Optimal Sobolev norms in the affine class

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J. Math. Anal. Appl. ••• (••••) •••–•••

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Journal of Mathematical Analysis and Applications www.elsevier.com/locate/jmaa

Optimal Sobolev norms in the affine class ✩ Qingzhong Huang a,∗ , Ai-Jun Li b a

College of Mathematics, Physics and Information Engineering, Jiaxing University, Jiaxing 314001, China b School of Mathematics and Information Science, Henan Polytechnic University, Jiaozuo City 454000, China

a r t i c l e

i n f o

Article history: Received 2 September 2015 Available online xxxx Submitted by A. Cianchi Keywords: Lp surface isotropic position Functional Lp surface isotropic position Lp affine energy Lp sine energy (p, 2) Fisher information matrix

a b s t r a c t Optimal Sobolev norms under volume preserving affine transformations are considered. It turns out that this minimal transform is equivalent to the (p, 2) Fisher information matrix defined by Lutwak, Lv, Yang, and Zhang. Furthermore, some analytic inequalities regarding to the Lp affine and Lp sine energies for the optimal function are investigated. © 2015 Elsevier Inc. All rights reserved.

1. Introduction For p ≥ 1 and n ≥ 2, the Lp Sobolev space W 1,p (Rn ) is the space of real-valued Lp functions on Rn with weak Lp partial derivatives. The sharp Lp Sobolev inequality states that if f ∈ W 1,p (Rn ) for 1 ≤ p < n, then there exists a best constant An,p such that np , ∇f p ≥ An,p f  n−p

(1.1)

where  · q denotes the Lq norm of the Euclidean norm of functions and ∇f is the gradient of f . Inequality (1.1) goes back to Federer and Fleming [12] and Maz’ya [31] for p = 1 and to Aubin [3] and Talenti [35] for 1 < p < n. For strengthened versions of (1.1), see, e.g., [6,7,9,10,32,39]. ✩ The first author was supported in part by the National Natural Science Foundation of China (No. 11371239). The second author was supported by NSFC-Henan Joint Fund (No. U1204102) and Doctoral Foundation of Henan Polytechnic University (No. B2011-024). * Corresponding author. E-mail addresses: [email protected] (Q. Huang), [email protected] (A.-J. Li).

http://dx.doi.org/10.1016/j.jmaa.2015.11.063 0022-247X/© 2015 Elsevier Inc. All rights reserved.

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A stronger version of the sharp Lp Sobolev inequality, known as sharp affine Lp Sobolev inequality, was introduced and established by Zhang [42] for p = 1 and by Lutwak, Yang and Zhang [23] for 1 < p < n. It states that if f ∈ W 1,p (Rn ) for 1 ≤ p < n, then np , Ep (f ) ≥ An,p f  n−p

(1.2)

where Ep (f ) is the Lp affine energy, defined, for f ∈ W 1,p (Rn ), by Ep (f ) = cn,p

 

Du f −n p du

−1/n

,

(1.3)

S n−1

with 1

cn,p =

1

1

1

1

p n n + p π 2p + 2 Γ( n+p 2 ) 1

1

1

1

p 2 p Γ(1 + n2 ) n + p Γ( p+1 2 )

and Du f = u · ∇f (x) is the directional derivative of f in the direction u. Note that the Lp affine energy is a fundamental concept in variants of affine Sobolev inequalities (see e.g., [2,8,17–19,23,27,36–38,42]). We emphasize the remarkable and important fact that Ep (f ) is invariant under volume preserving affine transformations on Rn , while ∇f p is invariant only under rigid motions. Hence the affine Lp Sobolev inequality (1.2) is invariant under affine transformations of Rn , while the classical Lp Sobolev inequality (1.1) is invariant only under rigid motions. Moreover, it was shown by Lutwak, Yang, and Zhang [23] that ∇f p ≥ Ep (f ).

(1.4)

Hence the affine Lp Sobolev inequality (1.2) is stronger than the Lp Sobolev inequality (1.1). Denote fT (x) = f (T x) for every T ∈ SL(n) and x ∈ Rn . From the above facts, we may ask the following question: Do there exist a Tp ∈ SL(n) and a constant Bn,p such that Bn,p ∇fTp p ≤ Ep (fTp ) = Ep (f )?

(1.5)

In other words, it asks whether there exists a transformation Tp ∈ SL(n) which minimizes ∇fT p for all T ∈ SL(n). We answer this question in the following theorem. Theorem 1.1. Given f ∈ W 1,p (Rn ). There exists a unique (up to orthogonal transformations) Tp ∈ SL(n) which minimizes {∇fT p : T ∈ SL(n)}. In particular, when p = 2, for the minimizer T2 , we have  ∇fT2 22 In

=n

∇fT2 (x) ⊗ ∇fT2 (x)dx,

(1.6)

Rn

where ∇fT2 ⊗ ∇fT2 is the rank-one orthogonal projection onto the space spanned by the vector ∇fT2 and In denotes the identity operator on Rn . It is shown in Section 6 that Tp in Theorem 1.1 is equivalent to the (p, 2) Fisher information matrix Jp,2 (X) defined by Lutwak, Lv, Yang, and Zhang [28] in the context of the information theory. For the related problem, please see [27] by Lutwak, Yang, and Zhang. Thus, the problem (1.5) can be answered as follows.

