ON SHARP HIGHER ORDER SOBOLEV EMBEDDINGS

May 18, 2004 16:50 WSPC/152-CCM 00138

Communications in Contemporary MathematicsVol. 6, No. 3 (2004) 495–511c© World Scientific Publishing Company

ON SHARP HIGHER ORDER SOBOLEV EMBEDDINGS

MARIO MILMAN

Department of Mathematics,

Florida Atlantic University,

Boca Raton, FL 33431, USA

[email protected]

EVGENIY PUSTYLNIK

Department of Mathematics,

Technion Israel Institute of Technology,

Haifa 32000, Israel

[email protected]

Received 30 December 2002Revised 5 August 2003

Let Ω be an open domain in Rn, let k ∈ N, p ≤ nk. Using a natural extension of

the L(p, q) spaces and a new form of the Polya–Szego symmetrization principle, we

extend the sharp version of the Sobolev embedding theorem: Wk,p0 (Ω) ⊂ L( np

n−kp, p)

to the limiting value p = nk. This result extends a recent result in [3] for the case

k = 1. More generally, if Y is a r.i. space satisfying some mild conditions, it is shownthat W

k,Y0 (Ω) ⊂ Yn(∞, k) = f : t−k/n(f∗∗(t) − f∗(t)) ∈ Y . Moreover Yn(∞, k) is

not larger (and in many cases essentially smaller) than any r.i. space X(Ω) such that

Wk,Y0 (Ω) ⊂ X(Ω). This result extends, complements, simplifies and sharpens recent

results in [10].

Keywords: Embedding; Sobolev; rearrangement.

Mathematics Subject Classification 2000: 46E30, 26D10

1. Introduction

Let W k,p0 (Ω) denote the usual Sobolev spaces of functions f defined on an open

domain Ω ⊂ Rn, such that f and all its distributional derivatives Dαf , |α| ≤ k, are

zero at ∂Ω, and, moreover, such that |Dαf | ∈ Lp(Ω), |α| = k. The classical Sobolev

embedding theorem asserts that if Ω an open domain in Rn then,

W k,p0 (Ω) ⊂ Lq(Ω),

1

q=

1

p−

k

n, 1 < p <

n

k. (1.1)

The norms of the embedding blow up as p → nk . In fact, if p = n

k , we formally have

q = ∞ and (1.1) is false. Thus, in the limiting case, it is necessary to go outside the

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May 18, 2004 16:50 WSPC/152-CCM 00138

496 M. Milman & E. Pustylnik

Lq scale to find the correct target spaces. Indeed, it was showna by Trudinger [25]

for k = 1, and Strichartz [23], for k > 1 (cf. also [9] for further historical references

and background), that, if |Ω| < ∞, we have

Wk, n

k

0 (Ω) ⊂ LΦ nk

(Ω) , (1.2)

where Φnk(x) is a Young’s function such that Φ n

k(x) ≈ ex

nn−k

for large x.

It is also known that neither (1.1) nor (1.2) are sharp. The sharp form of (1.1) is

provided by the O’Neil–Stein version of the Sobolev embedding theorem (cf. [19]),

which requires the use of the L(p, q) spaces:

W k,p0 (Ω) ⊂ L(q, p)(Ω) ,

1

q=

1

p−

k

n, 1 < p <

n

k. (1.3)

Again (1.3) fails when p = nk . Motivated in part by (1.3), Hansson [12], and Brezis–

Wainger [6] improved on (1.2) and obtained in the limiting case p = nk ,

Wk, n

k

0 (Ω) ⊂ Hnk(Ω) , (1.4)

where, for q > 1, Hq(Ω) is the Lorentz space defined by

Hq(Ω) =

f : ‖f‖Hq(Ω) =

∫ |Ω|

0

(

f∗∗(s)

1 + ln |Ω|s

)qds

s

1/q

< ∞

.

The result is optimal among all r.i. spaces; this was proved in [12] for Riesz poten-

tials and in [9] for Sobolev spaces themselves.

In particular, in [9] it is shown that, for any r.i. space X(Ω),

Wk, n

k

0 (Ω) ⊂ X(Ω) ⇒ H nk(Ω) ⊂ X(Ω) . (1.5)

It is customary to treat (1.3) and (1.4) (or (1.2)) as separate problems with

their corresponding separate proofs.b In this paper we shall show, extending the

methods developed in [3] for the case k = 1, a unified method to prove the Sobolev

embedding theorem and the corresponding sharp borderline cases. In fact, for the

borderline cases, we actually improve on the classical results since our target spaces

are rearrangement invariant sets which are strictly contained in the optimal spaces

described above.

