Proposition 3.5 (Calderón-Zygmund lemma for Sobolev functions). Let M be a complete non-compact Riemannian manifold satisfying (D). Let 1 ≤ ∞ ≤ ∞ ∈ 1 q< and assume that M satisfies (Pq ). Let q p< , f Wp and ∈ 1 α>0. Then one can find a collection of balls (Bi )i , functions bi Wq and a Lipschitz function g such that the following properties hold: (3.1)f= g + bi i (3.2) |g(x)|≤Cα and |∇g(x)|≤Cα, μ − a.e x ∈ M q q q (3.3) supp bi ⊂ Bi , (|bi | +|∇bi | )dμ≤ Cα μ(Bi ) Bi −p p (3.4) μ(Bi ) ≤ Cα (|f |+|∇f |) dμ i ≤ (3.5) χBi N. i real interpolation of sobolev spaces 241 The constants C and N only depend on q, p and on the constants in (D) and (Pq ). ∈ 1 ={ ∈ M | |+ Proof. Let f Wp , α>0. Consider x M : ( f |∇f |)q (x)>αq }.If =∅, then set g = f, bi = 0 for all i so that (3.2) is satisfied according to the Lebesgue differentiation theorem. Otherwise the maximal theorem – Theorem 2.2 – gives us p −p q q μ( ) ≤ Cα (|f |+|∇f |) p q (3.6) − ≤ Cα p |f |p dμ + |∇f |p dμ < +∞. In particular = M as μ(M) =+∞. Let F be the complement of . Since is an open set distinct of M, let (Bi ) be a Whitney decomposition of ([12]). The balls Bi are pairwise disjoint and there exist two constants C2 >C1 > 1, depending only on the metric, such that 1. =∪i Bi with Bi = C1Bi and the balls Bi have the bounded overlap property; = = 1 2. ri r(Bi ) 2 d(xi ,F)and xi is the center of Bi ; 3. each ball Bi = C2Bi intersects F (C2 = 4C1 works). For x ∈ , denote Ix ={i : x ∈ Bi }. By the bounded overlap property of the balls Bi , we have that Ix ≤ N. Fixing j ∈ Ix and using the properties of the 1 ≤ ≤ ∈ ⊂ Bi ’s, we easily see that 3 ri rj 3ri for all i Ix . In particular, Bi 7Bj for all i ∈ Ix . Condition (3.5) is nothing but the bounded overlap property of the Bi ’s and (3.4) follows from (3.5) and (3.6). The doubling property and the fact that Bi ∩ F =∅yield (|f |q +|∇f |q )dμ≤ (|f |+|∇f |)q dμ Bi Bi (3.7) q ≤ α μ(Bi ) q ≤ Cα μ(Bi ). Let us now define the functions bi . Let (χi )i be a partition of unity of subordinated to the covering (Bi ), such that for all i, χi is a Lipschitz function 242 nadine badr C supported in Bi with |∇χi | ∞ ≤ . To this end it is enough to choose χi (x) = ri −1 ψ C1d(xi ,x) ψ C1d(xk ,x) , where ψ is a smooth function, ψ = 1on ri k rk + = 1 C1 +∞ ≤ ≤ = − [0, 1], ψ 0on 2 , and 0 ψ 1. We set bi (f fBi )χi .It q q is clear that supp bi ⊂ Bi . Let us estimate |bi | dμ and |∇bi | dμ.We Bi Bi have | |q = | − |q ≤ | |q + | |q bi dμ (f fBi )χi dμ C f dμ fBi dμ B B B B i i i i ≤ C |f |q dμ Bi q ≤ Cα μ(Bi ). We applied Jensen’s inequality in the second estimate, and (3.7) in the last one. ∇ − = ∇ + − ∇ Since (f fBi )χi χi f (f fBi ) χi , the Poincaré inequality (Pq ) and (3.7) yield |∇ |q ≤ | ∇ |q + | − |q |∇ |q bi dμ C χi f dμ f fBi χi dμ B B B i i i Cq ≤ q + q |∇ |q Cα μ(Bi ) C q ri f dμ ri Bi q ≤ Cα μ(Bi ). Therefore (3.3) is proved. = − Set g f i bi . Since the sum is locally finite on , g is defined almost everywhere on M and g = f on F . Observe that g is a locally integrable function on M. Indeed, let ϕ ∈ L∞ with compact support. Since d(x,F) ≥ ri for x ∈ supp bi , we obtain |bi | |bi ||ϕ| dμ ≤ dμ sup d(x,F)|ϕ(x)| r ∈ i i i x M and 1 |b | |f − f | 1 q i Bi q q dμ = χi dμ ≤ μ(Bi ) |∇f | dμ ri Bi ri Bi ≤ Cαμ(Bi ). We used the Hölder inequality, (Pq ) and the estimate (3.7), q being the con- | || | ≤ | | jugate of q. Hence i bi ϕ dμ Cαμ( ) supx∈M d(x,F) ϕ(x) . Since f ∈ L1,loc, we deduce that g ∈ L1,loc. (Note that since b ∈ L1 in our case, real interpolation of sobolev spaces 243 we can say directly that g ∈ L1,loc. However, for the homogeneous case – ∈ Section5–weneed this observation to conclude that g L1,loc.) It remains = ∇ = ∈ to prove (3.2). Note that i χi (x) 1 and i χi (x) 0 for all x .We have ∇ =∇ − ∇ =∇ − ∇ − − ∇ g f bi f χi f (f fBi ) χi i i i = ∇ + ∇ 11F ( f) fBi χi . i From