ON IMPROVED SOBOLEV EMBEDDING THEOREMS. M. Ledoux University of Toulouse, France

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1 ON IMPROVED SOBOLEV EMBEDDING THEOREMS M. Ledox University of Tolose, France Abstract. We present a direct proof of some recent improved Sobolev ineqalities pt forward by A. Cohen, R. DeVore, P. Petrshev and H. X [C-DV-P-X] in their wavelet analysis of the space BV (R 2. These ineqalities are parts of the Hardy-Littlewood-Sobolev theory, connecting Sobolev embeddings and heat kernel bonds. The argment, relying on psedo-poincaré ineqalities, allows s to stdy the dependence of the constants with respect to dimension and to consider several extensions to manifolds and graphs. As part of the classical Sobolev ineqalities, it is well known that for every fnction f on R n vanishing at infinity in some mild sense, n n C f ( where q = and C > only depends on n. The Sobolev ineqality ( is invariant nder the ax + b (a >, b R n grop action, bt is not nder the Weyl-Heisenberg grop action. Namely, if f(x = f ω (x = e iω x ϕ(x where ϕ is in the Schwartz class, then f = ω ϕ + O( when ω, so that ( is not well-sited for sch modlated fnctions. In their stdy of the space BV (R 2, A. Cohen et al. [C-DV-P-X] (see also [C-M-O] improved the Sobolev ineqality ( into C f /q f (/q B (2 where B = B (n is the homogeneos Besov space of indices ( (n,, (see below. This improved Sobolev ineqality is easily seen to be sharper than (. Frthermore, if f(x = f ω (x = e iω x ϕ(x as above for some ϕ with enogh reglarity, it may easily be shown that f B = ω (n ϕ + O( ω n so that (2 amonts in this case to the trivial bond ϕ q ϕ /q ϕ (/q. The proof of (2 in [C-DV-P-X] and [C-M-O] is based on wavelet decompositions together with weak-l type estimates and interpolation reslts. The prpose of this note

2 is to propose a direct semigrop argment withot any se of wavelet decomposition. The approach we sggest also relies on weak-type estimates, bt emphasizes the se of psedo-poincaré ineqalities (cf. [SC] for families of operators (heat kernels for example. The argment easily extends to more general frameworks inclding manifolds or graphs, ot of reach of wavelets methods. It frthermore allows s to establish ineqalities with constants independent of the dimension (for a certain range of parameters. The first section of this note is devoted to the detailed proof of the improved Sobolev ineqality (2 in the Eclidean case. One step of the proof is reminiscent of the Marcinkiewicz interpolation theorem. The reslt is moreover mch in the spirit of the Varopolos theorem on the eqivalence between heat kernel bonds and Sobolev ineqalities. We then discss varios extensions of the argment to reach similar ineqalities in more general frameworks, inclding abstract Hardy-Littlewood-Sobolev theory and Varopolos theorem, manifolds with non-negative Ricci crvatre and Cayley graphs, and to stdy dependence of the constants with respect to dimension.. Improved Sobolev ineqality in R n Eqip R n with Lebesge measre λ, and denote by p the L p -norms. For simplicity, we only work with real-valed fnctions. We make essential se of the thermic description of Besov spaces (cf. [Tr]. To this task, let P t = e t, t, be the heat semigrop on R n. For α <, define then the Besov space B α as the space of tempered distribtions f on R n for which the Besov norm f B α = sp t α/2 P t f t> is finite. The space B α is the homogeneos Besov space of indices (α,, for which varios descriptions are available ([Tr], [Me], [C-M-]... The following theorem has been established first in [C-DV-P-X] for p =, q = 2 (α = in dimension 2, and for p =, q = n n (α = (n, that corresponds to (2, in [C-M-O]. The case p =, q = 2 (α = is extended in [C-D-D-DV] to in any dimension as a particlar case of some frther interpolation ineqalities between Besov spaces and the space of bonded variations (not covered by the method presented here. It shold be emphasized, according to the previos references, that Littlewood-Paley analysis may be developed to cover the case p >. The vale p =, of most interest, is not covered by these tools and ths reqires an independent investigation. Theorem. For every p < q < and every fnction f in the Sobolev space W,p, C f θ p f θ B θ/(θ where θ = p q and C > only depends on p and q (and n. 2