The two main ingredients of our method are: (a) the use of a very natural

extension of the L(p, q) spaces recently proposed in [3],c and (b) the use of a new

version of the Polya–Szego symmetrization principle that is valid for higher order

derivatives. The new method can be easily generalized to give sharp results in the

context of Sobolev spaces based on general rearrangement invariant spaces.

aWe have learned that (1.2) was also obtained by Yudovich [26]. In connection with (1.4) weshould also mention their connection with the capacitary inequalities of [18] (cf. also [17]).bThe exponential estimates (1.2) though are usually derived as a consequence of a quantified formof (1.3) by extrapolation (cf. [23]).cIn particular, this allows us to treat the case p = n

k, q = ∞ in a unified form with the other cases.

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On Sharp Higher Order Sobolev Embeddings 497

To better explain the ingredients of the proof we start by recalling the L(∞, q)

spaces recently introduced in [3] in connection with the case k = 1 of (1.3). We

observe that if we formally let p → nk in (1.3), and disregard the blow up constants,

one is naturally led to consider the space L(∞, nk ). However, it is well known, and

easy to see, that with the usual definition L(∞, q) = 0. On the other hand,

imitating Bennett–DeVore–Sharpley [4],d we modify the definition of the L(p, q)

spaces as follows. Let 1 < p ≤ ∞, 0 < q ≤ ∞, let Ω be a domain in Rn, and let

M(Ω) be the set of measurable functions on Ω. We let

L(p, q)(Ω) = f ∈ M(Ω) : ‖f‖L(p,q) < ∞ ,

with

‖f‖L(p,q) =

∫ ∞

0

[(f∗∗(t) − f∗(t))t1/p]qdt

t

1/q

, 1 < p ≤ ∞, 0 < q < ∞ ,

we use the usual modifications when q = ∞.

These spaces make sense (and are not trivial) for 1 < p ≤ ∞, 0 < q ≤ ∞. More-

over, the new spaces actually coincide with the classical L(p, q) spacese whenever

1 < p < ∞, 0 < q ≤ ∞. In fact for functions in the classical L(p, q) spaces we havef

∫ ∞

0

[(f∗∗(t) − f∗(t))t1/p]qdt

t

1/q

≈

∫ ∞

0

[f∗∗(t)t1/p]qdt

t

1/q

, 1 < p < ∞, 0 < q ≤ ∞ .

The second idea underlying our method is a suitable extension of the Polya–

Szego symmetrization principle for higher order derivatives. It is easy to see that the

standard form of this principle, comparing the first derivatives of a function f(x)

and its non increasing rearrangement f ∗(t), cannot be generalized, even to second

derivatives, because there are infinitely many smooth functions f(x) such that df∗

dt

is not differentiable, even in the sense of distributions. For example, the function

f(x) = 1 + sin x, 0 < x < 3π2 , has rearrangement f∗(t) = (1 + cos t

2 )χ(0,π)(t) + (1 +

sin t)χ[π, 3π2

)(t), thus df∗

dt has a “jump” at the point t = π (a more detailed discussion

devoted to this topic can be found in [7]). Nevertheless, we shall show that a suitable

modification of Polya–Szego holds for higher order derivatives. To state our result,

we recall an inequality for radial spherically decreasing rearrangements from [3] and

[2]. Suppose that f is a smooth function such that f and |∇f | vanish at infinity

and let f denote its radial spherically decreasing symmetric rearrangement. Then,

there exists a universal constant γn such that

t−1/n(f∗∗(t) − f∗(t)) ≤ γn|∇f|∗∗(t) . (1.6)

dIn their pionering paper, these authors defined the L(∞,∞) spaces.eThis justifies using the same notation, L(p, q), for the new spaces.f In what follows, we use the symbol ≈ to denote equivalence modulo constants, and the symbol. to indicate smaller or equal modulo constants.

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This leads to

‖t−1/n(f∗∗(t) − f∗(t))‖Y . ‖∇f‖Y , ∀ f ∈ C∞0 (Rn) , (1.7)

for any r.i. space Y satisfying the (P ) condition (cf. Definition 2.1 below). We will

showg (cf. Theorem 3.4 below) that, under mild conditions on a r.i. norm Y (Ω),

1 ≤ k < n,

‖t−k/n(f∗∗(t) − f∗(t))‖Y (Ω) . ‖∇kf‖Y (Ω), ∀ f ∈ C∞0 (Ω) . (1.8)

We prove that conditions of this type are optimal by means of reformulating a

necessary condition for Sobolev embeddings derived in [10] (cf. Theorem 3.6 below).