the definition of F and the Lebesgue differentiation theorem, we have | |+|∇ | ≤ − that 11F ( f f ) αμ a.e.. We claim that a similar estimate holds = ∇ | |≤ ∈ for h i fBi χi .Wehave h(x) Cα for all x M. For this, note first that h vanishes on F and is locally finite on . Then fix x ∈ and let Bj be ∈ | − |≤ some Whitney ball containing x. For all i Ix ,wehave fBi fBj Crj α. Indeed, since Bi ⊂ 7Bj ,weget | − |≤ 1 | − | fBi f7Bj f f7Bj dμ μ(Bi ) B i C ≤ |f − f | dμ μ(B ) 7Bj (3.8) j 7Bj 1 q q ≤ Crj − |∇f | dμ 7Bj ≤ Crj α | − where we used Hölder inequality, (D), (Pq ) and (3.7). Analogously f7Bj |≤ fBj Crj α. Hence | |= − ∇ ≤ | − | −1 ≤ h(x) (fBi fBj ) χi (x) C fBi fBj ri CNα. i∈Ix i∈Ix From these estimates we deduce that |∇g(x)|≤Cα μ − a.e.. Let us now ∞ = + | | ≤ estimate g .Wehave g f 11F i fBi χi . Since f 11F α, still need to estimate i fBi χi ∞. Note that q | |q ≤ 1 | | ≤ M | |+|∇ | q fBi C f dμ ( f f ) (y) μ(Bi ) Bi (3.9) ≤ M(|f |+|∇f |)q (y) ≤ αq 244 nadine badr where y ∈ Bi ∩ F since Bi ∩ F =∅. The second inequality follows from the fact that (Mf)q ≤ Mf q for q ≥ 1. Let x ∈ . Inequality (3.9) and the fact that Ix ≤ N yield | |= ≤ | |≤ g(x) fBi χi fBi Nα. i∈Ix i∈Ix We conclude that g ∞ ≤ Cα μ− a.e. and the proof of Proposition 3.5 is therefore complete. ∈ 1 Remark 3.6. 1. It is a straightforward consequence of (3.3) that bi Wr 1 ≤ r ≤ q b 1 ≤ Cαμ(B ) r for all 1 with i Wr i . ∈ 1 2. From the construction of the functions bi , we see that i bi Wp , with ≤ ∈ 1 |∇ | bi 1 C f W 1 . It follows that g Wp . Hence (g, g ) satisfies the i Wp p Poincaré inequality (Pp). Theorem 3.2 of [23] asserts that for μ − a.e. x, y ∈ M p 1 p 1 |g(x) − g(y)|≤Cd(x, y) (M|∇g| ) p (x) + (M|∇g| ) p (y) . From Theorem 2.2 with p =∞and the inequality |∇g| ∞ ≤ Cα,we deduce that g has a Lipschitz representative. Moreover, the Lipschitz constant is controlled by Cα. 3. We also deduce from this Calderón-Zygmund decomposition that g ∈ W 1 s 1 1 ≤ ≤∞ | |s +|∇ |s s ≤ s for p s .Wehave ( g g )dμ Cαμ( ) and (|g|s +|∇g|s )dμ = (|f |s +|∇f |s )dμ F F ≤ (|f |p|f |s−p +|∇f |p|∇f |s−p)dμ F ≤ s−p p ∞ α f 1 < . Wp Corollary 3.7. Under the same hypotheses as in the Calderón-Zygmund lemma, we have 1 ⊂ 1 + 1 ≤ ≤ ≤ ≤ ∞ Wp Wr Ws for 1 r q p s< . Proof of Theorem 1.2. To prove part 1., we begin applying Theorem 2.4, part 1. We have 1 t r 1 ∗ r K(f,t r ,Lr ,L∞) ∼ (f (s)) ds . 0 real interpolation of sobolev spaces 245 On the other hand 1 1 t r t r ∗ ∗ ∗∗ 1 f (s)r ds = |f(s)|r ds = t|f |r (t) r 0 0 where in the first equality we used the fact that f ∗r = (|f |r )∗ and the second ∗∗ 1 follows from the definition of f . We thus get K(f,t r ,Lr ,L∞) ∼ 1 r∗∗ 1 t r (|f | ) r (t). Moreover, 1 1 1 r 1 1 ≥ r + |∇ | r K(f,t ,Wr ,W∞) K(f,t ,Lr ,L∞) K( f ,t ,Lr ,L∞) since the linear operator ∇ 1 → ; × (I, ) : Ws (M) (Ls (M C TM)) is bounded for every 1 ≤ s ≤∞. These two points yield the desired inequality. ∈ 1 ≤ We will now prove part 2. We treat the case when f Wp , q p< ∞. Let t>0. We consider the Calderón-Zygmund decomposition of f of ∗ 1 q q Proposition 3.5 with α = α(t) = M(|f |+|∇f |) (t). We write f = + = + i bi g b g where (bi )i ,gsatisfy the properties of the proposition. From the bounded overlap property of the Bi ’s, it follows that for all r ≤ q r r ≤ | | ≤ | |r b r bi dμ N bi dμ M i i Bi r r ≤ Cα (t) μ(Bi ) ≤ Cα (t)μ( ). i 1 Similarly we have |∇b| r ≤ Cα(t)μ( ) r . Moreover, since (Mf)∗ ∼ f ∗∗ and (f + g)∗∗ ≤ f ∗∗ + g∗∗,weget 1 q ∗ q∗∗ 1 q∗∗ 1 α(t) = M(|f |+|∇f |) q (t) ≤ C |f | q (t) +|∇f | q (t) . Noting that μ( ) ≤ t, we deduce that 1 1 q∗∗ 1 q∗∗ 1 r 1 1 ≤ r | | q +|∇ | q (3.10)K(f,t,Wr ,W∞) Ct f (t) f (t) ∈ 1 ≤ ∞ for all t>0 and obtain the desired inequality for f Wp , q p< . Note that in the special case where r = q, we have the upper bound of K ∈ 1 for f Wq . Applying a similar argument to that of [14] – Euclidean