2. Basics of convex bodies

THE AFFINE SOBOLEV INEQUALITY

GAOYONG ZHANG

1. Introduction

The Sobolev inequality is one of the fundamental inequalities con- necting analysis and geometry. The literature related to it is vast (see, for example, 1], 5], 7], 3], 6], 11], 12], 19], 22], 20], 21], 23], 25], 27], 28], 37], and 45]). In this paper, a new inequality that is stronger than the Sobolev inequality is presented. A remarkable feature of the new inequality is that it is independent of the norm chosen for the ambient Euclidean space.

The Sobolev inequality in the Euclidean space ^Rn states that for any ^C1 function^f(^x) with compact support there is

(1^:1) ^Z

R n

jrfjdx n!

1=n

n ^kfkn;1n

where^jrfjis the Euclidean norm of the gradient of^f,^kfkp is the usual

Lp norm of ^f in ^Rn, and^!n is the volume enclosed by the unit sphere

Sn^;1 in ^Rn. The best constant in the inequality is attained at the characteristic functions of balls.

It is known that the sharp Sobolev inequality (1.1) is equivalent to the classical isoperimetric inequality (see, for instance, 2], 8], 13], 14], 33], 35], 40], and 41]). We prove an ane Sobolev inequality which is stronger than (1.1). This inequality is proved by using a generalization of the Petty projection inequality to compact sets that is established in this paper (see 30], 31], 16], 26], 38] and 42] for the classical Petty projection inequality of convex bodies).

Received February 2, 2000. Research supported, in part, by NSF Grant DMS- 9803261

183

Theorem 1.1.

(1^:2) 1

n Z

S^{n;1k r}u^fk;n

1

ducn^kfk;n

n

n;1

where^ru^f is the partial derivative of^f in direction^u,^duis the standard surface measure on the unit sphere, and the constant ^cn =^;2_!^!n;1ⁿ

n is best.

The inequality (1.2) is^GL(ⁿ) invariant while the inequality (1.1) is only ^SO(ⁿ) invariant. Thus, inequality (1.2) does not depend on the Euclidean norm of ^Rn. The best constant in (1.2) is attained at the characteristic functions of ellipsoids. Applying the Holder inequality and Fubini's theorem to the left-hand side of (1.2), one can easily see that inequality (1.2) is stronger than the Sobolev inequality (1.1). For radial functions, the inequality (1.2) reduces to (1.1). It is worth noting that the left-hand side of (1.2) is a natural geometric invariant. Specif- ically, for a ^C1 function^f(^x), there is an important norm of ^Rn given by

kuk=^kru^fk1 =

Z

R n

jhurf(^x)^{ijdx u2Rn}

where ^{h i} is the usual inner product in ^Rn. The volume of the unit ball of this norm is exactly the left-hand side of (1.2).

We will also prove a generalization of the Gagliardo-Nirenberg in- equality.

Theorem 1.2.

1 be a sequence of unit vectors in ^Rn and let ^fi^gm

1 be a sequence of positive numbers satisfying

jxj

2 =^Xm

i⁼¹

i^hxui^{i2 x2R}n:

If ^f is a ^C1 function with compact support in ^Rn, then

m

Y

i⁼¹

k ru^ifk1ni 2^kfkn;1ⁿ :

2. Basics of convex bodies

A convex body is a compact convex set with nonempty interior in

Rn.

A convex body ^K is uniquely determined by its support function dened by

hK(^u) = max_x

2K^{huxi u2S}n^;1:

If^K contains the origin in its interior, the polar body^K of^K is given by

K

=^fx2Rn :^hxyi1 for all^y2Kg:

Denote by ^V(^K) the volume of ^K. The mixed volume ^V(^{K L}) of convex bodies^K and ^L is dened by

V(^{K L}) = 1

n

"lim^!0+

V(^K+^"L)^;V(^K)

"

:

There is a unique nite measure ^SK on ^Sn;1 so that

V(^{K L}) = 1

n Z

S^n;1hL(^u)^dSK(^u)^:

The measure ^SK is called the surface area measure of ^K. When ^K has a ^C2 boundary ^@K with positive curvature, the Radon-Nykodim derivative of ^SK with respect to the Lebesgue measure on^Sn;1 is the reciprocal of the Gauss curvature of ^@K.