More precisely we show that if X(Ω) is a r.i. space such that

‖f‖X(Ω) . ‖∇kf‖Y (Ω)

and |Ω| < ∞, then

‖f∗‖X(Ω) ≤ c(Ω)‖t−k/n(f∗∗(t) − f∗(t))‖Y . (1.9)

We are now ready to give our proof of the following sharp form of the Sobolev

embedding theorem.h

Theorem 1.1. Let Ω be an open domain in Rn, let 1 < p ≤ nk , 1

q = 1p − k

n , with

the convention that q = ∞ if 1q = 0 (i.e. q = ∞ when p = k

n ). Then

W k,p0 (Ω) ⊂ L(q, p)(Ω) , (1.10)

with

‖f‖L(q,p) ≤ c‖∇kf‖Lp, ∀ f ∈ C∞0 (Ω) . (1.11)

Moreover, the result is optimal: if |Ω| < ∞, then for any r.i. space X(Ω), 1 < p ≤ nk ,

W k,p0 (Ω) ⊂ X(Ω) ⇒ L

(

np

n − kp, p

)

⊂ X(Ω) .

Proof. As a consequence of Example 4.1 below it follows that (1.8) holds with

Y = Lp, 1 < p ≤ nk . Therefore (1.10) and (1.11) follow from the following trivial

computation with indices,

L

(

np

n − kp, p

)

= f : (f∗∗(t) − f∗(t))t−k/n ∈ Lp , 1 < p ≤n

k,

with

‖f‖L( npn−kp

,p) = ‖(f∗∗(t) − f∗(t))t−k/n‖Lp .

gThe reason we are able to obtain a higher order result seems to be connected with the formulationof (1.6) and (1.7), which involves the quantity t−1/n(f∗∗(t) − f∗(t)) = −t1−1/n(f∗∗(t))′ . In theliterature, authors (cf. in particular Talenti [24] or Kolyada [15]) often formulate the first orderestimates in terms of −t1−1/n(f∗(t))′ , which leads to an estimate that is more difficult to iterate.hWe should note here the contribution of Maly–Pick [16], see Sec. 4.1 below.

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That this condition is optimal now follows directly from the previous calculation

and (1.9).

To extend Theorem 1.1 to more general r.i. spaces, it is therefore natural to

turn the left hand side of (1.8) into a definition. Given a r.i. space Y (Ω), let

Yn(∞, k)(Ω) = f : t−k/n(f∗∗(t) − f∗(t)) ∈ Y (Ω) ,

(1.12)‖f‖Yn(∞,k) = ‖t−k/n(f∗∗(t) − f∗(t))‖Y .

The preceding discussion leads to the following generalization of Theorem 1.1.

Theorem 1.2. Let Ω be an open domain in Rn, suppose that k < n. Let Y (Ω) be

a r.i. space satisfying the conditions Q(k − 1) (cf. Definition 2.2 below) and (2.1)

below. Then

W k,Y0 (Ω) ⊂ Yn(∞, k) , (1.13)

and

‖f‖Yn(∞,k) . ‖∇kf‖Y (Ω), ∀ f ∈ C∞0 (Ω) .

Moreover, if |Ω| < ∞, and X(Ω) is a r.i. space then

W k,Y0 (Ω) ⊂ X(Ω) ⇒ Yn(∞, k) ⊂ X(Ω) ,

and

‖f∗∗‖X . ‖f‖Yn(∞,k) .

Proof. If Y is a r.i. space satisfying the assumptions of the theorem then, by

Lemma 2.3 and Theorem 3.4 below, (1.8) holds. This is all we need to obtain

(1.13). The fact that the Yn(∞, k) spaces are optimal follows once again from (1.9).

The last theorem obtains a particularly simple form if the space Yn(∞, k) itself

is a r.i. space. For example, if Y satisfies the Q(k)-condition (see Definition 2.2

below), it follows by Lemma 2.6 that

‖f‖Yn(∞,k) ≈ ‖t−k/nf∗∗(t)‖Y . (1.14)

Corollary 1.3. Let Ω be an open domain in Rn, let Y (Ω) be a r.i. space satisfying

the Q(k)-condition. Then Yn(∞, k) is a r.i. space with norm provided by (1.14) and

moreover,

‖t−k/nf∗(t)‖Y . ‖f‖Yn(∞,k) . ‖f‖W k,Y0

(Ω) .