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Theorem 1.2. Given f ∈ W 1,p (Rn ), we have Bn,p ∇fTp p ≤ Ep (f ) ≤ ∇fTp p ,

(1.7)

where 1

Bn,p =

1

1

1

n n p π 2p + 2 Γ( n+p 2 ) Γ(1 + p ) 1

1

1

1

1 p 2 p +1 Γ(1 + n2 ) n + p Γ( p+1 2 ) Γ(1 + p )

.

Moreover, let K be an origin-symmetric convex body in Rn and let f (x) = g(xK ) with g ∈ C 1 (0, ∞). For p = 2, there is equality in the left inequality if and only if K is a parallelotope. If p is not an even integer, then there is equality in the right inequality if and only if K is an ellipsoid. Unlike the Lp Sobolev norm ∇f p , the quantity min ∇fT p

T ∈SL(n)

is clearly invariant under volume preserving affine transformations on Rn . Thus, the inequality (1.7) is invariant under affine transformations of Rn , while the inequality (1.4) is invariant only under rigid motions. Next, motivated by the work of Maresch, Schuster [29] and Guo, Leng [16], we define the Lp sine energy, for f ∈ W 1,p (Rn ), by Gp (f ) = dn,p

 

Du⊥ f −n p du

−1/n

,

(1.8)

S n−1

where 1

dn,p =

1

1

n+p n n + p π 2 Γ( n−1 2 )Γ( 2 ) 1

2Γ(1 + n2 )1+ n Γ( n+p−1 ) 2

and Du⊥ f = Pu⊥ ∇f (x). Here u⊥ is the central hyperplane perpendicular to u and Pu⊥ ∇f (x) is the orthogonal projection of the vector ∇f (x) onto u⊥ . Unfortunately, it is easy to see that Gp (f ) is not invariant under volume preserving affine transformations on Rn but is still invariant under rigid motions. Similarly to Theorem 1.2, the following theorem is established. Theorem 1.3. Given f ∈ W 1,p (Rn ), we have Cn,p ∇fTp p ≤ Gp (fTp ) ≤ ∇fTp p ,

(1.9)

where n+p 2 2 p n−1 n p −1 (n − 1) p − n−1 +1 Γ(1 + np ) n Γ( n−1 2 ) Γ( 2 ) 1

Cn,p =

1

1

1

1

1

n+p−1 2 n−1 Γ( n+2 )2 Γ( 4Γ( n−1 ) p ) 2 2

.

Moreover, let K be an origin-symmetric convex body in Rn and let f (x) = g(xK ) with g ∈ C 1 (0, ∞). If p is not an even integer, then there is equality in the right inequality if and only if K is an ellipsoid. Furthermore, we establish the following inequalities.

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Theorem 1.4. Given f ∈ W 1,p (Rn ), for every u ∈ S n−1 , n− p ∇fTp p ≤ Du fTp p ≤ n− 2 ∇fTp p , 1

1

1 ≤ p ≤ 2,

and n− p ∇fTp p ≥ Du fTp p ≥ n− 2 ∇fTp p , 1

1

p ≥ 2.

Theorem 1.5. Given f ∈ W 1,p (Rn ), for every u ∈ S n−1 ,  n − 1  p1 n

∇fTp p ≤ Du⊥ fTp p ≤

 n − 1  12 n

∇fTp p ,

1 ≤ p ≤ 2,

and  n − 1  p1 n

∇fTp p ≥ Du⊥ fTp p ≥

 n − 1  12 n

∇fTp p ,

p ≥ 2.

The rest of this paper is organized as follows: In Section 2 the basic notations and preliminaries are provided. To prove our main theorems, some facts about the Lp surface isotropic positions are given in Section 3. The interplay of the geometric inequalities and corresponding analytic inequalities is presented in Section 4 and Section 5. In Section 6, the equivalence of Tp in Theorem 1.1 and the (p, 2) Fisher information matrix Jp,2 (X) is explored. 2. Notations and preliminaries We refer to the book [34] for the basic facts about convex bodies and the Lp Brunn–Minkowski theory. Denote by p∗ ∈ [1, ∞) the Hölder conjugate of p ∈ [1, ∞); i.e., 1 1 + = 1. p p∗ Denote by κn (p) the volume of the unit ball of the np space, that is, κn (p) =

2n (Γ(1 + p1 ))n Γ(1 + np )

.

Abbreviate κn (2) by κn , the volume of n-dimensional Euclidean unit ball B2n . Define αn,p :=

Γ(1 + 12 )Γ( n+p 2 )

,

p n−1 n p −n (n − 1) p − n−1 +n Γ(1 + np ) n

βn,p :=

 Γ(1 + n )Γ( 1+p )  np 2 2

n

n

n

n

n−1 Γ( n−1 p )

,

and γn,p :=

n+p−1 ) 2Γ( n+2 2 )Γ( 2 n+p Γ( n−1 2 )Γ( 2 )

.

A convex body is a compact convex set in Rn which is throughout assumed to contain the origin in its interior. We denote by Kon the class of convex bodies. Let Ken denote the class of origin-symmetric members

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of Kon . Each convex body K is uniquely determined by its support function hK = h(K, ·) : Rn → R defined by hK (x) = h(K, x) := max{x · y : y ∈ K}, where x · y denotes the standard inner product of x and y in Rn . Let  · K : Rn → [0, ∞) denote the Minkowski functional of K ∈ Kon ; i.e., xK = min{λ ≥ 0 : x ∈ λK}. The polar set K ∗ of K ∈ Kon is the convex body defined by K ∗ = {x ∈ Rn : x · y ≤ 1 for all y ∈ K}. Clearly, for K ∈ Kon , hK ∗ (·) =  · K .