An important inequality of mixed volume is the Minkowski inequal- ity,

(2^:1) ^V(^{K L})^{n V}(^K)^n;1V(^L) with equality if and only if ^K and ^L are homothetic.

For a convex body ^K, let ^Kju? be the projection of ^K onto the 1-codimensional subspace^u? orthogonal to ^u. The projection function v(^{K u}) of ^K is dened by

v(^{K u}) = voln^;1(^Kju?) ^u2Sn;1:

The projection function v(^{K u}) of^K denes a new convex body ^K whose support function is given by

h

K(^u) = v(^{K u}) = voln^;1(^Kju?) ^u2Sn;1:

The convex body ^K is called the projection body of ^K. The volume of the polar of the projection body ^K is given by

V(^K) = 1

n Z

S^n;1v(^{K u})^;ndu:

The Petty projection inequality is (see 30] and 36]) (2^:2) ^V(^K)^n;1V(^K)

!n

!n^;1

n

with equality if and only if^K is an ellipsoid.

It was shown in 32] that the Petty projection inequality is stronger than the classical isoperimetric inequality of convex bodies.

Let^u1u2:::unbe an orthonormal basis of^Rn. For a convex body

K in^Rn, the Loomis-Whitney inequality is (see 29] and 8, p. 95])

V(^K)^n;1Yn

i⁼¹v(^{K u}i)^:

The Loomis-Whitney inequality was generalized by Ball 4]. Let

fui^gm

1 be a sequence of unit vectors in^Rn, and let^fci^gm

1 be a sequence of positive numbers for which

m

X

i⁼¹

ci^ui^ui =^In

where ^ui^ui is the rank-1 orthogonal projection onto the span of ^ui, and ^In is the identity on ^Rn. Then, for a convex body ^K in ^Rn, Ball proved the inequality

(2^:3) ^V(^K)^{n;1 Ym}

i⁼¹v(^{K u}i)^ci: The condition on^ui and ^ci is equivalent to

jxj

2 =^Xm

i⁼¹

ci^hxui^{i2 x2R}n

:

For details of convex bodies, see 16], 38] and 42].

3. Inequalities for compact domains

In this section, we generalize inequalities (2.1){(2.3) to compact do- mains. In this paper, a compact domain is the closure of a bounded open set. The generalization of the Minkowski inequality to compact

domains can be obtained from the Brunn-Minkowski inequality. This appears to be standard. See 10] and 8].

If^M and^N are compact domains in^Rn, then the Brunn-Minkowski inequality is

(3^:1) ^V(^M +^N)^{n1 V}(^M)¹ⁿ +^V(^N)¹ⁿ with equality if and only if ^M and ^N are homothetic.

Let^M be a compact domain with piecewise^C1 boundary^@M, and let^Kbe a convex body in^Rn. The mixed volume of^Mand^K,^V(^MK), is dened by

(3^:2) ^V(^MK) = 1

n Z

@M^hK((^x))^dSM(^x)

where ^dSM is the surface area element of ^@M, and(^x) is the exterior unit normal vector of ^@M at ^x.

If ^K is the unit ball ^Bn in^Rn, then ^nV(^MBn) is the surface area

S(^M) of ^M.

Lemma 3.1.

@M, and^K is a convex body in ^Rn, then (3^:3) ^nV(^MK) = lim_"

!0 +

V(^M+^"K)^;V(^M)

"

:

When^M is not convex, the limit of the right-hand side of (3.3) may not exist. Equation (3.3) holds when^Mis a convex body or is a compact domain with piecewise ^C1 boundary. We give a proof of Lemma 3.1 in the Appendix.

Lemma 3.2.

(3^:4) ^V(^MK)^{n V}(^M)^n;1V(^K) with equality if and only if ^M and ^K are homothetic.