case – ∈ 1 + we get (3.10) for f Wq W∞. Here we will omit the details. We were not able to show this characterization when r 3.2.2. The local case . Let M be a complete non-compact Riemannian man- ifold satisfying a local doubling property (Dloc) and a local Poincaré inequality (Pqloc) for some 1 ≤ q<∞. Denote by ME the Hardy-Littlewood maximal operator relative to a meas- urable subset E of M, that is, for x ∈ E and every locally integrable function f on M 1 MEf(x)= sup |f | dμ B:x∈B μ(B ∩ E) B∩E where B ranges over all open balls of M containing x and centered in E.We say that a measurable subset E of M has the relative doubling property if there exists a constant CE such that for all x ∈ E and r>0wehave μ(B(x, 2r) ∩ E) ≤ CEμ(B(x, r) ∩ E). This is equivalent to saying that the metric-measure space (E, d|E,μ|E) has the doubling property. On such a set ME is of weak type (1, 1) and bounded on Lp(E, μ),1 We have μ( ) ≤ t.If = M then |f |r dμ + |∇f |r dμ = |f |r dμ + |∇f |r dμ M M μ( ) μ( ) ≤ |f |r∗(s) ds + |∇f |r∗(s) ds 0 0 t t ≤ |f |r∗(s) ds + |∇f |r∗(s) ds 0 0 = t |f |r∗∗(t) +|∇f |r∗∗(t) . Therefore 1 1 1 1 r∗∗ 1 r∗∗ 1 K(f,t r ,W ,W ) ≤ f 1 ≤ Ct r |f | r (t) +|∇f | r (t) r ∞ Wr 1 q∗∗ 1 q∗∗ 1 ≤ Ct r |f | q (t) +|∇f | q (t) since r ≤ q. We thus obtain the upper bound in this case. real interpolation of sobolev spaces 247 = { } ⊂ = Now assume M. Pick a countable set xj j∈J M, such that M 1 ∈ j∈J B xj , 2 and for all x M, x does not belong to more than N1 balls j ∞ B := B(xj , 1). Consider a C partition of unity (ϕj )j∈J subordinated to the 1 j j ≤ ≤ ⊂ |∇ | ∞ ≤ balls 2 B such that 0 ϕj 1, supp ϕj B and ϕj C uniformly ∈ 1 ≤ ∞ = with respect to j. Consider f Wp , q p< . Let fj fϕj so that = ∈ ∈ ∇ = ∇ +∇ ∈ f j∈J fj .Wehaveforj J , fj Lp and fj f ϕj fϕj Lp. ∈ 1 j j Hence fj Wp (B ). The balls B satisfy the relative doubling property with constant independent of the balls Bj . This follows from the next lemma quoted from [4] p. 947. Lemma 3.8. Let M be a complete Riemannian manifold satisfying (Dloc). Then the balls Bj above, equipped with the induced distance and measure, satisfy the relative doubling property (D), with the doubling constant that may be chosen independently of j. More precisely, there exists C ≥ 0 such that for all j ∈ J (3.11) μ(B(x, 2r) ∩ Bj ) ≤ C μ(B(x, r) ∩ Bj ) ∀x ∈ Bj ,r>0, and (3.12) μ(B(x, r)) ≤ Cμ(B(x, r) ∩ Bj ) ∀x ∈ Bj , 0 Remark 3.9. Noting that the proof in [4] only used the fact that M is a length space, we observe that Lemma 3.8 still holds for any length space. Recall that a length space X is a metric space such that the distance between any two points x,y ∈ X is equal to the infimum of the lengths of all paths joining x to y (we implicitly assume that there is at least one such path). Here a path from x to y is a continuous map γ :[0, 1] → X with γ(0) = x and γ(1) = y. Let us return to the proof of the theorem. For any x ∈ Bj we have q MBj (|fj |+|∇fj |) (x) 1 = (|f |+|∇f |)q dμ sup j j j B:x∈B,r(B)≤2 μ(B ∩ B) Bj ∩B (3.13) μ(B) 1 ≤ C (|f |+|∇f |)q dμ sup j B:x∈B,r(B)≤2 μ(B ∩ B) μ(B) B ≤ CM(|f |+|∇f |)q (x) where we used (3.12) of Lemma 3.8. Consider now j q q j = x ∈ B : MBj (|fj |+|∇fj |) (x)>Cα (t) 248 nadine badr j where C is the constant in (3.13). j is an open subset of B , hence of M, and j ⊂ = M for all j ∈ J . For the fj ’s, and for all t>0, we have a Calderón-Zygmund decomposition similar to the one done in Proposition 3.5: j there exist bjk,gj supported in B , and balls (Bjk)k of M, contained in j , such that (3.14)fj = gj + bjk k (3.15) |gj (x)|≤Cα(t) and |∇gj (x)|≤Cα(t) for μ − a.e.x ∈ M (3.16) supp bjk ⊂ Bjk, r r r for 1 ≤ r ≤ q (|bjk| +|∇bjk| )dμ≤ Cα (t)μ(Bjk) Bjk −p p (3.17) μ(Bjk) ≤ Cα (t) (|fj |+|∇fj |) dμ j k B ≤ (3.18) χBjk N k with C and N depending only on q, p and the constants in (Dloc) and (Pqloc). The proof of this decomposition will be the same as in Proposition 3.5, taking for all j ∈ J a Whitney decomposition (Bjk)k of j = M and using the doubling property for balls whose radii do not exceed 3 We used the bounded overlap property of the ( j )j∈J ’s and that of the (Bjk)k’s 1 for all j ∈ J . It follows that b r ≤ Cα(t)μ( ) r . Similarly we get |∇b| r ≤ 1 Cα(t)μ( ) r . For g we have g ∞ ≤ sup |gj (x)|≤sup N1 sup |gj (x)|≤N1 sup gj ∞ ≤ Cα(t). x x ∈ ∈ j∈J j J j J Analogously |∇g| ∞ ≤ Cα(t). We conclude that 1 1 1 1 K(f,t r ,W ,W ) ≤ b 1 + t r g 1 r ∞ Wr W∞ 1 1 ≤ Cα(t)μ( ) r + Ct r α(t) 1 1 q∗∗ 1 q∗∗ 1 ≤ Ct r α(t) ∼ Ct r |f | q (t) +|∇f | q (t) which completes the proof of Theorem 1.2 in the case r 4. Interpolation Theorems In this section we establish our interpolation Theorem 1.1 and some con- sequences for non-homogeneous Sobolev spaces on a complete non-compact Riemannian manifold M satisfying (Dloc) and (Pqloc) for some 1 ≤ q<∞. ≤ ≤ ∞ 1 For 1 r q 1 1 1 W = (W ,W ) − r . p,r r ∞ 1 p ,p ∈ 1 + 1 From the previous results we know that for f Wr W∞ ∞ 1 p 1 r∗∗ 1 r∗∗ 1 p dt f − r ≥ C t p (|f | r +|∇f | r )(t) 1 p ,p 1 0 t ∈ 1 and for f Wp ∞ 1 p 1 q∗∗ 1 q∗∗ 1 p dt f − r ≤ C t p (|f | q +|∇f | q )(t) . 1 p ,p 2 0 t 250 nadine badr 1 = 1 We claim that Wp,r Wp , with equivalent norms. Indeed, ∞ 1 p r∗∗ 1 r∗∗ 1 p f − r ≥ C |f | r (t) +|∇f | r (t) dt 1 p ,p 1 0 1 1 r∗∗ r r∗∗ r ≥ C f p + |∇f | p r r 1 1 r r r r ≥ C f p + |∇f | p r r = C f p + |∇f | p = C f 1 , Wp and ∞ 1 p q∗∗ 1 q∗∗ 1 p f − r ≤ C |f | q (t) +|∇f | q (t) dt 1 p ,p 2 0 1 1 q∗∗ q q∗∗ q ≤ C f p + |∇f | p q q 1 1 q q q q ≤ C f p + |∇f | p q q = C f p + |∇f | p = C f 1 , Wp ∗∗ where we used that for l>1, f l ∼ f l (see [34], ChapterV: Lemma 3.21 p. 191 and Theorem 3.21, p. 201). Moreover, from Corollary 3.7, we have 1 ⊂ 1 + 1 ∞ 1 Wp Wr W∞ for r Theorem 4.1. Let (X,d,μ)be a complete metric-measure space with μ locally doubling and admitting a local Poincaré inequality (Pqloc), for some 1 Here, the definition of (Pqloc) is that of Section 7. It reduces to the one of Section 3 when the metric space is a Riemannian manifold. real interpolation of sobolev spaces 251 Comment on the proof of this theorem. The proof goes as in [28] where this theorem is proved for X satisfying (D) and admitting a global Poincaré inequality (Pq ). By using the same argument and choosing suffi- ciently small radii for the considered balls, (Pqloc) will give us (P(q− )loc) for every ball of radius less than r2, for some r2 < min(r0,r1), r0,r1 being the constants given in the definitions of local doubling property and local Poincaré inequality. ={ ∈ ∞ } = Define AM q [1, [:(Pqloc) holds and q0M inf AM . When no confusion arises, we write q0 instead of q0M . As we mentioned in the intro- duction, this improvement of the exponent of a Poincaré inequality together with the reiteration theorem yield another version of our interpolation result: Corollary 1.3. Proof of Corollary 1.3. Let 0 <θ<1 such that 1 = 1−θ + θ . p p1 p2 1. Case when p1 >q0. Since p1 >q0, there exists q ∈ AM such that q q q q0 1 1 1 1 1 1 1 1 (W ,W ) = (W ,W ) = (W ,W∞) − q = W = W . p1 p2 θ,p p1,q p2,q θ,p q 1 p ,p p,q p p1 p1 2. Case when 1 ≤ p1 ≤ q0. Let θ = θ 1 − = 1 − . The reiteration p2 p theorem applied this time only to the second exponent yields 1 1 1 1 1 1 1 1 (W ,W ) = (W ,W ) = (W ,W ) = W = W . p1 p2 θ,p p1 p2,p1 θ,p p1 ∞ θ ,p p,p1 p Theorem 4.2. Let M and N be two complete non-compact Riemannian manifolds satisfying (Dloc). Assume that q0M and q0N are well defined. Take 1 ≤ p1 ≤ p2 ≤∞, 1 ≤ r1, r, r2 ≤∞. Let T be a bounded linear operator from W 1 (M) to W 1(N) of norm L , i = 1, 2. Then for every couple (p, r) pi ri i 1 1 1 1 1 1 such that p ≤ r, p>q0 , r>q0 and , = (1 − θ) , + θ , , M N p r p1 r1 p2 r2 1 1 ≤ 1−θ θ 0 <θ<1, T is bounded from Wp (M) to Wr (N) with norm L CL0 L1. Proof. Tf W 1(N) ≤ C Tf (W 1 (N),W 1 (N)) r r1 r2 θ,r 1−θ θ ≤ CL L f (W 1 (M),W 1 (M)) 0 1 p1 p2 θ,r 1−θ θ ≤ CL L f (W 1 (M),W 1 (M)) 0 1 p1 p2 θ,p 1−θ θ ≤ CL L f 1 . 