(2.1)

For K, L ∈ Kon , and ε > 0, the Lp Minkowski–Firey combination K +p ε · L is the convex body whose support function is given by h(K +p ε · L, ·)p = h(K, ·)p + εh(L, ·)p . The Lp mixed volume Vp (K, L) of K, L ∈ Kon was defined in [21] by Vp (K, L) =

p V (K +p ε · L) − V (K) lim , n ε→0+ ε

where V is the n-dimensional volume (i.e. Lebesgue measure in Rn ). In particular, Vp (K, K) = V (K). It was shown in [21] that corresponding to each K ∈ Kon , there is a positive Borel measure Sp (K, ·) on S n−1 , called the Lp surface area measure of K, such that for every L ∈ Kon , Vp (K, L) =

1 n

 hpL (u)dSp (K, u). S n−1

Moreover, the Lp surface area measure is absolutely continuous with respect to the surface area measure S(K, ·) of K: 1−p (·)dS(K, ·). dSp (K, ·) = hK

Recall that for a Borel set ω ⊂ S n−1 , S(K, ω) is the (n − 1)-dimensional Hausdorff measure Hn−1 of the set of all boundary points of K for which there exists a normal vector of K belonging to ω. When L = B2n , the Lp surface area Sp (K) of K is given by Sp (K) = nV (K, B2n ). Clearly, Sp (tK, ·)/V (tK) = t−p Sp (K, ·)/V (K) for all t > 0 and K ∈ Kon . In [24], Lutwak, Yang, and Zhang established the following remarkable result.

(2.2)

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The solution of normalized even Lp Minkowski problem. Let p ≥ 1. If μ is an even Borel measure on S n−1 whose support is not contained in a great subsphere of S n−1 , then there exists a unique origin-symmetric convex body K such that Sp (K, ·)/V (K) = μ. A finite nonnegative Borel measure μ on S n−1 is called isotropic if  v ⊗ vdμ(v) = In .

(2.3)

S n−1

The measure μ is said to be even if it assumes the same value on antipodal sets. The two most important examples of even isotropic measures on S n−1 are suitably normalized spherical Lebesgue measure and the cross measure, i.e., measures concentrated uniformly on {±b1 , . . . , ±bn }, where b1 , . . . , bn denote orthonormal basis vectors of Rn . 3. The Lp surface isotropic position A convex body K in Rn is said in Lp surface isotropic position if its normalized Lp surface area measure nSp (K, ·)/Sp (K) is isotropic on S n−1 . This concept was first introduced by Lutwak, Yang, Zhang [26], while the L1 case was first defined in [33]. Moreover, Lutwak, Yang, Zhang [26] obtained the following result (see also [40] by Yu). For p = 1, it was due to Petty [33] and Giannopoulos, Papadimitrakis [14]. Proposition 3.1. Suppose p ∈ [1, ∞) and K ∈ Kon . Then Sp (K) = min{Sp (T K) : T ∈ SL(n)} if and only if K is in Lp surface isotropic position. The minimal Lp surface area position is unique up to orthogonal transformations. Other extensions of this notion can be found in [1,13–15,30,41]. Define the convex body Cp (μ) in Rn whose support function, for u ∈ S n−1 , is given by   hCp (μ) (u) =

1/p

|u · v|p dμ(v)

.

(3.1)

S n−1

The following lemma, extending the results of Ball [4] and Barthe [5], was due to Lutwak, Yang and Zhang [25]. Lemma 3.2. Suppose p ∈ [1, ∞). If μ is an even isotropic measure on S n−1 , then κn ≤ V (Cp∗ μ) ≤ κn (p). αn,p

(3.2)

If p is not an even integer, then there is equality in the left inequality if and only if μ is suitably normalized Lebesgue measure. For p = 2, there is equality in the right inequality if and only if μ is a cross measure. Define the convex body Sp (μ) in Rn whose support function, for u ∈ S n−1 , is given by   hSp (μ) (u) = S n−1

(1 − |u · v| )

2 p/2

 

1/p dμ(v)

=

1/p

|Pv⊥ u|p dμ(v)

.

S n−1

The following lemma, extending the case p = 1 due to Maresch and Schuster [29], was proved by Guo and Leng [16].

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Lemma 3.3. Suppose p ∈ [1, ∞). If μ is an even isotropic measure on S n−1 , then n κn γn,p κn ∗ ≤ V (S μ) ≤ . p n γn,p βn,p

(3.3)

If p is not an even integer, then there is equality in the left inequality if and only if μ is suitably normalized Lebesgue measure. As shown in [29] and [16], the right inequality in (3.3) is asymptotically optimal. For each convex body K ∈ Kon , the Lp projection body, Πp K, of K was introduced by Lutwak, Yang, and Zhang [22], which is the origin-symmetric convex body whose support function, for u ∈ S n−1 , is given by  hΠp K (u) =

1 V (K)



1/p

|u · v|p dSp (K, v)

.

(3.4)

S n−1

From (2.2), we have Πp (cK) = c−1 Πp K.