Proof. For ^" 0, consider the function

f(^") =^V(^M +^"K)^{n1 ;V}(^M)^{1n ;"V}(^K)^n1:

From the Brunn-Minkowski inequality (3.1), the function^f(^") is non- negative and concave. By Lemma 3.1, we have

"lim^!0+

f(^")^;f(0)

"

=^V(^M)^{1;nn V}(^MK)^;V(^K)ⁿ¹ 0^:

This proves the inequality (3.4). If the equality holds, ^f(^") must be linear, and^M and ^K are homothetic. q.e.d.

Lemma 3.2 was proved in 10] when^M has a ^C11 boundary.

Let^M be a compact domain in^Rn with piecewise^C1 boundary^@M and exterior unit normal vector(^x). For any continuous function^f on

Sn^;1, dene a linear functionalM on the space of continuous functions

C(^Sn;1) on ^Sn;1 by

(3^:5) M(^f) =

Z

@M^f((^x))^dSM(^x)

where ^dSM is the surface area element of ^M. The linear functional

M is a non-negative linear functional on ^C(^Sn;1). Since the sphere is compact,M is a nite measure on^Sn;1. The measureM is called the surface area measureof the compact domain^M.

The Minkowski existence theorem states that for every nite non- negative measureon ^Sn;1 such that

(3^:6)

Z

S^n;1ud(^u) = 0 and

Z

S^n;1jhuvijd(^v)^>0 ^u2Sn;1 there exists a unique convex body^K (up to translation) whose surface area measure is. See 38], pp. 389-393.

We verify that the surface area measure M of a compact domain

M dened above satises (3.6). By Green's formula, for any^C1 vector eld (^x) in^Rn, there is

Z

@M^h (^x)(^x)^idSM(^x) =

Z

Mdiv (^x)^dx:

Choose (^x) =^ei,ⁱ= 12ⁿ, the coordinate vectors, then div = div^ei =0. Therefore, Green's formula yields

Z

@M^hei(^x)^idSM(^x) = 0^:

Let ^f(^u) =^hei^ui. Then (3.5) gives

Z

S^n;1hei^uidM(^u) =

Z

@M^hei(^x)^idSM(^x) = 0^: Since ^M has non-empty interior, one has

Z

@M^jhu(^x)^ijdSM(^x)^>0

that is, ^Z

S^n;1jhuvijdM(^v)^>0^: Hence,M satises the condition (3.6).

Let^M be a compact domain in^Rnwith piecewise^C1 boundary^@M. A convexication ^M of^M is a convex body whose surface area measure

S

M is dened by

(3^:7) ^S_M =M^:

Note that a convex body is determined by its surface area measure only up to translation. Therefore, the convexication ^M is unique up to translation.

Let^M be a compact domain in^Rnwith piecewise^C1 boundary^@M. The projection function v(^Mu) of ^M on ^Sn;1 is dened by

v(^Mu) = 12

Z

@M^jhu(^x)^ijdSM(^x)

= 12

Z

S^n;1jhuvijdM(^v) ^u2Sn;1 where (^x) is the exterior unit normal vector of^M at ^x.

The following lemma is obvious.

Lemma 3.3.

v(^Mu) = v( ^Mu) ^u2Sn;1

V(^MK) =^V( ^MK) for any convex body ^K in ^Rn.

Lemma 3.4.

(3^:8) ^V( ^M) ^V(^M)

with equality if and only if^M is convex.

Proof. From Lemma 3.3, for any convex body^K, we have

V(^MK) =^V( ^MK)^: Let^K= ^M. Then

V(^MM) =^V( ^M)

and the generalized Minkowski inequality (3.4) yields

V( ^M) =^V(^MM) ^V(^M)^{n;1n V}( ^M)^1n: This proves (3.8). q.e.d.

The convexication in ^Rn was introduced in 9]. Lemma 3.4 for polytopes was proved in 9] see also 43].