0 1 Wp (M) We used the fact that Kθ,q is an exact interpolation functor of exponent θ, that W 1(M) = (W 1 (M), W 1 (M)) , W 1(N) = (W 1 (N), W 1 (N)) with p p1 p2 θ,p r r1 r2 θ,r 252 nadine badr equivalent norms and that (W 1 (M), W 1 (M)) ⊂ (W 1 (M), W 1 (M)) if p1 p2 θ,p p1 p2 θ,r p ≤ r. Remark 4.3. Let M be a Riemannian manifold, not necessarily complete, satisfying (Dloc). Assume that for some 1 ≤ q<∞, a weak local Poin- ∞ caré inequality holds for all C functions, that is there exists r1 > 0,C= ∞ C(q,r1), λ ≥ 1 such that for all f ∈ C and all ball B of radius r 5. Homogeneous Sobolev spaces on Riemannian manifolds Definition 5.1. Let M be a C∞ Riemannian manifold of dimension n.For ≤ ≤∞ ˙ 1 1 p , we define Ep to be the vector space of distributions ϕ with |∇ϕ|∈Lp, where ∇ϕ is the distributional gradient of ϕ. It is well known that ˙ 1 ˙ 1 the elements of Ep are in Lploc. We equip Ep with the semi-norm ϕ ˙ 1 = |∇ϕ| . Ep p ˙ 1 Definition 5.2. We define the homogeneous Sobolev space Wp as the ˙ 1 quotient space Ep/R. ˙ 1 Remark 5.3. For all ϕ ∈ E , ϕ ˙ 1 = |∇ϕ| , where ϕ denotes the class p Wp p of ϕ. ˙ 1 Proposition 5.4 ([20]). 1. Wp is a Banach space. 2. Assume that M satisfies (D) and (Pq ) for some 1 ≤ q<∞ and for all f ∈ Lip, that is there exists a constant C>0 such that for all f ∈ Lip and for every ball B of M of radius r>0 we have 1 1 q q q q (Pq ) − |f − fB | dμ ≤ Cr − |∇f | dμ . B B ∩ ˙ 1 ˙ 1 ≤ ∞ Then Lip(M) Wp is dense in Wp for q p< . Proof. The proof of item 2 is implicit in the proof of Theorem 9 in [17]. We obtain for the K-functional of the homogeneous Sobolev spaces the following homogeneous form of Theorem 1.2, weaker in the particular case r = q, but again sufficient for us to interpolate. real interpolation of sobolev spaces 253 Theorem 5.5. Let M be a complete Riemannian manifold satisfying (D) and (Pq ) for some 1 ≤ q<∞. Let 1 ≤ r ≤ q. Then ∈ ˙ 1 + ˙ 1 1. there exists C1 such that for every F Wr W∞ and t>0 1 1 ∗∗ 1 r ˙ 1 ˙ 1 ≥ r |∇ |r r K(F,t , Wr , W∞) C1t f (t) ∈ ˙ 1 + ˙ 1 = where f Er E∞ and f F ; ≤ ∞ ∈ ˙ 1 2. for q p< , there exists C2 such that for every F Wp and t>0 1 1 q∗∗ 1 r ˙ 1 ˙ 1 ≤ r |∇ | q K(F,t , Wr , W∞) C2t f (t) ∈ ˙ 1 = where f Ep and f F . Before we prove Theorem 5.5, we give the following Calderón-Zygmund decomposition that will be also in this case our principal tool to estimate K. Proposition 5.6 (Calderón-Zygmund lemma for Sobolev functions). Let M be a complete non-compact Riemannian manifold satisfying (D) and (Pq ) ≤ ∞ ≤ ∞ ∈ ˙ 1 for some 1 q< . Let q p< , f Ep and α>0. Then there is ∈ ˙ 1 a collection of balls (Bi )i , functions bi Eq and a Lipschitz function g such that the following properties hold: (5.1)f= g + bi i (5.2) |∇g(x)|≤Cα μ − a.e. r r (5.3) supp bi ⊂ Bi and for 1 ≤ r ≤ q |∇bi | dμ ≤ Cα μ(Bi ) Bi −p p (5.4) μ(Bi ) ≤ Cα |∇f | dμ i ≤ (5.5). χBi N i The constants C and N depend only on q, p and the constant in (D). Proof. The proof goes as in the case of non-homogeneous Sobolev spaces, q q but taking ={x ∈ M : M(|∇f | )(x) > α } as f p is not under control. 254 nadine badr We note that in the non-homogeneous case, we used that f ∈ Lp only to control g ∈ L∞ and b ∈ Lr . Remark 5.7. It is sufficient for us that the Poincaré inequality holds for all ∈ ˙ 1 f Ep. Corollary 5.8. Under the same hypotheses as in the Calderón-Zygmund lemma, we have ˙ 1 ⊂ ˙ 1 + ˙ 1 ≤ ≤ ≤ ∞ Wp Wr W∞ for 1 r q p< . Proof of Theorem 5.5. The proof of item 1 is the same as in the non- ∈ ˙ 1 ∈ ˙ 1 homogeneous case. Let us turn to inequality 2. For F Wp we take f Ep ∗ 1 with f = F . Let t>0 and α(t) = M(|∇f |q ) q (t). By the Calderón- Zygmund decomposition with α = α(t), f can be written f = b + g, 1 hence F = b + g, with b ˙ 1 = |∇b| ≤ Cα(t)μ( ) r and g ˙ 1 = Wr r W∞ |∇g| ∞ ≤ Cα(t). Since for α = α(t) we have μ( ) ≤ t, then we get 1 1 q∗∗ 1 r ˙ 1 ˙ 1 ≤ r |∇ | q K(F,t , Wr , W∞) Ct f (t). We can now prove our interpolation result for the homogeneous Sobolev spaces. Proof of Theorem 1.4. The proof follows directly from Theorem 5.5. Indeed, item 1 of Theorem 5.5 yields ˙ 1 ˙ 1 ˙ 1 (W , W ) − r ⊂ W r ∞ 1 p ,p p with F ˙ 1 ≤ C F − r , while item 2 gives us that Wp 1 p ,p ˙ 1 ˙ 1 ˙ 1 W ⊂ (W , W ) − r p r ∞ 1 p ,p with F − r ≤ C F ˙ 1 . We conclude that 1 p ,p Wp ˙ 1 ˙ 1 ˙ 1 W = (W , W ) − r p r ∞ 1 p ,p with equivalent norms. Corollary 5.9 (The reiteration theorem). Let M be a complete non- compact Riemannian manifold satisfying (D) and (Pq ) for some 1 ≤ q<∞. Define q0 = inf{q ∈ [1, ∞[:(Pq ) holds}. Then for p>q0 and 1 ≤ p1 < p Application. Consider a complete non-compact Riemannian manifold M satisfying (D) and (Pq ) for some 1 ≤ q<2. Let be the Laplace-Beltrami 1 operator. Consider the linear operator 2 with the following resolution ∞ 1 − dt ∞ 2 = t √ ∈ f c e f ,fC0 0 t − 1 1 where c = π 2 . Here 2 f can be defined for f ∈ Lip as a measurable function (see [3]). In [3], Auscher and Coulhon proved that on such a manifold, we have 1 C μ x ∈ M : | 2 f(x)| >α ≤ |∇f | αq q ∈ ∞ ∈ for f C0 , with q [1, 2[. In fact one can check that the argument applies 1 1 ∈ ∩ ˙ 1 2 = 2 ∩ ˙ 1 to all f Lip Eq and since 1 0, can be defined on Lip Wq by 1 taking quotient which we keep calling 2 . Moreover, Proposition 5.4 gives us 1 ˙ 1 that 2 has a bounded extension from Wq to Lq,∞. Since we already have 1 2 f 2 ≤ |∇f | 2 then by Corollary 5.9, we see at once 1 (5.6) 2 f p ≤ Cp |∇f | p ≤ ∈ ˙ 1 for all q 6. Sobolev spaces on compact manifolds Let M be a C∞ compact manifold equipped with a Riemannian metric. Then M satisfies the doubling property (D) and the Poincaré inequality (P1). Theorem 6.1. Let M be a C∞ compact Riemannian manifold. There exist ∈ 1 + 1 C1,C2 such that for all f W1 W∞ and all t>0 we have ∗∗ ∗∗ 1 1 C1t |f | (t) +|∇f | (t) ≤ K(f,t,W ,W∞) ∗ 1 ( comp) ∗∗ ∗∗ ≤ C2t |f | (t) +|∇f | (t) . Proof. It remains to prove the upper bound for K as the lower bound is trivial. Indeed, let us consider for all t>0 and for α(t) = (M(|f |+|∇f |))∗(t), ={x ∈ M; M(|f |+|∇f |)(x) ≥ α(t)}.If = M, we have the Calderón- Zygmund decomposition as in Proposition 3.5 with q = 1 and the proof will be the same as the proof of Theorem 1.2 in the global case. Now if = M,we 256 nadine badr prove the upper bound by the same argument used in the proof of Theorem 1.2 in the local case. Thus, in the two cases we obtain the right hand inequality of ∗ ∈ 1 + 1 ( comp) for all f W1 W∞. It follows that ≤ ≤∞ 1 Theorem 6.2. For all 1 p1 7. Metric-measure spaces In this section we consider (X,d,μ)a metric-measure space with μ doubling. 7.1. Upper gradients and Poincaré inequality Definition 7.1 (Upper gradient [26]). Let u : X → R be a Borel function. → +∞ We say that a Borel function g : X [0, ] is an upper gradient of u if | − |≤ b → u(γ (b)) u(γ (a)) a g(γ(t))dt for all 1-Lipschitz curve γ :[a,b] X1. Remark 7.2. If X is a Riemannian manifold, |∇u| is an upper gradient of u ∈ Lip and |∇u|≤g for all upper gradients g of u. Definition 7.3. For every locally Lipschitz continuous function u defined on a open set of X, we define ⎧ ⎪ |u(y) − u(x)| ⎨ lim sup if x is not isolated, Lip u(x) = y→x d(y,x) ⎩⎪ y =x 0 otherwise. Remark 7.4. Lip u is an upper gradient of u. Definition 7.5 (Poincaré Inequality). A metric-measure space (X,d,μ) admits a weak local Poincaré inequality (Pqloc) for some 1 ≤ q<∞, if there exist r1 > 0, λ ≥ 1, C = C(q,r1)>0, such that for every continuous function u and upper gradient g of u, and for every ball B of radius 0 1 1 q q q q (Pqloc) − |u − uB | dμ ≤ Cr − g dμ . B λB If λ = 1, we say that we have a strong local Poincaré inequality. 