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Yn(∞, k) is the optimal target space for the Sobolev embedding W k,Y0 (Ω) ⊂

X(Ω) among all r.i. spaces. The quasi-normed space defined by the quasi-norm

‖t−k/nf∗(t)‖Y is the optimal target space among the class of quasi-normed

r.i. spaces.

Remark 1.4. The optimal target spaces for embeddings of generalized Sobolev

spaces are described in the recent paper [10]. The description obtained in [10] is

indirect and does not imply the previous Corollary. A version of our Corollary 1.3

(with stronger assumptions on Y and without a study of optimal conditions) was

claimed much earlier in [14]. Unfortunately, the proof indicated there is based on

two incorrect arguments (a higher order Polya–Szego principle as a mere iteration

of first order results, and a theorem on interpolation of r.i. spaces that was later

shown to be false).

The rest of the paper is organized as follows: in Sec. 2 we establish some basic

facts about r.i. spaces that are used in Sec. 3 to establish the main symmetrization

estimates to which we have referred in the course of the proof of the main theorems.

In the final Sec. 4 we collect a few remarks and open problems.

2. Preliminaries

Let Ω ⊂ Rn be a domain, and let Y = Y (Ω) be a rearrangement invariant space

(r.i. space). Let k ∈ N and define the Sobolev spaces

W k,Y0 (Ω) = f : |Dαf | ∈ Y, Dαf vanishes on ∂Ω, |α| ≤ k ,

where |Dαf | is the length of the vector whose components are all the derivatives of

f of order |α|. W k,Y0 (Ω) is provided with the seminorm

‖f‖W k,Y0

(Ω) =∑

|α|=k

‖Da(f)‖Y .

Let (cf. [1])

∇f =

(

∂f

∂x1, . . . ,

∂f

∂xn

)

, ∆f =

n∑

i=1

∂2f

∂x2i

and

∇kf =

∆k/2f for even k ,

∇(∆(k−1)/2)f for odd k .

We shall usually formulate conditions on r.i. spaces Y in terms of the Hardy

operators

Pf(t) =1

t

∫ t

0

f(s)ds ; Qf(t) =

∫ ∞

t

f(s)ds

s.

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Recall that a r.i. space has a representation as a function space on Y ˆ(0, |Ω|)

such that

‖f‖Y (Ω) = ‖f∗‖Y ˆ(0,|Ω|) .

Moreover, since the measure space will be always clear from the context it is conve-

nient to “drop the hat” and use the same letter Y to indicate the different versions

of the space Y that we use. We shall also set our functions equal to zero for x /∈ Ω.

Let (σsf)(t) = f( ts ), t, s > 0, and define the dilation function dY (s) by

dY (s) = ‖(σsf)(t)‖Y

= ‖(σsf∗)(t)‖Y (since (σsf)∗ = (σsf

∗)) .

Definition 2.1. We shall say that Y satisfies the (P )-condition if P : Y (0,∞) →

Y (0,∞) defines a bounded operator (equivalently Y satisfies the (P )-condition if

and only if the upper Boyd index of Y is less than 1). In particular, if Y satisfies

the (P )-condition then ‖f∗∗‖Y is an equivalent norm on Y ,

‖f‖Y ≈ ‖f∗∗‖Y . (2.1)

Likewise we shall say that Y satisfies the (Q)-condition if Q : Y (0,∞) →

Y (0,∞) defines a bounded operator (equivalently Y satisfies the (Q)-condition if

and only if the lower Boyd index of Y is greater than 0).

In what follows we also need to consider weighted norm inequalities for Q with

power weights. This leads to conditions on our spaces.

Definition 2.2. Let k > 0, and let Y be a r.i. space. We shall say that Y satisfies

the Q(k)-condition if

Q(k, Y ) =

∫ ∞

1

sk/ndY

(

1

s

)

ds

s< ∞ . (2.2)

Note that if Y satisfies the Q(k)-condition for some k then it satisfies a Q(b)-

condition for every b < k, including the Q(0) = Q-condition.

The following known result (cf. [22, 17]) will be useful in what follows.

Lemma 2.3. Let Ω be an open domain in Rn, and let Y (Ω) be a r.i. space satisfying

the (P ) and (Q)-conditions. Then

‖|∆f |‖Y (Ω) ≈∑

|α|=2

‖Da(f)‖Y (Ω), ∀ f ∈ C∞0 (Ω) , (2.3)

holds with constants of equivalence independent of f .

Proof. We only need to remark that the arguments in [22, 17] for Lp extend to r.i.

spaces satisfying the (P ) and (Q) conditions by a well known result due to Boyd

(cf. [5]).

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Example 2.4. In particular if Ω is an open domain in Rn, 1 < p < ∞, then

‖|∆f |‖Lp(Ω) ≈∑

|α|=2

‖Da(f)‖Lp(Ω), ∀ f ∈ C∞0 (Ω) .

Lemma 2.5. If Y is a r.i. space satisfying the Q(k)-condition, then Q is bounded

on the space Y provided with weight t−k/n. More precisely,

‖t−k/nQf(t)‖Y ≤ Q(k, Y )‖t−k/nf(t)‖Y . (2.4)

Proof. Let α = kn . We have

t−α|Qf(t)| =

∫ ∞

t

f(s)t−α ds

s=

∫ ∞

1

f(st)t−α ds

s

=

∫ ∞

1

f(st)(st)−αsα−1ds .

Applying Minkowski’s inequality we obtain

‖t−αQf(t)‖Y ≤

∫ ∞

1

‖f(st)(st)−α‖Y sα−1ds

≤

∫ ∞

1

dY

(

1

s

)

sα−1ds‖t−αf(t)‖Y .

Lemma 2.6. Let Y be a r.i. space satisfying the Q(k)-condition for some k > 0.

Then, for all measurable functions f with f ∗∗(∞) = 0,

‖t−k/n(f∗∗(t) − f∗(t))‖Y ≤ ‖t−k/nf∗∗(t)‖Y ≤ Q(k, Y )‖t−k/n(f∗∗(t) − f∗(t))‖Y .

(2.5)

Proof. The first inequality is trivial. To prove the second inequality note thatddtf

∗∗(t) = f∗(t)−f∗∗(t)t . Therefore, by the fundamental theorem of calculus, we

have

f∗∗(t) =

∫ ∞

t

(f∗∗(s) − f∗(s))ds

s= Q(f∗∗ − f∗)(t) . (2.6)

The desired result follows from Lemma 2.5.

It is useful to remark here that, for the operator P, the corresponding weighted

norm inequalities for power weights are automatically true.

Lemma 2.7. Let Y be a r.i. space. Then, for any α > 0

‖t−αP (f)(t)‖Y ≤1

α‖t−αf(t)‖Y .

Proof. Computing dY (s) for Y = L1, and Y = L∞, we find, by interpolation, that

for any r.i. space Y

dY (s) ≤ max1, s

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(cf. [5] for the details). Since P is a positive operator we may suppose that f ≥ 0.

Now,

t−αPf(t) =1

t

∫ t

0

f(s)t−αds =

∫ 1

0

f(st)(st)−αsαds .

Thus,

‖t−αPf(t)‖Y ≤

∫ 1

0

dY

(

1

s

)

sαds‖t−αf(t)‖Y

≤

∫ 1

0

sα−1ds‖t−αf(t)‖Y ,

as desired.

3. Symmetrization Inequalities for Higher Order Sobolev Spaces

The main tool for our analysis is the following result from [3].

Lemma 3.1. Suppose that Y is a r.i. space satisfying a (P )-condition. Then for

all smooth functions f such that f ∗∗(∞) = 0,

‖t−1/n(f∗∗(t) − f∗(t))‖Y . ‖∇f‖Y . (3.1)

Proof. The proof we give is the same as the one given in [3] for Y = Lp. However,

it is important for our development to provide the complete details here. Recall

that from Lemma 1 in [3] we have the pointwise inequality

t−1/n(f∗∗(t) − f∗(t)) . |∇f|∗∗(t) , (3.2)

where f(x) = f∗(cn|x|n), denotes the radial spherically symmetrically decreasing

rearrangement of f , cn = measure of the unit ball. Recall also that the Polya–Szego

symmetrization principle holds for r.i. spaces (cf. [11, 13], a more recent reference

is [8]):

‖∇f‖Y . ‖∇f‖Y . (3.3)

Applying the Y norm to (3.2), and using successively the (P )-condition and (3.3),

we obtain the desired result.

Our main result in this section is the higher order version of Lemma 3.1. The

first step of the induction process that leads to higher order estimates is provided

by the next result.

Theorem 3.2. Let Ω be a domain in Rn, and let Y (Ω) be a r.i. space satisfying

the (P ) and Q(1) conditions. Then for all smooth functions f such that f ∗∗(∞) =

|∇f |∗∗(∞) = 0,

‖t−2/n(f∗∗(t) − f∗(t))‖Y . ‖∆f‖Y . (3.4)

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Proof. We shall prove below the elementary estimate

|∇|∇f || ≤ |D2f | . (3.5)

Applying Lemma 3.1 to |∇f | and combining with (3.5), we obtain

‖t−1/n(|∇f |∗∗(t) − |∇f |∗(t))‖Y . ‖|∇|∇f ||‖Y

. ‖|D2f |‖Y .

Therefore combining with (2.3) we get

‖t−1/n(|∇f |∗∗(t) − |∇f |∗(t))‖Y . ‖∆f‖Y .

Combining the previous inequality with Lemma 2.6 we obtain

‖t−1/n|∇f |∗∗(t)‖Y . ‖∆f‖Y .

By the generalized Polya–Szego symmetrization principle (3.3) (applied to the r.i.

norm defined by ‖g‖B = ‖t−1/ng∗∗(t)‖Y ) we see that

‖t−1/n|∇f|∗∗(t)‖Y . ‖t−1/n|∇f |∗∗(t)‖Y . ‖∆f‖Y . (3.6)

Combining the last inequality with the pointwise inequality (3.2) we find

‖t−2/n(f∗∗(t) − f∗(t))|Y . ‖t−1/n|∇f|∗∗(t)‖Y .

Inserting the last estimate in (3.6) gives us the desired estimate

‖t−2/n(f∗∗(t) − f∗(t))‖Y . ‖∆f‖Y .

It remains to prove (3.5). Using subindexes to indicate partial derivatives, we have

|∇g|j =1

|∇g|

n∑

i=1

gigij

|∇|∇g|| =1

|∇g|

n∑

j=1

(

n∑

i=1

gigij

)2

1/2

.

Therefore, by Cauchy–Schwartz applied to inner sum, we have

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|∇|∇g|| ≤1

|∇g|

n∑

i=1

g2i

n∑

j=1

n∑

i=1

g2ij

1/2

=

n∑

i,j=1

g2ij

1/2

= |D2g| .

Remark 3.3. The previous theorem could be considered as a form of the Polya–

Szego principle for the Laplacian.

A higher order version of Theorem 3.2 can now be obtained by iteration. The

main problem here is to check that all requirements to implement the iteration are

fulfilled.

Theorem 3.4. Let Ω be an open domain in Rn, let k ∈ N, k < n, and let Y (Ω) be a

r.i. space satisfying the (P ) and Q(k−1)-conditions. Then for all smooth functions

f such that |∇mf |∗∗(∞) = 0, m = 0, 1, . . . , k − 1, the following inequality holds

‖t−k/n(f∗∗(t) − f∗(t))‖Y . ‖∇kf‖Y . (3.7)

Proof. Lemma 3.1 and Theorem 3.2 prove the cases k = 1, and k = 2. For the

general case we now proceed by finite induction. Induction turns out to be the same

for odd or even numbers k. Let m be an arbitrary number such that 2 < m ≤ k,

m ≡ k (mod 2). Suppose that (3.7) is true for k = m − 2. Suppose that g is any

function such that ‖∇kg‖Y < ∞, and let f = ∆g. Applying (3.7) for k = m − 2,

we get

‖t−(m−2)/n(∆g∗∗(t) − ∆g∗(t))‖Y . ‖∇m−2∆g‖Y .

Combining with Lemma 2.6 and the definition of ∇m gives

‖t−(m−2)/n(∆g)∗∗(t)‖Y . ‖∇mg‖Y . (3.8)

Consider now the r.i. space B defined by the norm

‖h‖B = ‖t−(m−2)/nh∗∗(t)‖Y .

By computation it is readily seen that dB(s) = dY (s)s−(m−2)/n. Therefore B sat-

isfies the Q(1)-condition. Moreover, by Lemma 2.3, (2.3) holds in B norm. Conse-

quently we may now apply Theorem 3.2 to derive

‖t−2/n(g∗∗(t) − g∗(t))‖B . ‖∆g‖B = ‖t−(m−2)/n(∆g)∗∗‖Y . (3.9)

On the other hand, using the property ‖uv‖Y ≤ 2‖u∗v∗‖Y , which is valid for any

r.i. space Y , we derive

‖h‖B ≥ ‖t−(m−2)/nh∗(t)‖Y ≥1

2‖t−(m−2)/nh(t)‖Y .

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Thus

‖t−2/n(g∗∗(t) − g∗(t))‖B ≥1

2‖t−m/n(g∗∗(t) − g∗(t))‖Y .

Combining the last estimate with (3.8) and (3.9) yields

‖t−m/n(g∗∗(t) − g∗(t))‖Y . ‖∇mg‖Y ,

for any admissible m. It remains to put m = k.

Example 3.5. Let Ω be an open domain in Rn, and let 1 < p < nk−1 . Then Lp(Ω)

satisfies all the hypotheses of Theorem 3.4.

Proof. We give the details for the convenience of the reader. By Hardy’s inequality

we see that Lp(Ω) satisfies the (P )-condition. Moreover since dY (s) = s1/p we see

that for 1 < p < nk−1 ,

Q(k, Y ) =

∫ ∞

1

s(k−1)/ns−1/p ds

s< ∞

proving that Lp also satisfies the Q(k − 1) condition. In particular note that we

may take p = nk .

We end this section with a result that shows that our conditions are best pos-

sible. For this purpose we recall a result of [10] (see p. 322, formulae (3.11), and

p. 324, (3.16)) which states that: if X(Ω) is a r.i. space, k < n, then

‖f‖X(Ω) . ‖∇kf‖Y (Ω) , for all f ∈ Ck0 (Ω) ,

implies that∥

∥

∥

∥

∥

∫ |Ω|

t

f(s)skn

ds

s

∥

∥

∥

∥

∥

X

. ‖f‖Y , for all f ≥ 0 . (3.10)

As a consequence we prove

Theorem 3.6. Suppose that k < n, |Ω| < ∞, and X(Ω) is a r.i. space such that

‖f‖X(Ω) . ‖∇kf‖Y (Ω) , for all f ∈ C∞0 (Ω) .

Then,

Yn(∞, k)(Ω) ⊂ X(Ω) , (3.11)

and moreover there exists a constant c = c(|Ω|, X) such that

‖f∗‖X(0,|Ω|) ≤ c‖f‖Yn(∞,k) .

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Proof. By the fundamental theorem of Calculus we have

f∗∗(t) =

∫ ∞

t

(f∗∗(s) − f∗(s))ds

s

=

∫ |Ω|

t

(f∗∗(s) − f∗(s))ds

s+

∫ ∞

|Ω|

f∗∗(s)ds

s

=

∫ |Ω|

t

(f∗∗(s) − f∗(s))ds

s+

1

|Ω|

∫ |Ω|

0

f∗(s)ds .

Therefore,

‖f∗‖X(0,|Ω|) ≤ ‖f∗∗‖X(0,|Ω|)

≤

∥

∥

∥

∥

∥

∫ |Ω|

t

(f∗∗(s) − f∗(s))ds

s

∥

∥

∥

∥

∥

X(0,|Ω|)

+1

|Ω|

∫ |Ω|

0

f∗(s)ds‖χΩ‖X(0,|Ω|) .

(3.12)

It remains to estimate the first term in the previous inequality. Replacing f by

s−k/n(f∗∗(s) − f∗(s)) in (3.10) we find∥

∥

∥

∥

∥

∫ |Ω|

t

(f∗∗(s) − f∗(s))ds

s

∥

∥

∥

∥

∥

X

. ‖s−k/n(f∗∗(s) − f∗(s))‖Y .

Therefore, inserting the last estimate in (3.12), we find

‖f∗‖X(0,|Ω|) . ‖f‖Yn(∞,k) +1

|Ω|

∫ |Ω|

0

f∗(s)ds‖χΩ‖X(0,|Ω|) ,

and (3.11) follows.

4. Final Remarks

4.1. Equivalent conditions

In [16] Maly–Pick introduced the space

MPn(Ω) =

f : f∗(t/2) − f∗(t) ∈ Ln((0, |Ω|),dt

t

,

and showed with ad-hoc methods that for |Ω| < ∞,

W 1,n0 (Ω) ⊂ MPn(Ω) Hn(Ω) (4.1)

(see also [15, Lemma 5.1]). In [3] it was proved that

MPn(Ω) = L(∞, n)(Ω) . (4.2)

For pk = n the first half of Theorem 1.1 gives

W k,p0 (Ω) ⊂ L

(

∞,n

k

)

(Ω) .

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Therefore we could have used the second half of (4.1) combined with (4.2) and (1.5)

to prove that the L(∞, nk ) spaces are optimal.

Likewise, since it is also shown independently in [3] that for |Ω| < ∞, q > 1,

L(∞, q)(Ω) ⊂ Hq(Ω) ,

we have still another method to prove that these spaces are optimal.

In [16] the proof of the fact that MPq(Ω) 6= Hq(Ω) (and therefore by (4.2) that

L(∞, q)(Ω) 6= Hq(Ω)) is indirect. The authors show that L(∞, q)(Ω) is not a linear

space. We prefer to give here a simple direct proof. To this end we now exhibit

f ∈ Hq(Ω)\L(∞, q)(Ω).

Let f(t) = Σ∞i=1χ(0,ci)(t), with ci = e−2i

. Then a computation shows that

f∗∗(t) − f∗(t) =1

t

∞∑

i=1

ciχ(ci,∞)(t) .

Therefore, for any q ∈ (1,∞),∫ 1

0

(f∗∗(t) − f∗(t))q dt

t≥

∞∑

i=1

∫ 1

0

cqi χ(ci,1)(t)

tqdt

t

≥

∞∑

i=1

∫ 1

ci

cqi

tq+1dt

=1

q

∞∑

i=1

(1 − cqi ) = ∞ .

On the other hand, by Hardy’s inequality,

‖f‖Hq≈

(∫ 1

0

(

f∗(t)

ln et

)qdt

t

)1/q

≤∞∑

i=1

(

∫ ci

0

(

ln1

t

)−qdt

t

)1/q

=1

q − 1

∞∑

i=1

2−i(q−1)/q < ∞ .

More generally, the method of proof of (4.2) given in [3] can be easily extended

to the spaces Yn(∞, k) introduced in this paper so that we can readily show that

‖f‖Yn(∞,k) ≈

∥

∥

∥

∥

t−k/n

(

f∗

(

t

2

)

− f∗(t)

)∥

∥

∥

∥

Y

.

The interesting paper by Y. Sagher and P. Shvartsman [21] (see also the refer-

ences therein) contains an extensive comparison of the expressions f ∗∗(t) − f∗(t)

and f∗( t2 )− f∗(t), and other analogous expressions obtained using Peetre’s K and

E-functionals.

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4.2. Some questions

In this section we indicate some questions and open problems raised by our reasearch

for this paper.

It is interesting to observe that the conditions of our embedding theorems allow

us to describe the r.i. sets containing W k,p0 even for p bigger than the critical value

nk . As is well known in this case, W k,p

0 ⊂ L∞. Our rearrangement invariant sets are

contained in L∞, which is a somewhat surprising property, since for finite measure

spaces L∞ is the smallest rearrangement invariant space. It could be of interest to

study this phenomena in detail. We now give an example of a result in this direction

Example 4.1. Let nk < p < n

k−1 , f ∈ W k,p0 (Ω). Suppose that f∗( t

2 ) − f∗(t) is

a.e. equivalent to a monotone function on (0, 1). Then

limt→0

f∗( t2 ) − f∗(t)

tk/n−1/p= 0 .

The reason we were able to improve, in some cases, well known optimal embed-

ding theorems, is the fact that the spaces Yn(∞, k) are only r.i. sets. It is therefore

of interest to ask if the Yn(∞, k) spaces are the optimal target for the Sobolev

embedding theorem among all r.i. sets. In particular, we conjecture that for pk = n

we have

r.i. hull(W k,n0 (Ω)) = L

(

∞,n

k

)

(Ω) .

Here the rearrangement invariant hull of W k,p0 (Ω) = f ∈ M(Ω) : ∃ g ∈ W k,p

0 (Ω)

s.t. g∗ = f∗. In this respect we refer the reader to [4] where it is shown that for a

cube Q,

r.i. hull BMO(Q)) = L(∞,∞)(Q) .

It could also be of interest to consider the modified rearrangement invariant

hull, as defined by Netrusov [20]

modified r.i. hull(W k,n0 (Ω)) = f : ∃ g ∈ W k,p

0 (Ω) s.t. f∗ ≤ g∗ .

A closely connected problem is to give necessary and sufficient conditions on Y

for the space Yn(∞, k) to be a r.i. space. It follows from our results that a sufficient

condition is for Y to satisfy the Q(k)-condition. We conjecture that the condition

is necessary as well. This conjecture is obviously true for L(p, q) spaces and can be

readily verified for other spaces studied in [9].

Finally we would like to suggest that the Yn(∞, k) spaces introduced in this

paper should also be of interest in interpolation theory (cf. [21].)

Acknowledgments

We thank the referee for his/her critical reading of the paper and helpful sugges-

tions. We are thankful to A. Cianchi for useful information. This paper is in final

form and no version of it will be submitted for publication elsewhere.

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ON SHARP HIGHER ORDER SOBOLEV EMBEDDINGS

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