Let^M be a compact domain in^Rn with piecewise^C1boundary^@M. The projection body ^M of ^M is dened by

h

M(^u) = v(^Mu) ^u2Sn;1:

It is easily seen that ^M is an origin-symmetric convex body. Let

`u be a line parallel to the unit vector <sup>u</sup>, and let <sup>d`</sup>u be the volume element of the subspace^u? orthogonal to^u. Then

h

M(^u) = 12

Z

#(^{M\`</sup>u)<sup>d`}u^:

This is the projection that counts (geometric) multiplicity. For the projection bodies of more general compact sets, see 39].

The volume of the polar projection body ^M is

V(^M) = 1

n Z

S^n;1v(^Mu)^;ndu:

Note that the arithmetic average of the projection function v(^Mu) over ^Sn;1 is the surface area of ^M, up to a constant factor. One can view the^SL(ⁿ)-invariant^V(^M)^;1=n as an ane surface area of^M.

Lemma 3.5.

(3^:9) ^V(^M)^n;1V(^M)

!n

!n^;1

n

with equality if and only if ^M is an ellipsoid.

Proof. By (3.8), Lemma 3.3, and the Petty projection inequality (2.2) for convex bodies, we have

V(^M)^n;1V(^M)^V( ^M)^n;1V(^M)

!n

!n^;1

n

with equalities if and only if ^M is convex and ^M is an ellipsoid, and hence ^M is an ellipsoid. q.e.d.

As noted earlier for convex bodies, the generalized Petty projection inequality (3.9) is stronger than the classical isoperimetric inequality for compact domains. From the Holder inequality, one can easily see

S(^M) ^n!1+

1

n n

!n^{;1 V}(^M)^;n1

which and (3.9) imply the classical isoperimetric inequality

S(^M) ^n!_n^1=nV(^M)^{n;1n :}

Lemma 3.6.

1 be a sequence of non-zero vectors in ^Rn which are not contained in one hyperplane. Then for any compact do- main ^M in ^Rn

(3^:10) ^Ym

i⁼¹v(^Mwi)^{i cV}(^M)^n;1 where i = ^hA;1wi^wiⁱ, ^c = (det^A)^12=Qm_i^=1pjwij

i ⁱ

, and ^A is the positive denite matrix given by ^hAxxi=^Pm_i^=1hxwi^i2:

Proof. First, we show the case that ^M is a convex body^K. Let ^Q be a non-singular matrix so that ^A=^QTQ, and let^y=^Qx. Then

jyj

2 =^hAxxi=^Xm

i⁼¹

hwi^xi2 =^Xm

i⁼¹

i^hui^yi2

where^ui =^;_i ^12Q;Twi.

It can be easily veried that

v(^{QTK w}i)^jwi^j= det(^Q)v(^{K Q;Tw}i)^jQ;Twi^j:

From (2.3), we have

V(^K)^{n;1 Ym}

i⁼¹v(^{K u}i)ⁱ =^Ym

i⁼¹v(^{K Q;Tw}i)ⁱ

=^Ym

i⁼¹v(^{QTK w}i)ⁱ

jwi^j

jQ

;Twi^jdet^Q

ⁱ

=^Ym

i⁼¹v(^{QTK w}i)ⁱ

jwi^j

p

idet^Q

ⁱ

:

Using the fact that^V(^QTK) =^V(^K)det^Qand ^Pm_i⁼¹i =ⁿwe obtain the inequality (3.10) for convex ^M.

When^M is not convex, let ^M be a convexication of^M. By Lemma 3.3, ^M and ^M have the same projection function. From (3.8) and the convex case it follows that

cV(^M)^n;1cV( ^M)^n;1Ym

i⁼¹v( ^Mwi)ⁱ

=^Ym

i⁼¹v(^Mwi)^i: q.e.d.

4. The ane Sobolev inequality

In this section , we prove the results stated in the Introduction.

Theorem 4.1.