(3.5)

The following lemma without the equality conditions was due to Yu [40], which extended the case p = 1 by Giannopoulos and Papadimitrakis [14]. Here we characterize the equality conditions. Lemma 3.4. Suppose p ∈ [1, ∞). If K ∈ Ken is in Lp surface isotropic position, then  S (K) n/p nn/p κn p ≤ V (Π∗p K) ≤ nn/p κn (p). αn,p V (K)

(3.6)

If p is not an even integer, then there is equality in the left inequality if and only if K is a ball. For p = 2, there is equality in the right inequality if and only if K is a cube. Proof. Define the even measure μ on S n−1 by μ=

n Sp (K, ·). Sp (K)

Since K is in Lp surface isotropic position, it follows that μ is isotropic. By the definitions of Cp μ (3.1) and Πp K (3.4), we have Cp∗ μ =

 S (K) 1/p p Π∗p K. nV (K)

Then, inequalities (3.6) immediately follows from (3.2). Now, we deal with the characterization of the equalities in (3.6). The equality conditions of (3.2) yield that if p is not an even integer, then there is equality in the left inequality if and only if nSp (K, ·)/Sp (K) is suitably normalized Lebesgue measure; For p = 2, there is equality in the right inequality if and only if nSp (K, ·)/Sp (K) is a cross measure. On the other hand, it is easy to verify that nSp (B2n , ·)/Sp (B2n ) is a suitably normalized Lebesgue measure and nSp (OC0 , ·)/Sp (OC0 ) is a cross measure on S n−1 for some O ∈ O(n), where C0 = [−1, 1]n . It follows from (2.2) that Sp (λ1 B2n , ·) nSp (B2n , ·) = Sp (B2n ) V (λ1 B2n )

with λ1 =

 nV (B n ) − p1 2

Sp (B2n )

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and Sp (λ2 OC0 , ·) nSp (OC0 , ·) = Sp (OC0 ) V (λ2 OC0 )

with λ2 =

 nV (C ) − p1 0 . Sp (C0 )

Thus, the equality conditions follow from the uniqueness of the solution of the normalized even Lp Minkowski problem and the fact that  nV (tK) − p1 Sp (tK)

tK =

 nV (K) − p1 Sp (K)

K

for all t > 0. 2 Corresponding to the Lp projection body Πp K, Maresch, Schuster [29] and Guo, Leng [16] introduced a new convex body Ψp K whose support function, for u ∈ S n−1 , is given by  hΨp K (u) =

1 V (K)



1/p

|Pv⊥ u|p dSp (K, v)

.

(3.7)

S n−1

From (2.2), we have Ψp (cK) = c−1 Ψp K.

(3.8)

Using the similar argument to Lemma 3.4, Guo and Leng [16] established the following result. Lemma 3.5. Suppose p ∈ [1, ∞). If K ∈ Ken is in Lp surface isotropic position, then n  S (K) n/p nn/p κn γn,p nn/p κn p ∗ ≤ V (Ψ K) ≤ . p n γn,p V (K) βn,p

(3.9)

If p is not an even integer, then there is equality in the left inequality if and only if K is a ball. The following observation, extending the case p = 1 by Giannopoulos and Papadimitrakis [14], was due to Yu [40]. For the completeness, we present the proof here. Lemma 3.6. Suppose p ∈ [1, ∞). If K ∈ Kon is in Lp surface isotropic position, then for every u ∈ S n−1 , n− p 1

 S (K)  p1  S (K)  p1 1 p p ≤ h(Πp K, u) ≤ n− 2 , V (K) V (K)

1 ≤ p ≤ 2,

and n− p 1

 S (K)  p1  S (K)  p1 1 p p ≥ h(Πp K, u) ≥ n− 2 , V (K) V (K)

p ≥ 2.

Proof. For p = 2, by the definition of Πp K (3.4), we have  h(Π2 K, u) =

1 V (K)



 12

|u · v|2 dS2 (K, v) S n−1

for every u ∈ S n−1 . For 1 ≤ p < 2, by (3.4) and the Hölder inequality, we obtain

=

 S (K)  12 2 , nV (K)

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 h(Πp K, u) =

≤



1 V (K)



9

 p1

|u · v|p dSp (K, v) S n−1



1 V (K)

 p2  S (K) 1− p2  p1 p V (K)

|u · v|2 dSp (K, v) S n−1

 S (K)  p1 1 p = n− 2 , V (K) for every u ∈ S n−1 . On the other hand, we have 



1 h(Πp K, u) ≥ V (K)

 p1

|u · v|2 dSp (K, v)

=

 S (K)  p1 p . nV (K)

S n−1

For p > 2, the argument is similar to the above arguments. 2 To obtain the similar result to Lemma 3.6 about Ψp K, we need the following lemma. Lemma 3.7. If μ is an isotropic measure on S n−1 , then 1 n−1

 |Pv⊥ u|2 dμ(v) = 1,

for every u ∈ S n−1 .

S n−1

Proof. It follows from the isotropicity assumption (2.3) that  Pu⊥ v ⊗ Pu⊥ vdμ(v).

Pu⊥ =

(3.10)

S n−1

Notice that Pu⊥ has rank n − 1. Taking traces in (3.10), we get  |Pu⊥ v|2 dμ(v) = n − 1,

for every u ∈ S n−1 .

S n−1

The result now follows from the fact that |Pv⊥ u| = |Pu⊥ v| for every u, v ∈ S n−1 .

2

Then we have Lemma 3.8. Suppose p ∈ [1, ∞). If K ∈ Kon is in Lp surface isotropic position, then for every u ∈ S n−1 ,  n − 1  12  S (K)  p1  n − 1  p1  S (K)  p1 p p ≤ h(Ψp K, u) ≤ , n V (K) n V (K)

1 ≤ p ≤ 2,

and  n − 1  12  S (K)  p1  n − 1  p1  S (K)  p1 p p ≥ h(Ψp K, u) ≥ , n V (K) n V (K)

p ≥ 2.

Proof. Using Lemma 3.7 with the isotropic measure μ = ndSp (K, ·)/Sp (K), we have 1 n−1

 |Pv⊥ u|2 S n−1

n dSp (K, v) = 1, Sp (K)

for every u ∈ S n−1 .

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For p = 2, by (3.7), we have  h(Ψ2 K, u) =

1 V (K)



 12

|Pv⊥ u|2 dS2 (K, v)

=

 (n − 1)S (K)  12 2 , nV (K)

S n−1

for every u ∈ S n−1 . For 1 ≤ p < 2, by (3.7) and the Hölder inequality, we obtain  h(Ψp K, u) =



1 V (K)

 p1

|Pv⊥ u|p dSp (K, v) S n−1

≤





1 V (K)

 p2  S (K) 1− p2  p1 p V (K)

|Pv⊥ u|2 dSp (K, v) S n−1

=

 n − 1  12  S (K)  p1 p , n V (K)

for every u ∈ S n−1 . On the other hand, we have 

1 h(Ψp K, u) ≥ V (K)



 p1

|Pv⊥ u|2 dSp (K, v)

=

 (n − 1)S (K)  p1 p . nV (K)

S n−1

For p > 2, the argument is similar to the previous. 2 4. The functional Lp surface isotropic position The even functional Minkowski problem on W 1,p (Rn ) was solved by Lutwak, Yang, and Zhang [27]. Theorem 4.1. Given a function f ∈ W 1,p (Rn ), there exists a unique origin-symmetric convex body f p such that   1 Φp (∇f (x))dx = Φ(v)p dSp ( f p , v), (4.1) V ( f p ) Rn

S n−1

for every even continuous function Φ : Rn → [0, ∞) that is homogeneous of degree 1. The lemma below (see [27, Proposition 5.4.]) describes how f p behaves if f is composed with a volume preserving linear transformation. Lemma 4.2. Suppose f ∈ W 1,p (Rn ). Then for T ∈ SL(n), f ◦ T −1 p = T f p . We say that a function f ∈ W 1,p (Rn ) is in the functional Lp surface isotropic position if f p is in the Lp surface isotropic position. Theorem 4.3. Suppose f ∈ W 1,p (Rn ) is in the functional Lp surface isotropic position. Then ∇f p = min{∇(f ◦ T )p : T ∈ SL(n)}.

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The minimal position is unique up to orthogonal transformations. Moreover, if f is in the functional L2 surface isotropic position, then  ∇f 22 In

∇f (x) ⊗ ∇f (x)dx.

=n Rn

Proof. Taking Φ(v) = |v| in (4.1), we get  ∇f pp =

|∇f (x)|p dx = Rn

Sp ( f p ) . V ( f p )

(4.2)

From (4.2), Proposition 3.1 with K = f p , Lemma 4.2, and (4.2) again, we have ∇f pp = Sp ( f p )/V ( f p ) = min{Sp (T f p )/V (T f p ) : T ∈ SL(n)} = min{Sp ( f ◦ T −1 p )/V ( f ◦ T −1 p ) : T ∈ SL(n)} = min{∇(f ◦ T −1 )p : T ∈ SL(n)}. Now, we consider the case p = 2. Taking Φ(v) = |u · v| in (4.1), we get 

1 V ( f 2 )

|u · ∇f (x)|2 dx = Rn

 |u · v|2 dS2 ( f 2 , v).

(4.3)

S n−1

Since f is in the functional L2 surface isotropic position, it follows that nS2 ( f 2 , ·)/S2 ( f 2 ) is isotropic. Thus, from (4.3) and (4.2), we have  |u · ∇f (x)|2 dx = Rn

∇f 22 S2 ( f 2 ) = . nV ( f 2 ) n

2

Theorem 4.4. Suppose f ∈ W 1,p (Rn ) is in the functional Lp surface isotropic position. Then cn,p n

1 1 n+p

1

κn (p) n

∇f p ≤ Ep (f ) ≤ ∇f p .

(4.4)

For p = 2, there is equality in the left inequality if and only if f p is a cube. If p is not an even integer, then there is equality in the right inequality if and only if f p is a ball. Proof. Taking Φ(v) = |u · v| in (4.1) and (3.4), we have  Du f pp

|u · ∇f (x)|p dx

= Rn

=

1 V ( f p )

 |u · v|p dSp ( f p , v) = h(Πp f p , u)p . S n−1

From (1.3) and the polar coordinate formula for volume, we obtain

(4.5)

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Ep (f ) = cn,p

 

Du f −n p du

−1/n

S n−1

  = cn,p

h(Πp f p , u)−n du

−1/n

S n−1

= cn,p n− n V (Π∗p f p )− n . 1

1

(4.6)

The inequalities (4.4) and their equality conditions follow by combining (4.2), (4.6) and Lemma 3.4.

2

Theorem 4.5. Suppose f ∈ W 1,p (Rn ) is in the functional Lp surface isotropic position. Then 1 n dn,p βn,p 1

1

1

n n + p κnn γn,p

∇f p ≤ Gp (f ) ≤ ∇f p .

(4.7)

If p is not an even integer, then there is equality in the right inequality if and only if f p is a ball. Proof. Taking Φ(v) = |Pu⊥ v| in (4.1) and (3.7), we have  p Du⊥ f p = |Pu⊥ ∇f (x)|p dx Rn

=

1 V ( f p )

1 = V ( f p ) =

 |Pu⊥ v|p dSp ( f p , v) S n−1



|Pv⊥ u|p dSp ( f p , v)

S n−1 h(Ψp f p , u)p .

(4.8)

The third equality follows from the fact that |Pu⊥ v| = |Pv⊥ u| for every u, v ∈ S n−1 . From (1.8) and the polar coordinate formula for volume, we obtain −1/n   Gp (f ) = dn,p Du⊥ f −n p du S n−1

  = dn,p

h(Ψp f p , u)−n du

−1/n

S n−1

= dn,p n− n V (Ψ∗p f p )− n . 1

1

(4.9)

The inequalities (4.7) and their equality conditions follow by combining (4.2), (4.9) and Lemma 3.5.

2

By Lemmas 3.6 and 3.8, together with (4.2), (4.5) and (4.8), we have Theorem 4.6. Suppose f ∈ W 1,p (Rn ) is in the functional Lp surface isotropic position. Then for every u ∈ S n−1 , n− p ∇f p ≤ Du f p ≤ n− 2 ∇f p , 1

1

1 ≤ p ≤ 2,

and n− p ∇f p ≥ Du f p ≥ n− 2 ∇f p , 1

1

p ≥ 2.

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Theorem 4.7. Suppose f ∈ W 1,p (Rn ) is in the functional Lp surface isotropic position. Then for every u ∈ S n−1 ,  n − 1  p1 n

∇f p ≤ Du⊥ f p ≤

 n − 1  12 n

∇f p ,

1 ≤ p ≤ 2,

and  n − 1  p1 n

 n − 1  12

∇f p ≥ Du⊥ f p ≥

n

∇f p ,

p ≥ 2.

Therefore, the previous theorems together with Proposition 3.1 immediately yield our main results (Theorems 1.1–1.5). 5. From analytic to geometric inequalities To obtain geometric inequalities from their analytic inequalities, the following lemma will be needed (see e.g., [20,37,38]). Since the argument is slightly different, we give the proof for completeness. Lemma 5.1. Suppose K ∈ Ken and f = g(xK ) with g ∈ C 1 (0, ∞). Then f p is a dilate of K; that is f p = c(f )− p K, 1

where c(f ) = V (K)

∞ 0

(5.1)

tn−1 |g  (t)|p dt.

Proof. Since hK ∗ is a Lipschitz function and hK ∗ (x) = 1 on ∂K, for almost every x ∈ ∂K, we have νK (x) =

∇hK ∗ (x) , |∇hK ∗ (x)|

where νK (x) is the outer unit normal vector of K at the point x. Note that hK (∇hK ∗ (x)) = 1 for almost every x ∈ Rn . Hence hK (νK (x)) =

1 |∇hK ∗ (x)|

.

(5.2)

Then by (4.1) and (2.1), the co-area formula (see, e.g., [11, p.258]), the fact that ∇hK ∗ is homogeneous of degree 0, and (5.2), we get 1 V ( f p ) 

 Φp (v)dSp ( f p , v) S n−1

Φp (∇f (x))dx

= Rn



=

Φp (g  (hK ∗ (x)∇hK ∗ (x)))dx

Rn



=

|g  (hK ∗ (x))|p Φp

Rn

∞  = 0 ∂K

 ∇h ∗ (x)  K |∇hK ∗ (x)|p dx |∇hK ∗ (x)|

tn−1 |g  (t)|p Φp

 ∇h ∗ (x)  K |∇hK ∗ (x)|p−1 dHn−1 (x)dt |∇hK ∗ (x)|

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∞ =

tn−1 |g  (t)|p dt

0

∞ =

Φp (νK (x))hK (νK (x))1−p dHn−1 (x)

∂K

tn−1 |g  (t)|p dt

0

=



1 V (K0 )

 Φp (v)dSp (K, v)

S n−1



Φp (v)dSp (K0 , v) S n−1

for every even continuous function Φ that is homogeneous of degree 1. From (2.2), we get K0 = K/c(f )1/p ∞ with c(f ) = V (K) 0 tn−1 |g  (t)|p dt. Thus, the uniqueness of the solution of the normalized even Lp Minkowski problem yields f p = K0 = K/c(f )1/p .

2

By Lemma 5.1, we see that the function f is in the functional Lp surface isotropic position if and only if the convex body K is in the Lp surface isotropic position when f = g(xK ) with g ∈ C 1 (0, ∞) and K ∈ Ken . The assumption that f (x) = g(xK ), together with (4.2), (5.1) and (2.2), give that  S ( f )  p1  S (c(f )− p1 K)  p1  S (K)  p1 1 p p p p p ∇f p = = = c(f ) . 1 −p V ( f p ) V (K) V (c(f ) K) From (4.5), (4.6), (5.1) and (3.5), we have Du f p = h(Πp f p , u) = h(Πp (c(f )− p K), u) 1

1

= c(f ) p h(Πp K, u), for u ∈ S n−1 , and Ep (f ) = cn,p n− n V (Π∗p f p )− n 1

1

= cn,p n− n V (Π∗p (c(f )− p K))− n 1

1

1

= c(f ) p cn,p n− n V (Π∗p K)− n . 1

1

1

Similarly, from (4.8), (4.9), (5.1) and (3.8), we get 1

Du⊥ f p = c(f ) p h(Ψp K, u), for u ∈ S n−1 , and Gp (f ) = c(f ) p dn,p n− n V (Ψ∗p K)− n . 1

1

1

Consequently, we see that our analytic inequalities in Section 4 could imply corresponding geometry inequalities in Section 3. Moreover, the equality conditions of Theorem 1.2 and Theorem 1.3 follow from Lemma 5.1 and the equality conditions of Theorem 4.4 and Theorem 4.5.

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6. Relation to the (p, 2) Fisher information matrix The last section is dedicated to explain that Tp in Theorem 1.1 and the (p, 2) Fisher information matrix Jp,2 (X) are actually equivalent. The following two minimization problems will be considered: (i) Given f ∈ W 1,p (Rn ), define m1 (f ) = inf{T ∇f pp : T ∈ SL(n)}. Does there exist Q1 ∈ SL(n) such that Q1 ∇f pp = m1 (f )? (ii) Given f ∈ W 1,p (Rn ), define m2 (f ) = inf{|T |−1 : T ∇f pp = n and T ∈ GL(n)}. Does there exist Q2 ∈ GL(n) such that |Q2 |−1 = m2 (f )? Here |T | denotes the absolute value of the determinant of the matrix T . The extremal problems (i) and (ii) are actually equivalent. 1

Proposition 6.1. If Q1 is a solution of the problem (i), then Q2 = n p Q1 ∇f −1 p Q1 is a solution of the 1 problem (ii). Conversely, if Q2 is a solution of the problem (ii), then Q1 = |Q2 |− n Q2 is a solution of the problem (i). 1

p Proof. Suppose Q1 is a solution of (i). Clearly, the matrix Q2 = n p Q1 ∇f −1 p Q1 satisfies Q2 ∇f p = n. p Then for each T ∈ GL(n) such that T ∇f p = n, we have

|Q2 |−1 =

Q1 ∇f np |Q1 |−1 nn/p

= |Q1 |−1/n Q1 ∇f np n−n/p ≤ |T |−1/n T ∇f np n−n/p =

T ∇f np |T |−1 nn/p

= |T |−1 . Thus, Q2 is a solution of (ii). 1 Conversely, suppose Q2 is a solution of (ii), then the matrix Q1 = |Q2 |− n Q2 ∈ SL(n). Note that for each 1 p ¯ T ∈ SL(n), the matrix T¯ = n p T ∇f −1 p T satisfies T ∇f p = n. Since Q2 is a solution of (ii), then 1 n −1 |Q2 |−1 ≤ |T¯|−1 = |n p T ∇f −1 = n− p T ∇f np . p T|

Therefore,

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p

Q1 ∇f pp = |Q2 |− n Q2 ∇f pp p

= n|Q2 |− n ≤ T ∇f pp , which means Q1 is a solution of (i).

2

For T ∈ GL(n), there exist an orthogonal matrix O and a positive definite symmetric matrix A such that T = OA. Let S denote the class of positive definite symmetric n-by-n matrices. Then the problem (ii) can be rewritten as: given f ∈ W 1,p (Rn ), m2 (f ) = inf{|A|−1 : A∇f pp = n and A ∈ S} = inf{|A| : A−1 ∇f pp = n and A ∈ S}. p It follows from [28, Theorem 4.3] that there is a unique Ap ∈ S such that A−1 p ∇f p = n and |Ap | = m2 (f ). Thus,

Q2 = O2 A−1 p , for some O2 ∈ O(n). In [28, Definition 4.2], the (p, 2) Fisher information matrix Jp,2 (X) is defined as Jp,2 (X) = App . Hence Q2 = O2 Jp,2 (X)− p . 1

(6.1)

However, comparing the problem (i) and Theorem 1.1, we have Q1 = O1 Tpt ,

(6.2)

for some O1 ∈ O(n). Therefore, it follows from Proposition 6.1, (6.1) and (6.2) that t Jp,2 (X)− p = n p Tpt ∇f −1 p O3 T p ,

(6.3)

Tpt = |Jp,2 (X)| np O4 Jp,2 (X)− p ,

(6.4)

1

1

and 1

1

for some O3 , O4 ∈ O(n). Moreover, if we rewrite (6.3) and (6.4) with the relation Ap = Jp,2 (X)1/p , then t −1 t p A−1 p = n Tp ∇f p O3 Tp , 1

and Tpt = |Ap | n O4 A−1 p . 1

When p = 2, it is easy to verify that the identity (1.6):  T2t ∇f 22 In = n Rn

T2t ∇f (x) ⊗ T2t ∇f (x)dx,

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is equivalent to [28, (16)]:  In =

−1 A−1 2 ∇f (x) ⊗ A2 ∇f (x)dx.

Rn

Acknowledgment The authors are indebted to the referee for many valuable suggestions, especially for the last section. References [1] D. Alonso-Gutierrez, On a reverse Petty projection inequality for projections of convex bodies, Adv. Geom. 14 (2014) 215–223. [2] D. Alonso-Gutierrez, J. Bastero, J. Bernués, Factoring Sobolev inequalities through classes of functions, Proc. Amer. Math. Soc. 140 (2012) 3557–3566. [3] T. Aubin, Problèmes isopérimétriques et espaces de Sobolev, J. Differential Geom. 11 (1976) 573–598. [4] K. Ball, Volume ratios and a reverse isoperimetric inequality, J. London Math. Soc. 44 (1991) 351–359. [5] F. Barthe, On a reverse form of the Brascamp–Lieb inequality, Invent. Math. 134 (1998) 685–693. [6] W. Beckner, Sharp Sobolev inequalities on the sphere and the Moser–Trudinger inequality, Ann. of Math. 138 (1993) 13–242. [7] S. Bobkov, M. Ledoux, From Brunn–Minkowski to sharp Sobolev inequalities, Ann. Mat. Pura Appl. 187 (2008) 369–384. [8] A. Cianchi, E. Lutwak, D. Yang, G. Zhang, Affine Moser–Trudinger and Morrey–Sobolev inequalities, Calc. Var. Partial Differential Equations 36 (2009) 419–436. [9] D. Cordero-Erausquin, D. Nazaret, C. Villani, A mass-transportation approach to sharp Sobolev and Gagliardo–Nirenberg inequalities, Adv. Math. 182 (2004) 307–332. [10] M. Del Pino, J. Dolbeault, Best constants for Gagliardo–Nirenberg inequalities and applications to nonlinear diffusions, J. Math. Pures Appl. 81 (2002) 847–875. [11] H. Federer, Geometric Measure Theory, Springer, Berlin, 1969. [12] H. Federer, W. Fleming, Normal and integral currents, Ann. of Math. (2) 72 (1960) 458–520. [13] A. Giannopoulos, V.D. Milman, Extremal problems and isotropic positions of convex bodies, Israel J. Math. 117 (2000) 29–60. [14] A. Giannopoulos, M. Papadimitrakis, Isotropic surface area measures, Mathematika 46 (1999) 1–13. [15] A. Giannopoulos, V.D. Milman, M. Rudelson, Convex bodies with minimal mean width, in: Geom. Aspects of Funct. Analysis, in: Lecture Notes in Math., vol. 1745, 2000, pp. 81–93. [16] L. Guo, G. Leng, Volume inequalities for the Lp -sine transform of isotropic measures, Bull. Korean Math. Soc. 52 (2015) 837–849. [17] C. Haberl, F. Schuster, Asymmetric affine Lp Sobolev inequalities, J. Funct. Anal. 257 (2009) 641–658. [18] C. Haberl, F. Schuster, J. Xiao, An asymmetric affine Pólya–Szegö principle, Math. Ann. 352 (2012) 517–542. [19] J. Haddad, C.H. Jiménez, M. Montenegro, Sharp affine Sobolev type inequalities, arXiv:1505.07763v1. [20] M. Ludwig, J. Xiao, G. Zhang, Sharp convex Lorentz–Sobolev inequalities, Math. Ann. 350 (2011) 169–197. [21] E. Lutwak, The Brunn–Minkowski–Firey theory. I. Mixed volumes and the Minkowski problem, J. Differential Geom. 38 (1993) 131–150. [22] E. Lutwak, D. Yang, G. Zhang, Lp affine isoperimetric inequalities, J. Differential Geom. 56 (2000) 111–132. [23] E. Lutwak, D. Yang, G. Zhang, Sharp affine Lp Sobolev inequalities, J. Differential Geom. 62 (2002) 17–38. [24] E. Lutwak, D. Yang, G. Zhang, On the Lp -Minkowski problem, Trans. Amer. Math. Soc. 356 (2004) 4359–4370. [25] E. Lutwak, D. Yang, G. Zhang, Volume inequalities for subspaces of Lp , J. Differential Geom. 68 (2004) 159–184. [26] E. Lutwak, D. Yang, G. Zhang, Lp John ellipsoids, Proc. London Math. Soc. 90 (2005) 497–520. [27] E. Lutwak, D. Yang, G. Zhang, Optimal Sobolev norms and the Lp Minkowski problem, Int. Math. Res. Not. IMRN (2006) 62978. [28] E. Lutwak, S. Lv, D. Yang, G. Zhang, Extensions of Fisher information and Stam’s inequality, IEEE Trans. Inform. Theory 58 (2012) 1319–1327. [29] G. Maresch, F. Schuster, The sine transform of isotropic measures, Int. Math. Res. Not. IMRN (2012) (4) 717–739. [30] E. Markessinis, G. Paouris, C. Saroglou, Comparing the M-position with some classical positions of convex bodies, Math. Proc. Cambridge Philos. Soc. 152 (2012) 131–152. [31] V.G. Maz’ya, Classes of domains and imbedding theorems for function spaces, Soviet Math. Dokl. 1 (1960) 882–885. [32] V.G. Maz’ya, Conductor and capacitary inequalities for functions on topological spaces and their applications to Sobolevtype imbeddings, J. Funct. Anal. 224 (2005) 408–430. [33] C.M. Petty, Surface area of a convex body under affine transformations, Proc. Amer. Math. Soc. 12 (1961) 824–828. [34] R. Schneider, Convex Bodies: The Brunn–Minkowski Theory, second expanded edition, Encyclopedia Math. Appl., Cambridge University Press, Cambridge, 2014. [35] G. Talenti, Best constant in Sobolev inequality, Ann. Mat. Pura Appl. 110 (1976) 353–372. [36] T. Wang, The affine Sobolev–Zhang inequality on BV (Rn ), Adv. Math. 230 (2012) 2457–2473. [37] T. Wang, The affine Pólya–Szegö principle: equality cases and stability, J. Funct. Anal. 265 (2013) 1728–1748.

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T. Wang, On the discrete functional Lp Minkowski problem, Int. Math. Res. Not. IMRN (2015) rnu256. J. Xiao, The sharp Sobolev and isoperimetric inequalities split twice, Adv. Math. 211 (2007) 417–435. W. Yu, Isotropic p-surface area measure, Acta Math. Sci. Ser. A Chin. Ed. 31 (2011) 644–651 (in Chinese). J. Yuan, G. Leng, W. Cheung, Convex bodies with minimal p-mean width, Houston J. Math. 36 (2010) 499–511. G. Zhang, The affine Sobolev inequality, J. Differential Geom. 53 (1999) 183–202.