1 Since every rectifiable curve admits an arc-length parametrization that makes the curve 1- Lipschitz, the class of 1-Lipschitz curves coincides with the class of rectifiable curves, modulo a parameter change. real interpolation of sobolev spaces 257 Moreover, X admits a global Poincaré inequality or simply a Poincaré inequality (Pq ) if one can take r1 =∞. 1 7.2. Interpolation of the Sobolev spaces Hp 1 Before defining the Sobolev spaces Hp it is convenient to recall the following proposition. Proposition 7.6 (see [22] and [10] Theorem 4.38). Let (X,d,μ)be a com- plete metric-measure space, with μ doubling and satisfying a weak Poincaré inequality (Pq ) for some 1 u 1 = u + |Du| ≡ u + u . Hp p p p Lip p 1 We denote H∞ for the set of all bounded Lipschitz functions on X. Remark 7.7. Under the hypotheses of Proposition 7.6, the uniqueness of ∈ 1 ≥ the gradient holds for every f Hp with p q. By uniqueness of gradient we mean that if un is a locally Lipschitz sequence such that un → 0inLp and Dun → g ∈ Lp then g = 0 a.e.. Then D extends to a bounded linear operator 1 from Hp to Lp. In the remaining part of this section, we consider a complete non-compact metric-measure space (X,d,μ)with μ doubling. We also assume that X ad- mits a Poincaré inequality (Pq ) for some 1 Define q0 = inf{q ∈ ]1, ∞[:(Pq ) holds}. 258 nadine badr ∞ ∩ 1 Lemma 7.8. Under these hypotheses, and for q0 ∗ ∈ ∩ 1 Proof. We have ( met) for all f Lip Hq from the Calderón-Zygmund ∈ 1 = decomposition that we have done. Now for f Hq , by Lemma 7.8, f 1 1 lim f in H , with f Lipschitz and f −f 1 < for all n. Since for all n, n n q n n Hq n 1 fn ∈ Lip, there exist gn,hn such that fn = hn +gn and hn 1 +t q gn 1 ≤ Hq H∞ 1 q∗∗ 1 q∗∗ 1 Ct q |fn| q (t) +|Dfn| q (t) . Therefore we find 1 1 f − g 1 + t q g 1 ≤ f − f 1 + h 1 + t q g 1 n Hq n H∞ n Hq n Hq n H∞ 1 1 q∗∗ 1 q ∗∗ 1 ≤ + Ct q |f | q (t) +|Df | q (t) . n n n q q q q Letting n →∞, since |fn| −→ |f | in L1 and |Dfn| −→ |Df | in L1,it n→∞ n→∞ q∗∗ q∗∗ q∗∗ q∗∗ comes |fn| (t) −→ |f | (t) and |Dfn| (t) −→ |Df | (t) for all t>0. n→∞ n→∞ real interpolation of sobolev spaces 259 ∗ ∈ 1 ∗ ∈ 1 + 1 Hence ( met) holds for f Hq . We prove ( met) for f Hq H∞ by the same argument of [14]. Theorem 7.11 (Interpolation Theorem). Let (X,d,μ)be a complete non- compact metric-measure space with μ doubling, admitting a Poincaré inequal- ∞ ≤∞2 1 ity (Pq ) for some 1 2 We allow p1 = 1ifq0 = 1. 260 nadine badr 8. Applications 8.1. Carnot-Carathéodory spaces An important application of the theory of Sobolev spaces on metric-measure spaces is to a Carnot-Carathéodory space. We refer to [23] for a survey on the theory of Carnot-Carathéodory spaces. n Let ⊂ R be a connected open set, X = (X1,...,Xk) a family of vector fields defined on , with real locally Lipschitz continuous coefficients and 1 | |= k | |2 2 L n Xu(x) j=1 Xj u(x) . We equip with the Lebesgue measure and the Carnot-Carathéodory metric ρ associated to the Xi . We assume that ρ defines a distance. Then, the metric space ( , ρ) is a length space. ≤ ∞ 1 Definition 8.1. Let 1 p< . We define Hp,X( ) as the completion of locally metric3 Lipschitz functions (equivalently of C∞ functions) for the norm f 1 = f + |Xf | . Hp,X Lp ( ) Lp ( ) 1 We denote H∞,X for the set of bounded metric Lipschitz function. ≤ ≤∞ 1 = 1 ={ ∈ Remark 8.2. For all 1 p , Hp,X Wp,X( ) : f Lp( ) : |Xf |∈Lp( )}, where Xf is defined in the distributional sense (see for ex- ample [19] Lemma 7.6). Adapting the same method, we obtain the following interpolation theorem 1 for the Hp,X. Theorem 8.3. Consider ( , ρ, L n) where is a connected open subset of Rn. We assume that L n is locally doubling, that the identity map id : ( , ρ) → ( , |.|) is an homeomorphism. Moreover, we suppose that the space admits a local weak Poincaré inequality (Pqloc) for some 1 ≤ q<∞. Then, for ≤ ≤∞ 1 1 p1 8.2. Weighted Sobolev spaces We refer to [24], [29] for the definitions used in this subsection. Let be an n n open subset of R equipped with the Euclidean distance, w ∈ L1,loc(R ) with w>0,dμ= wdx. We assume that μ is q-admissible for some 1 3 that is relative to the metric ρ of Carnot-Carathéodory. real interpolation of sobolev spaces 261 doubling and there exists C>0 such that for every ball B ⊂ Rn of radius r>0 and for every function ϕ ∈ C∞(B), q q q (Pq ) |ϕ − ϕB | dμ ≤ Cr |∇ϕ| dμ B B = 1 with ϕB μ(B) B ϕdμ. The Aq weights, q>1, satisfy these two conditions (see [25], Chapter 15). ≤ ∞ 1 Definition 8.4. For q p< , we define the Sobolev space Hp ( , μ) to be the closure of C∞( ) for the norm u 1 = u + |∇u| . Hp ( ,μ) Lp (μ) Lp (μ) 1 We denote H∞( , μ) for the set of all bounded Lipschitz functions on . Using our method, we obtain the following interpolation theorem for the 1 Sobolev spaces Hp ( , μ): Theorem 8.5. Let be as in above. Then for q0 As in Section 7, we were not able to get our interpolation result for p1 ≤ q0 since again in this case the uniqueness of the gradient does not hold for p1 ≤ q0. Remark 8.6. Equip with the Carnot-Carathéodory distance associated to a family of vector fields with real locally Lipschitz continuous coefficients instead of the Euclidean distance. Under the same hypotheses used in the be- ginning of this section, just replacing the balls B by the balls B˜ with respect to ρ, and ∇ by X and assuming that id : ( , ρ) → ( , |.|) is an homeomorph- ism, we obtain our interpolation result. As an example we take vectors fields satisfying a Hörmander condition or vectors fields of Grushin type [16]. 8.3. Lie Groups In all this subsection, we consider G a connected unimodular Lie group equipped with a Haar measure dμ and a family of left invariant vector fields X1,...,Xk such that the Xi ’s satisfy a Hörmander condition. In this case the Carnot-Carathéodory metric ρ is is a distance, and G equipped with the dis- tance ρ is complete and defines the same topology as that of G as a manifold (see [13] page 1148). From the results in [21], [32], it is known that G satisfies (Dloc). Moreover, if G has polynomial growth it satisfies (D). From the results in [33], [35], G admits a local Poincaré inequality (P1,loc).IfG has polynomial growth, then it admits a global Poincaré inequality (P1). 262 nadine badr Interpolation of non-homogeneous Sobolev spaces. We define the 1 non-homogeneous Sobolev spaces on a Lie group Wp in the same manner as in Section 3 on a Riemannian manifold replacing ∇ by X (see Definition 3.1 and Proposition 3.2). To interpolate the W 1 , we distinguish between the polynomial and the pi exponential growth cases. If G has polynomial growth, then we are in the global case. If G has exponential growth, we are in the local case. In the two cases we obtain the following theorem. Theorem 8.7. Let G be as above. Then, for all 1 ≤ p1 9. Appendix In view of the hypotheses in the previous interpolation results, a naturel ques- tion to address is whether the properties (D) and (Pq ) are necessary. The aim of the appendix is to exhibit an example where at least Poincaré is not needed. Consider = ∈ n; 2 +···+ 2 ≤ 2 X (x1,x2,...,xn) R x1 xn−1 xn equipped with the Euclidean metric of Rn and with the Lebesgue measure. X consists of two infinite closed cones with a common vertex. X satisfies the doubling property and admits (Pq ) in the sense of metric-measure spaces if 1 and only if q>n([23] p. 17). Denote by the interior of X. Let Hp (X) be the closure of Lip0(X) for the norm f 1 = f + |∇f | . Hp (X) Lp ( ) Lp ( ) 1 ∈ ∇ ∈ We define Wp ( ) as the set of all functions f Lp( ) such that f Lp( ) and equip this space with the norm f 1 = f 1 . 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