(4^:1) 1

n Z

S^n;1kru^fk;n

1

du

;

!n

2^!n^;1

n

kfk

;n

n

n;1 :

Proof. For^t>0, consider the level sets of ^f in^Rn ,

Mt=^{fx 2Rn} :^jf(^x)^j>tg

St=^{fx 2Rn} :^jf(^x)^j=^tg:

Since^f is of class^C1, for almost all^t>0,^St is a^C1 submanifold which has non-zero normal vector ^rf. Let^dSt be the surface area element of

St. Then one has the formula of volume elements, (4^:2) ^dx=^jrfj;1dSt^dt:

We have

(4^:3)

k ru^fk1=

Z

R n

jru^f(^x)^jdx

=

Z

1

0 Z

S^tjhrfuijjrfj

;1

dSt^dt

= 2

Z

1

0

v(^Mt^u)^dt:

On the other hand,

(4^:4)

Z

R n

jfj n

n;1

dx=

Z

R n

Z

jf^j

0 n

n;1^t

1

n;1

dt

!

dx

= ⁿ

n;1

Z

1

0 t

1

n;1 Z

M^tdx

dt

= ⁿ

n;1

Z

1

0 t

1

n;1

V(^Mt)^dt:

Since^V(^Mt) is decreasing with respect to ^t, it follows that

t 1

n;1

V(^Mt) =^tV(^Mt)^{n;1n n;11 V}(^Mt)^n;1n

Z t

0

V(^M)^{n;1n d}

1

n;1

V(^Mt)^n;1n

= ^n;1

n d

dt Z t

0

V(^M)^{n;1n d}

n

n;1

so that (4^:5)

Z

1

0 t

1

n;1

V(^Mt)^{dt n;}1

n Z

1

0

V(^Mt)^{n;1n dt}

n

n;1

:

Combining (4.4) and (4.5) gives (4^:6)

Z

R n

jfj n

n;1

dx Z

1

0

V(^Mt)^{n;1n dt}

n

n;1

:

By the generalized Petty projection inequality (3.9), we obtain

(4^:7)

V(^Mt)^{n;1n !n}

!n^;1V(^Mt)^;1n

= ^!n

!n^;1

1

n Z

S^n;1v(^Mt^u)^;ndu

; 1

n

:

From (4.6) and (4.7), it follows that

kfk n

n;1

!n

!n^;1

Z

1

0 1

n Z

S^n;1v(^Mt^u)^;ndu

; 1

n

dt:

Thus Minkowski's inequality for integrals yields

Z

1

0 Z

S^n;1v(^Mt^u)^;ndu

; 1

n

dt

Z

S^n;1

Z

1

0

v(^Mt^u)^dt

;n

du

!

; 1

n

:

By (4.3) and the last two inequalities, we nally obtain

kfk n

n;1

!n

2^!n^;1

1

n Z

S^{n;1k r}u^fk;n

1 du

; 1

n

which proves the Theorem. q.e.d.

We observe that inequality (4.1) is stronger than the Sobolev in- equality (1.1). Indeed, the Holder inequality and Fubini's theorem to- gether yield

(4^:8)

1

n!n

Z

S^n;1kru^fk;n

1 du

; 1

n

1

n!n

Z

S^{n;1k r}u^fk1du

= 1

n!n

Z

S^n;1

Z

R n

jhrfuijdxdu

= 1

n!n

Z

R n

Z

S^{n;1jhrfuijdudx}

= 2^!n;1

n!n

Z

R n

jrfjdx:

Thus inequalities (4.1) and (4.8) are combined to give the Sobolev in- equality (1.1).

Let us show that the generalized Petty projection inequality (3.9) can be proved by using the inequality (4.1).

For compact domain^M and for small^">0, we dene

f"(^x) =

(0 dist(^xM) ^"

1^;dist(_"^xM) dist(^xM)^<":

If ^"is small and dist(^xM) ^<", then there exists a unique^{x0 2@M} so

that dist(^xM) =^jx0;xj:

Let

(^x0) = ^x0;x

jx 0

;xj :

Consider

M" =^fx2Rn : 0^<dist(^xM)^<"g and its closure ^M". One has

rf"(^x) =

(

"

;1

(^x0) ^x2M"

0 ^x2=M_"^: It follows that

Z

R n

jhrf"^uijdx=^";1

Z

M^"jh(^x0)^uijdx:

Let^t= dist(^xM), 0^<t<". Then ^dx=^dSM^dt+^o(^t)^:Therefore, as

"! 0, we have

"

;1 Z

M^"jh(^x0)^uijdx;!