3 By the heat kernel embeddings P t r C t n/2r, r, one easily recovers from Theorem the classical Sobolev ineqalities C f p, q = p n, p < n, as well as the Gagliardo-Nirenberg ineqalities C f p/q p f (p/q r, q = p r qn. Theorem, p =, may be extended by standard approximations to fnctions of bonded variations (cf. [C-DV-P-X], [C-D-D-DV]. The rest of this section is devoted to the proof of Theorem. We divide the proof into three steps for the prpose of the extensions we discss next. Proof. st step. We note first that the weak-type ineqality, C f θ p f θ B θ/(θ (3 is very easy to prove. Assme by homogeneity that f θ/(θ B let t = t = 2(θ /θ so that P t f. Then, q λ ({ f 2 } p λ ( f P t f q p. For every >, f P t f p dλ. The main tool of the approach is the following psedo-poincaré ineqality: for any f in W,p and t, f P t f p C t f p (4 (where C > only depends on p and n. Ineqality (4 is classical and may be established in a rather easy way. For example, write f P t f = [f( f( y ] pt (ydy where p t (y is the Gassian kernel. Therefore, f P t f p f( f( y p p t (ydy. Since f( f( y p y f p when f W,p, the conclsion follows with C = y e y 2 /2 (2π n/2 dy A n where A > is nmerical. By means of (4, and since q p + p(θ /θ =, we then get that q λ ({ f 2 } C q p t p/2 3 f p dλ = C f p dλ

4 which amonts to the weak-type ineqality (3 by definition of,. 2nd step. We now wold like to replace the weak L q -norm in (3 by the strong one. Standard argments involving ct-off fnctions are available to this task (see e.g. [B-C- L-SC]. One difficlty however in this case is that we may not redce to non-negative fnctions. The argment we present is a refinement of the preceding step by means of some interpolation tools inspired from proof of the Marcinkiewicz theorem. Start first with a fnction f W,p sch that f L q. We will see in the third step how to get rid of the latter. Assme as before that f θ/(θ B and let s show that f q dλ C f p dλ (5 for some constant C > (depending on q and n, which amonts to the ineqality of the theorem. For >, let again t = t = 2(θ /θ. Let c 5 (depending on q and p to be specified later. Write For every >, set 5 q f q q = λ ({ f 5 } d( q. f = (f + ( (c + (f + ( (c. On { f 5}, f 4. Therefore, λ ({ f 5 } d( q where we sed that P t (f. By (4 applied to f, + λ ({ f 4 } d( q λ( { f P t ( f } d( q λ( {Pt ( f f 2 } d( q Hence { λ( f P t ( f } p f P t ( f p dλ C p t p/2 f p dλ C q f p dλ. { λ( f P t ( f } d( q C q { f c} = C q log c f p ( f f /c f p dλ. d dλ 4

5 On the other hand, since it follows that f f = f f { f c} + f f { f >c} + f { f >c}, λ( {Pt ( f f 2 } d( q = q q = q q {Pt ( } λ( f { f >c} d( q ( f { f >c} dλ d( q ( f { f >c} d( q dλ c q f q q. Combining the preceding estimates, 5 q f q q C q log c f p dλ + q q c q f q q. Choose then c large enogh depending only on q > to dedce (5. 3rd step. In this final step, one jst wold like to get rid of the assmption f L q in the preceding argment. Some approximation argment shold be enogh, however we have not been able to prodce a reasonably simple one. The following is a simple modification of the preceding proof that is enogh for this prpose. Let ths f be sch that f p < and f θ/(θ B (by homogeneity. Since by the weak-type ineqality (3, <, we may define for every ε, < ε <, N ε (f = /ε The 2nd step shows at some point that ε λ ({ f 5 } d( q <. N ε (f C q log c f p dλ + /ε ε ( f { f >c} dλ d( q. It is easy to check by integration by parts and Fbini s theorem that /ε ε ( f { f >c} dλ d( q cq /ε λ ({ f c } d( q q ε + cq q 5 ε q λ ({ f c } d. /ε

6 One may then get the bond /ε ( f { f >c} dλ d( q ε q q 5 q c q N ε(f + q q ( ( c c q f q q, q log + 5. q Hence, for c large enogh, it will follow that sp <ε< N ε (f <, and ths that <. The previos argment then shows that the ineqality of the theorem holds for any f sch that f L p and f B θ/(θ. The theorem is established. 2. Extensions. Dimension free bonds Besides the 3rd (approximation step, the proof in the Eclidean case emphasizes one main tool in the argment, namely the psedo-poincaré bond (4 on f P t f. The terminology psedo-poincaré is taken from the reference [SC] where it is applied for averages on balls (see below. D. Robinson [Ro] was the first to se this property to prove a Nash type ineqality, and it has then been scessflly applied in in varios contexts [C-SC], [V-SC-C] (cf. [SC]. Under (4 and domination of P t f by the Besov norm, the weak-type ineqality (3 immediately follows. The interpolation argment of the second step develops similarly in a rather large generality. Therefore, once we are given a family (P t t of niformly bonded operators on all L p -spaces satisfying a psedo-poincaré ineqality, and once we consider the corresponding Besov norm on (P t t, the main reslt of Theorem immediately extends. We describe in this section varios instances where (4, and ths the reslt of Theorem, generalizes. We frthermore inspect dependence of the constants pon dimension. For simplicity, we only state the ineqalities for classes of smooth fnctions. Sobolev embeddings and Varopolos s theorem. For the Laplace-Beltrami operator on a manifold, or a second order differential operator A (withot constant term, the psedo-poincaré ineqality (4 for the heat semigrop P t = e ta, t, for p = 2 is essentially spectral and always satisfied. Indeed, more generally, if P t = e ta is a Markov self-adjoint semigrop on L 2 (E, µ, then it is classical (cf. [Da], [SC] that f P t f t E(f, f where E is the associated Dirichlet form E(f, f = f( Afdµ. As a conseqence of the method developed in Section, we may ths conclde to the following statement. Corollary 2. Under the preceding notation, for every q, 2 < q < and every fnction f in the domain of A, C E(f, f θ/2 f θ B θ/(θ 6

7 where θ = 2 q and where C > only depends on q. Corollary 2 is a self-contained version of one half of the celebrated Varopolos theorem on the eqivalence between Sobolev ineqalities and heat kernel decay ([Va], [Va2], [Da]: namely, nder the heat kernel bond P t C t n/2, t >, the Sobolev ineqality f 2 2n/(n 2 C E(f, f = C f( Afdµ (n > 2 holds. As sch, the improved Sobolev ineqality of Corollary 2 is part of the abstract Hardy-Littlewood-Sobolev theory (cf. [B-C-L-SC], [SC]. 2 Riemannian manifolds with non-negative Ricci crvatre. Let M be a Riemannian manifold with dimension n and non-negative Ricci crvatre. Denote by the Laplace-Beltrami operator on M and by P t = e t, t, the associated heat semigrop. The following lemma shows how (4 extends to this context and frthermore describes the dependence of the constants with respect to dimension. Lemma 3. Under the previos notation and hypotheses, for every smooth fnction f on M and every t, f P t f p C t f p where C > is nmerical for p 2 and only depends on n for 2 < p. Proof. For every smooth fnction g with g p where p is the conjgate of p, t ( g(f P t fdx = g P s fdx ds = t ( P s g fdx ds t f p P s g p ds. We next provide two distinct argments to bond P s g p according as p 2 or 2 p. The Li-Ya ineqality [L-Y] in manifolds with non-negative Ricci crvatre asserts that for every smooth fnction g >, and every s >, Mltiplying by (P s g p, P s g 2 (P s g 2 P sg P s g n 2s. (6 (P s g p 2 P s g 2 (P s g p P s g n 2s (P sg p. 7

8 By integration by parts, p (P s g p 2 P s g 2 dx n 2s g p dx. Now, provided that p 2, we get from Hölder s ineqality that P s g p ( n /2 g p ( n /2 g p 2p s 2s from which the lemma easily follows in this case. When 2 p, we make se of the Poincaré type ineqality for heat kernel measres in Riemannian manifolds with non-negative Ricci crvatre (see e.g. [Le] that indicates that, for every smooth fnction g, and at every point, 2s P s g 2 P s (g 2 (P s g 2. Hence, by Hölder s ineqality again, P s g p ( /2 g p 2s with ths a nmerical constant. The proof of the lemma is then easily completed. As a conseqence of this lemma and the proof of Theorem in the previos section, we may state the following conseqence, with some frther aspects on the dependence of constants pon dimension. Define as in the Eclidean case the Besov norm, for α <, f B α = sp t α/2 P t f. t> Corollary 4. Let M be a Riemannian manifold with non-negative Ricci crvatre. Then, for every p < q < and every fnction f in the Sobolev space W,p (M, C f θ p f θ B θ/(θ where θ = p q and where C > only depends on q and p whenever p 2, and on q, p and n whenever 2 < p <. In particlar, one recovers in this way that nder the heat kernel bond P t C t n/2, t >, on a manifold M with non-negative Ricci crvatre, the classical Sobolev ineqality ( holds on M (cf. [C-L]. Moreover, in the context of Markov operators as in the preceding section, the fnctional dimension is only reflected at the level of the heat kernel bonds. One may similarly consider localized versions of the preceding reslt. For example, on a compact Riemannian manifold, the Li-Ya ineqality (6 holds (p to constants 8

9 depending on the lower bond of the Ricci crvatre for all < t. The preceding argments then show similarly that for the localized Besov norm f B α = sp t α/2 P t f, <t one has the localized improved Sobolev ineqalities C ( f p + f p θ f θ B θ/(θ. 3 Averages on balls and Morrey spaces. The argments developed in Section work similarly with averages on balls. For a locally integrable fnction f on R n, set f r (x = fdλ, r >, x R n, V (x, r B(x,r where B(x, r is the ball with center x and radis r and V (x, r (= V (, r n its volme. One may then consider the norm f M α = sp r α fr (x x R n,r> that corresponds to a Morrey space rather than a Besov space [Me], [Tr], and which is strictly smaller in the Eclidean case (cf. [Ta] and the references therein. The proof of Theorem may be repeated completely similarly with the averages f r in place of the Gassian convoltions P t f. In analogy with Lemma 2, the key ingredient is a family of local Poincaré ineqalities f f r p dλ C r p f p dλ (7 B(x,r B(x,r for p, some constant C > and all r > and f. Sch families are known (cf. [C-SC], [SC] to yield the psedo-poincaré ineqality for the averages f r f f r p C r f p (8 analogos to (4. With this tool, the proof of Theorem develops completely analogosly. While weaker than the Besov type reslt of Theorem in R n, it allows varios extensions of independent interest. Namely, local Poincaré ineqalities (7 and their associated psedo-poincaré ineqalities (8 have been investigated in a nmber of varios settings in connection with Sobolev type embeddings and Harnack ineqalities (we refer to [V-SC-C], [SC] for an accont on the sbject. In particlar, (7 has been 9

10 shown to hold in [B], by geometric tools, on manifolds of non-negative Ricci crvatre. We ths get a Morrey version of Corollary 3, sally not directly comparable. The Poincaré and psedo-poincaré ineqalities have been investigated also on related strctres sch as Lie grops and graphs. For example, the psedo-poincaré ineqality (8 is satisfied by the natral metrics on Lie grops associated with invariant vector fields [Ro], [V-SC-C], [C-SC], prodcing ths a version of the improved Sobolev embeddings in this context. If X is contable connected graph sch that each vertex has at most a finite nmber N of neighbors, define by d(x, y the minimal nmber of edges that connect x to y. If f is a fnction (with finite spport on X, let for x X and k, f k (x = f(y V (x, k y B(x,k where B(x, k is the ball with center x and radis k and V (x, k its volme (= cardinal. We also define the (length of the gradient f (x of f at the point x as f (x = y x f(x f(y where the sm rns over all neighbors y of x. If X is the Cayley graph of a finitely generated grop, it has been shown in [D-SC] (see also [C-SC], [SC], as a conseqence of local Poincaré ineqalities, that for every p and some C >, f f k p Ck f p (9 for every f and k. L p -norms are defined here with respect to the conting measre on X. As a conseqence, we may state the following improved Sobolev ineqalities in this case. We let f M α = sp k α fk (x. x X,k Corollary 5. Let X be the Cayley graph of a finitely generated grop. Then, for p < q <, some C > and any finitely spported fnction f on X, where θ = p q. f p C f θ p f θ M θ/(θ Acknowledgement. Thanks are de to A. Cohen and Y. Meyer for their interest and to P. Ascher, Th. Colhon and L. Saloff-Coste for helpfl comments. Thanks also to A. Ayache for making several references available.

11 References [Ba] [B-C-L-SC] [C-D-D-DV] [C-DV-P-X] D. Bakry. L hypercontractivité et son tilisation en théorie des semigropes. Ecole d Eté de Probabilités de St-Flor. Lectre Notes in Math. 58, 4 (994. Springer. D. Bakry, T. Colhon, M. Ledox, L. Saloff-Coste. Sobolev ineqalities in disgise. Indiana J. Math. 44, (995. [B] P. Bser. A note on the isoperimetric constant. Ann. scient. Éc. Norm. Sp. 5, (982. [C-M-O] [C-L] [C-SC] A. Cohen, W. Dahmen, I. Dabechies, R. DeVore. Harmonic analysis of the space BV. Rev. Mat. Iberoamericana 9, (23. A. Cohen, R. DeVore, P. Petrshev, H. X. Non linear approximation and the space BV (R 2. Amer. J. Math. 2, (999. A. Cohen, Y. Meyer, F. Or. Improved Sobolev ineqalities. Séminaires X-EDP, École Polytechniqe (998. Th. Colhon, M. Ledox. Isopérimétrie, décroissance d noya de la chaler et transformations de Riesz : n contre-exemple. Ark. Mat. 32, (994. Th. Colhon, L. Saloff-Coste. Isopérimétrie por les gropes et les variétés. Revista Math. Iberoamericana 9, (993. [Da] E. B. Davies. Heat kernels and spectral theory. Cambridge University Press (989. [D-SC] P. Diaconis, L. Saloff-Coste. Comparison theorems for reversible Markov chains. Ann. Applied Probab. 3, (993. [Le] M. Ledox. The geometry of Markov diffsion generators. Ann. Fac. Sci. Tolose IX, (2. [L-Y] P. Li, S.-T. Ya. On the parabolic kernel of the Schrödinger operator. Acta Math. 56, 53 2 (986. [Me] Y. Meyer. Ondelettes et opératers. Hermann (99. [Ro] D. Robinson. Elliptic operators on Lie grops. Oxford University Press (99. [SC] [Ta] L. Saloff-Coste. Aspects of Sobolev-type ineqalities. London Math. Soc. Lectre Notes Series 289. Cambridge University Press (22. M. Taylor. Analysis on Morrey spaces and applications to Navier-Stokes and other evoltion eqations. Comm. Partial Differential Eqations 7, (992. [Tr] H. Triebel. Theory of fnction spaces II. Birkhäser (992. [Va] N. Varopolos. Hardy-Littlewood theory for semigrops. J. Fnct. Anal. 63, (985. [Va2] [V-SC-C] N. Varopolos. Analysis and geometry on grops. Proceedings of the International Congress of Mathematicians, Kyoto (99, vol. II, (99. Springer-Verlag. N. Varopolos, L. Saloff-Coste, T. Colhon. Analysis and geometry on grops. Cambridge University Press (993. Institt de Mathématiqes, Université Pal-Sabatier, 362 Tolose, France address: ledox@math.ps-tlse.fr