Sharp affine Sobolev type inequalities via the Lp Busemann–Petty centroid inequality

arXiv:1505.07763v2 [math.FA] 29 Mar 2016 SHARP AFFINE SOBOLEV TYPE INEQUALITIES VIA THE Lp BUSEMANN-PETTY CENTROID INEQUALITY J. HADDAD, C.H. JIM´ ENEZ, M. MONTENEGRO Abstract. We show that the LpBusemann-Petty centroid inequality provides an elementary and powerful tool to the study of some sharp affine functional inequalities with a geometric content, like log-Sobolev, Sobolev and Gagliardo-Nirenberg inequalities. Our approach allows also to characterize directly the corresponding equality cases. 1. Introduction and previous results The goal of the present work is to discuss a new approach for the study of well-known affine Sobolev type inequalities such as log-Sobolev, Sobolev and Gagliardo-Nirenberg inequalities with sharp constants. More precisely, we wish to (1) give a unified and elementary treatment of sharp affine log-Sobolev, Sobolev and Gagliardo- Nirenberg inequalities; (2) illustrate the efficiency of the LpBusemann-Petty centroid inequality for the study of such inequalities, and by this method reveal in a more explicit way their geometrical nature; (3) determine all cases of equality in these sharp inequalities as a by-product of our method. Our main contribution consists in providing a new point of view for such inequalities by combining tools of Convex Geometry and Analysis. We address these inequalities in a more elementary way by avoiding the use of the solution of the LpMinkowski problem or a strengthen P´olya-Szeg¨o principle, which were fundamental ingredients when said inequalities were first obtained. Isoperimetric inequalities provide us with tools to compare parameters associated (in general) to convex bodies. In recent years a lot of attention has been paid to extending the classical Euclidean versions of said inequalities. Among these extensions we find the class of affine isoperimetric inequalities, that remain invariant under the action of non-degenerate (often volume preserving) linear maps. One of the main ingredients used here is the LpBusemann-Petty centroid inequality. This affine isoperimetric inequality, first obtained by Lutwak, Yang and Zhang [29], generalizes the classical Busemann-Petty centroid inequality due to Petty [37] that compares the ratio between the volume of a convex body Kand that of its centroid body. A more recent proof was obtained in [7] using the technique of shadow systems. Affine functional inequalities, as those treated in this work, have been obtained using essentially the Lpextension of the Minkowski problem, P´olya-Szeg¨o type results and the LpPetty projection inequality. The strong connection between the Petty projection and the Busemann-Petty (see [27]) inequalities makes the use of the latter in this type of approach far from surprising, however, it provides us with some additional geometric information. 2010 Mathematics Subject Classification. Primary 46E35, Secondary 51M16. Key words and phrases. LpBusemann-Petty centroid inequality, affine logarithmic inequalities, affine Sobolev inequalities. 1 2 J. HADDAD, C.H. JIM´ ENEZ, M. MONTENEGRO In order to show the efficiency of our approach, we choose three prototypes of sharp affine functional inequalities, namely sharp affine Lplog-Sobolev, Sobolev and Gagliardo-Nirenberg inequalities. Our method extends to other situations as can be quoted in the recent works [36], where Nguyen obtains a P´olya-Szeg¨o type principle by using some central ideas of this work, and [12], where the authors introduce and prove the sharp affine L2Sobolev trace inequality. Logarithmic Sobolev type inequalities (or log-Sobolev inequalities) are a powerful tool in Real Analysis, Complex Analysis, Geometric Analysis, Convex Geometry and Probability. The pioneer work by L. Gross [21] put forward the equivalence between a class of sharp log-Sobolev inequalities involving probability measures and hypercontractivity of the associated heat semigroup. For the Gaussian measure, the Gross’s inequality can be placed into another form known as L2log-Sobolev inequality (or L2-entropy inequality). Later, several authors established the sharp Lpversion of this latter one as well as its extremal functions. For a list of references concerning the Lplog-Sobolev inequality (or Lp-entropy inequality), we refer to [4], [8], [14], [20], [21], [25] and [45]. For some applications on the study of optimal estimates for solutions of certain nonlinear diffusion equations, we quote [5], [6], [15], [19], [20], [45] and references therein. Let n≥1 and p > 1. The Euclidean sharp Lplog-Sobolev inequality, proved by Gross [21] and Carlen [8] for p= 2, Del Pino and Dolbeault [14] for 1 < p < n and Gentil [20] for any p > 1, states that for any smooth function fon Rn, such that R|f|pdx = 1, we have (1) Ent(|f|p) = Z|f|plog |f|pdx ≤n plog Ln,p Z|∇f|pdx, where Ln,p =p np−1 ep−1 π−p 2  Γ(n 2+ 1) Γ(n(p−1) p+ 1)  p n Moreover, the inequality (1) is sharp and equality holds if and only if for some σ > 0 and x0∈Rn f(x) = ±cσe−1 σ|x−x0| p p−1 for all x∈Rn, where cσ=σ−n(p−1) p ZRn e−|x| p2 p−1!−1 p . More specifically, extremal functions of (1) were characterized in [8] for p= 2, [14] for 1 < p < n and [20] for p > 1, being the uniqueness part, up to scalings and translations, proved in [14] for any p > 1. Inequality (1) was proved by Ledoux [25] for p= 1 and its extremal functions were characterized by Beckner [4]. In order to state the sharp affine Sobolev type inequalities, for each p > 1, let Ep(f) = cn,p ZSn−1k∇ξfk−n pdξ−1 n with cn,p = (nωn)1 nnωnωp−1 2ωn+p−21 p , where ∇ξf(x) = ∇f(x)·ξand ωkdenotes the volume of the unit ball in Rk. SHARP AFFINE SOBOLEV TYPE INEQUALITIES VIA THE LpBUSEMANN-PETTY CENTROID INEQUALITY 3 The sharp affine Lplog-Sobolev inequality, proved independently by Haberl, Schuster, Xiao [24] and Zhai [46] for 1 < p < n, states that Theorem 1.1. Let n≥1and p > 1. Then for any smooth function fon Rnsatisfying R|f|pdx = 1, we have (2) Ent(|f|p) = Z|f|plog |f|pdx ≤n plog (Ln,pEp(f)p), where Ln,p is the best log-Sobolev constant given in (1). Moreover, equality holds if and only if for some σ > 0,x0∈Rnand A∈SLn, f(x) = ±cσe−1 σ|A(x−x0)| p p−1 for all x∈Rn, where SLndenotes the set of n×n-matrices preserving orientation and volume and cσ=σ−n(p−1) p ZRn e−|x| p2 p−1!−1 p . Theorem 1.1 has been stated in [24] only for 1 < p < n, although their approach, based on inequalities involving rearrangements of functions and inequality (1), allows to extend this theorem to any p > 1. We present a new and arguably shorter proof for any p > 1 based on the LpBusemann-Petty centroid inequality and Lplog-Sobolev inequality involving arbitrary norms (see [20]). The Euclidean sharp L∞log-Sobolev inequality, proved by Fujita in [17], states for any Lipschitz function fon Rnsatisfying Reβf dx = 1 that (3) Ent(eβf )≤nlog βkn e||∇f||∞, where β > 0 is a constant and kn=1 n!ωn1 n . Moreover, equality holds for normalized functions of the form f(x) = c−b|x−a|, where a∈Rn,b > 0 and c∈R. Let E∞(f) = cn,∞ZSn−1k∇ξfk−n ∞dξ−1 n with cn,∞= (nωn)1 n. Our method also produces a new and important L∞affine version of the sharp inequality (3) stated as follows. Theorem 1.2. Let n≥1and β > 0. Then for any Lipschitz function fon Rnsatisfying Reβf dx = 1, we have Ent(eβf )≤nlog βkn eE∞(f). Moreover, equality holds for normalized functions of the form f(x) = c−b|Ax −a|, where a∈Rn, b > 0,c∈Rand A∈SLn. 4 J. HADDAD, C.H. JIM´ ENEZ, M. MONTENEGRO This theorem can be seen as a limiting case of Theorem 1.1 and its proof is based on this idea. It is easy to see that the sharp affine Lplog-Sobolev inequality is stronger than its Euclidean counterpart for any p > 1 since Ep(f)≤ k∇fkp(see page 33 of [30]). Note also that the above affine inequality is invariant under volume preserving affine transformations. A well-known family of sharp interpolation inequalities is the Euclidean Gagliardo-Nirenberg inequality which states that for any smooth function fon Rn, (4) kfkr≤ Gn,p,m,r||∇f||θ pkfk1−θ m, where 1 ≤p < n, 1 ≤m < r < np n−p,θ=np(r−m) r(m(p−n)+np)∈(0,1) and Gn,p,m,r is the best Gagliardo- Nirenberg constant which is well defined thanks to a standard combination between the H¨older’s inequality and the Sobolev inequality. Inequality (4) was introduced independently by Gagliardo and Nirenberg in [18] and [35]. Some particular cases are quite famous. For instance, in the limiting case r=np n−p, (4) yields the Sobolev inequality proved by Sobolev in [40] with extremal functions and best constants obtained by Federer and Fleming [16] and Maz’ya [32] for p= 1 and by Aubin [1] and Talenti [42] for 1 < p < n. Indeed, in that latter one, fis an extremal function of the sharp Sobolev inequality if and only if f(x) = a1 + b|x−x0|p p−11−p n for some a∈R,b > 0 and x0∈Rn, and the best Sobolev constant, denoted by Sn,p, is given by (5) Sn,p =π−1 2n−1 pp−1 n−p1−1 p  Γn 2+ 1Γ(n) Γ−n p+n+ 1Γn p  1 n Two other celebrated ones are: the Nash inequality, introduced by Nash in [34], corresponds to p= 2, m= 1 and r= 2, with best constant computed by Carlen and Loss [9], and the Moser inequality, introduced by Moser in [33], corresponds to p= 2, m= 2 and r=2(n+2) n. Later, Del Pino and Dolbeault ([13] for p= 2 and [14] for 1 < p < n) and Cordero-Erausquin, Nazaret and Villani ([11] for 1< p < n) independently obtained extremal functions and best constants for the family of parameters r=αp and m=α(p−1) + 1 with α∈(1,n n−p). Namely, in this cases, fis an extremal function if and only if f(x) = a1 + b|x−x0|p p−1−p−1 m−p for some a∈R,b > 0 and x0∈Rn, and so Gn,p,α(p−1)+1,αp can be explicitly computed and its value will be presented and used later. A stronger version of the sharp Sobolev inequality, known as sharp affine LpSobolev inequality, was introduced and proved by Zhang [47] for p= 1 and by Lutwak, Yang and Zhang [30] for 1 < p < n. Recently, Zhai (Corollary 1.5 of [46]) proved an affine version of the sharp Gagliardo-Nirenberg inequality for the same family considered in [13], [14] and [11]. An alternative proof has been carried out by Haberl, Schuster and Xiao (Corollary 12 of [24]) by using their affine P´olya-Szeg¨o principle as the key tool. The sharp affine Sobolev and Gagliardo-Nirenberg inequalities are the following SHARP AFFINE SOBOLEV TYPE INEQUALITIES VIA THE LpBUSEMANN-PETTY CENTROID INEQUALITY 5 Theorem 1.3. Let n > 1and 1< p < n. Then for any smooth function fon Rn, we have kfknp n−p≤ Sn,pEp(f). Moreover, equality holds if and only if f(x) = a1 + b|A(x−x0)|p p−11−p n for some a∈R,b > 0,x0∈Rnand A∈GLn, where GLndenotes the set of invertible n×n-matrices. Theorem 1.4. Let n > 1and 1< p < n,r=αp and m=α(p−1) + 1. Assume α∈(1,n n−p). Then for any smooth function fon Rn, we have (6) kfkr≤ Gn,p,αEp(f)θkfk1−θ m, where Gn,p,α =Gn,p,α(p−1)+1,αp. Moreover, equality holds if and only if f(x) = a1 + b|A(x−x0)|p p−1−p−1 m−p for some a∈R,b > 0,x0∈Rnand A∈GLn. Theorem 1.3 was established in [30] where the main ingredients used by the authors were the Lp Petty-Projection inequality and the solution of the Lpextension of the classical Minkowski problem. Our approach makes use of the LpBusemann-Petty centroid inequality (see Section 2 for precise statements) but replaces the second one by the main result of Cordero, Nazaret and Villani [11] on general sharp Sobolev inequalities obtained by means of a mass transportation method. The connection between the Lpextension of the Minkowski problem and sharp affine Sobolev inequalities was already studied in [31]. Our approach also allows us to prove Theorem 1.4 with the same two ingredients mentioned above, whereas the known proofs presented in [24] and [46] use rearrangements and the P´olya-Szeg¨o principle. The paper is organized as follows. In section 2 we fix some notations to be used in the paper and state the LpPetty-Projection inequality. In section 3 we provide the proofs of Theorems 1.1 and 1.4. Proof of Theorem 1.3 is carried out in the same spirit of the latter, so we omit the details. Both proofs heavily rely on a central tool involving convex bodies (Theorem 3.2) and general sharp Sobolev and Gagliardo-Nirenberg inequalities proved in [20] and [11]. The proof of Theorem 3.2 requires three auxiliary lemmas. The statement of these inequalities involving abstract norms is included in this section as well for the sake of completeness. Finally, section 4 is dedicated to the proof of Theorem 1.2. 2. Notations and key tools We recall that a convex body K⊂Rnis a convex compact subset of Rnwith non-empty interior. As usual Sn−1and Bnstand for the unit Euclidean sphere and ball, respectively. It is known that vol(Bn) = ωn=πn 2 Γn 2+ 1. For K⊂Rnas before, its support function hKis defined as hK(x) = max{hx, yi:y∈K}. 6 J. HADDAD, C.H. JIM´ ENEZ, M. MONTENEGRO The support function, which describes the (signed) distances of supporting hyperplanes to the origin, uniquely characterizes K. We also have the gauge k · kKand radial rK(·) functions of Kdefined respectively as kxkK:= inf{λ > 0 : x∈λK}, x ∈Rn\{0}, rK(x) := max{λ > 0 : λx ∈K}, x ∈Rn\{0}. Clearly, kxkK=1 rK(x). For K⊂Rnwe define its polar body, denoted by K◦, by K◦:= {x∈Rn:hx, yi ≤ 1∀y∈K}. Evidently, hK=rK◦. It is also easy to see that (λK)◦=1 λK◦for λ > 0. A simple computation using polar coordinates shows that vol(K) = 1 nZSn−1 rn K(x)dx =1 nZSn−1kxk−n Kdx. For a given convex body K⊂Rnwe find in the literature many associated bodies to it, in particular Lutwak and Zhang introduced [28] for a body Kits Lp-centroid body denoted by ΓpK. This body is defined by hp ΓpK(x) := 1 a1vol(K)ZK|hx, yi|pdy for x∈Rn, where a1=ωn+p ω2ωnωp−1 . There are some other normalizations of the Lp-centroid body in the literature, the previous one is made so that ΓpBn=Bn. Inequalities (usually affine invariant) that compare the volume of a convex body Kand that of an associated body are common in the literature. For the specific case of Kand ΓpK, Lutwak, Yang and Zhang [29] (see also [7] for an alternative proof) came up with what it is known as the Lp Busemann-Petty centroid inequality, namely vol(ΓpK)≥vol(K). This inequality is sharp if and only if Kis a 0-symmetric ellipsoid. For a comprehensive survey on LpBrunn-Minkowski theory and other topics within Convex Geometry we refer to [39] and references therein. 3. The strategy of proofs In this section we will illustrate how the idea behind the proof provides a direct, elementary and unified approach to the sharp affine functional inequalities stated in Theorems 1.1, 1.3 and 1.4. The general strategy for the proof of these theorems will be based on Theorem 3.2 below combined with some known results concerning each inequality that we present for completeness. SHARP AFFINE SOBOLEV TYPE INEQUALITIES VIA THE LpBUSEMANN-PETTY CENTROID INEQUALITY 7 Sharp affine Lplog-Sobolev inequalities Let C:Rn→R+be a q-homogeneous function for q > 1, that is, C(x) satisfies C(λx) = λqC(x),∀λ≥0, x ∈Rn. It is easy to prove that for Cas before, its Legendre transform, defined by C∗(x) = sup y∈Rn{hx, yi−C(y)} is p-homogeneous where 1 p+1 q= 1. The starting point to prove Theorem 1.1 is the following result proved by Gentil (Theorem 1.1 in [20]). Theorem 3.1. Let q > 1,n≥1, and Caq-homogeneous, even, strictly convex function. Suppose that 1 p+1 q= 1. Then for any smooth function fon Rnsuch that R|f|pdx = 1, we have Ent(|f|p) = Z|f|plog |f|pdx ≤n plog LCZC∗(∇f)dx, where LC=pp+1 nep−1Re−C(x)dxp n . Moreover, the above log-Sobolev inequality is sharp and equality holds if f(x) = aexp (−bC(x−¯x)) for x∈Rn, where a, b > 0and ¯x∈Rnsatisfy a−p=Rexp (−pbC(x−¯x)))dx. Before stating the referred Theorem 3.2, some preliminary results will be necessary and a little bit of notation should be introduced. Definition 1. Let fbe a non-zero real smooth function on Rnwith compact support. We define C∗ f(x) := ZSn−1k∇ξfk−n−p p|hx, ξi|pdξ and consider its Legendre transform Cf. Proposition 2. Let fbe a non-zero real smooth function on Rnwith compact support. The function C∗ fgiven in the definition above is well defined, strictly convex, p-homogeneous and C∗ f(x) = 0 if and only if x= 0. Proof. We only prove the good definition and strict convexity of C∗ f, since the other statements are evident. First note that the equality k∇ξfkp= 0 implies ∂f ∂ξ (x) = 0 for every x∈Rnwhich contradicts the fact that fis non-zero and has compact support. Then, continuity with respect to ξgives k∇ξfkp≥ε > 0 for every ξ∈Sn−1which proves the good definition. For the strict convexity, take x1, x2∈Rnwith x16=x2and λ∈(0,1). We have C∗ f(λx1+ (1 −λ)x2) = ZSn−1k∇ξfk−n−p p|λhx1, ξi+ (1 −λ)hx2, ξi|pdξ <ZSn−1k∇ξfk−n−p p(λ|hx1, ξi|p+ (1 −λ)|hx2, ξi|p)dξ =λC∗ f(x1) + (1 −λ)C∗ f(x2). 8 J. HADDAD, C.H. JIM´ ENEZ, M. MONTENEGRO  Denote by Lf={ξ∈Rn:k∇ξfkp≤1}, Kf={x∈Rn:Cf(x)≤1}. We note that Kfand Lfare convex bodies with gauge functions kxkKf=Cf(x)1 qand kξkLf=k∇ξfkp and radial functions rKf(x) = Cf(x)−1 qand rLf(ξ) = k∇ξfk−1 p. For convenience, we set Zp(f) = ZSn−1k∇ξfk−n pdξ−1 n and notice we have the identities nvol(Lf) = Zp(f)−nand Ep(f) = cn,pZp(f). Lemma 3. Let fbe a non-zero real smooth function on Rnwith compact support. We have ZRn C∗ f(∇f(x))dx =nvol(Lf). Proof. ZRn C∗ f(∇f(x))dx =ZRnZSn−1k∇ξfk−n−p p|h∇f(x), ξi|pdξ dx =ZSn−1ZRnk∇ξfk−n−p p|h∇f(x), ξi|pdx dξ =ZSn−1k∇ξfk−n−p pk∇ξfkp pdξ =ZSn−1k∇ξfk−n pdξ =nvol(Lf).  Lemma 4. Let fbe a non-zero real smooth function on Rnwith compact support. We have C∗ f(v) = (n+p)a1vol(Lf)hp ΓpLf(v). Proof. We compute a1vol(Lf)hp ΓpLf(v) = ZLf|hv, xi|pdx =ZSn−1ZrLf(ξ) 0 rn−1|hv, rξi|pdr dξ =ZSn−1|hv, ξi|pZrLf(ξ) 0 rn+p−1dr dξ =ZSn−1|hv, ξi|prLf(ξ)n+p n+pdξ =C∗ f(v)1 n+p.  Lemma 5. Let fbe a non-zero real smooth function on Rnwith compact support. We have K◦ f={x∈Rn:C∗ f(x)≤ℓq}, where ℓq=q−p qp−1. SHARP AFFINE SOBOLEV TYPE INEQUALITIES VIA THE LpBUSEMANN-PETTY CENTROID INEQUALITY 9 Proof. Firstly, note that ℓq= maxt≥0{t1 q−t}. Take x∈K◦ f. By definition of Kf,hx, zi ≤ 1 whenever Cf(z)≤1. Since Cfis q-homogeneous, we have for any y∈Rnthat Cf(Cf(y)−1 qy) = 1, thus hx, Cf(y)−1 qyi ≤ 1 and hx, yi−Cf(y)≤Cf(y)1 q−Cf(y)≤ℓq, so that C∗ f(x)≤ℓq. Now take x∈Rn\K◦ f. By definition of K◦ f, there exists y6= 0 such that hx, yi>1≥Cf(y), so that hx, yi> Cf(y)1 q. For any t > 0, we have hx, tyi>(tqCf(y))1 q and hx, tyi−Cf(ty)>(tqCf(y))1 q−tqCf(y). Since Cf(y)>0, we may choose t > 0 such that the right-hand side is maximized. Thus, we get C∗ f(x)> ℓq. We are in conditions to finally state our key theorem announced above. Theorem 3.2. Let fbe a non-zero real smooth function on Rnwith compact support. (7) vol(Kf)−p nZRn C∗ f(∇f(x))dx ≤a2Zp(f)p, where a2=ℓqnn+p n a1(n+p). Moreover, if equality holds in (7) then there exists an invertible matrix Msuch that (8) Ep(f)p= det(M)p−n nZ|∇g|pdx, where g(x) = f(M−1x). Proof. In view of Lemma 3 and the identity nvol(Lf) = Zp(f)−n, (7) becomes vol(Kf)−p n≤a2Zp(f)n+p=a2(nvol(Lf))−p+n n. Note that since hp ΓpLfis p-homogeneous, we have (n+p) vol(Lf)a1hp ΓpLf(x)≤ℓq if and only if hp ΓpLf  ((n+p) vol(Lf)a1)1 px ℓ 1 p q ≤1. 16 J. HADDAD, C.H. JIM´ ENEZ, M. MONTENEGRO E-mail address:[email protected] E-mail address:[email protected] E-mail address:monten[email protected].br Universidade Federal de Minas Gerais Pontif ´ ıcia Universidade Cat´ olica do Rio de Janeiro Universidade Federal de Minas Gerais