Z

@M^jh(^x0)^uijdSM(^x0) = 2v(^Mu)^: On the other hand, since^f" converges to the characteristic function

M of ^M, we have

Z

R n

jf"^{jn+1n dx;!V}(^M)^: It follows that (4.1) implies (3.9).

We have seen that the ane Sobolev inequality (4.1) is equivalent to the generalized Petty projection inequality (3.9). The constant in the inequality (4.1) is sharp. It is attained at the characteristic functions of ellipsoids.

5. A generalization of the Gagliardo-Nirenberg inequality

Let^f be a^C1 function with compact support in^Rn. Gagliardo 15]

and Nirenberg 34] proved the inequality

(5^:1) ^Yn

i⁼¹

k rx^ifk11n 2^kfkn;1ⁿ :

This inequality implies the Sobolev embedding theorem. See 1, p. 38].

We give a generalization of inequality (5.1) which is equivalent to the inequality (3.10) for compact domains.

Theorem 5.1.

1 be a sequence of vectors in ^Rn not con- tained in one hyperplane. If^f(^x) is a^C1 function with compact support in ^Rn, then

(5^:2) ^Ym

i⁼¹

krw^{ifk1ni ckfk}n;1ⁿ

where the constants i = ^hA;1wi^wiⁱ and ^c = 2det^AQm_i⁼¹_iⁱ²ⁿ¹ depend only on the sequence of vectors, and ^A is the positive denite matrix given by^hAxxi=^Pm_i^=1hxwi^i2:

Proof. We use the notation in Theorem 4.1. From (4.6), we get

kfk n

n;1

Z

1

0

V(^Mt)^{n;1n dt:}

Using (3.10), ^Pm_i⁼¹i=ⁿ, and the Holder inequality, we have

c 1

n Z

1

0

V(^Mt)^{n;1n dt}

Z

1

0

m

Y

i⁼¹v(^Mt^wi)^nidt

m

Y

i⁼¹

Z

1

0

v(^Mt^wi)^dt

i

n

where the constant^c is from (3.10). Thus

(5^:3) ^cn1kfkn;1ⁿ

m

Y

i⁼¹

Z

1

0

v(^Mt^ui)^dt

c

i

n

:

Similar to (4.3), one has

krw^ifk1 = 2^jwi^j

Z

1

0

v(^Mt^wi)^dt:

From this and (5.3), inequality (5.2) follows. q.e.d.

Corollary 5.2.

1 be a sequence of unit vectors in ^Rn and let ^fi^gm

1 be a sequence of positive numbers satisfying

jxj

2 =^Xm

i⁼¹

i^hxui^{i2 x2R}n:

If ^f is a ^C1 function with compact support in ^Rn, then

m

Y

i⁼¹

kru^ifk1ni 2^kfkn;1ⁿ :

Similar to the equivalence of (3.9) and (4.1), the geometric inequality (3.10) is equivalent to the analytic inequality (5.2). A similar argument can also be carried out. The constant of the inequality (5.2) is best it is attained at the characteristic functions of parallelepipeds.

6. Appendix

Proof of Lemma 3.1. Let ^k be a positive integer. Since ^@M is compact and of class ^C1 piecewise, one can choose ^>0 such that

jhx;x 0

(^x0)^{ijk;1jx;x0j j}(^x)^;(^x0)^jk;1

jhK((^x))^;hK((^x0))^{j<k;1 xx0 2@M jx;x0j<:}

For^{x 2@M}, consider a point ^{y 2=M} but^y2x+^"K. We estimate the distance of ^y to ^@M. Let ^{x0 2@M} be the point which attains the distance. Then

jy;x 0

j=^jhy;x0(^x0)^ij=^jhy;x(^x0)ⁱ+^hx;x0(^x0)^ij

=^jhy;x(^x)ⁱ+^hy;x(^x0)^;(^x)ⁱ+^hx;x0(^x0)^ij (6.1)

jhy;x(^x)^ij+^jy;xjj(^x0)^;(^x)^j+^jhx;x0(^x0)^ij: