on the approximation numbers of sobolev embeddings on singular

ON THE APPROXIMATION NUMBERS OF SOBOLEV EMBEDDINGS ON SINGULAR DOMAINS AND TREES by W. D. EVANS† , D. J. HARRIS (School of Mathematics, Cardiff University, 23 Senghennydd Road, Cardiff CF24 4YH) ¯‡ and Y. SAITO (Department of Mathematics, University of Alabama at Birmingham, Birmingham, AL 35294, USA) [Received 27 May 2003. Revised 12 November 2003] Abstract Upper and lower bounds are determined for a function which counts the approximation numbers of the Sobolev embedding W 1, p (
)/C → L p (
)/C, for a wide class of domains
of finite volume in Rn and 1 < p < ∞. Results on the distribution of the eigenvalues of the Neumann Laplacian in L 2 (
) are special consequences.

We dedicate this paper to Jerry Goldstein on his 60th birthday. 1. Introduction Let
be a domain (that is, a connected open set) in Rn , n
1, of finite volume 1, p p 1, p |
|, and let E C (
) : WC (
) → L C (
), E 0 (
) : W0 (
) → L p (
) be the natural embed1, p

p

dings, where WC (
), L C (
) are the quotient spaces W 1, p (
)/C, L p (
)/C respectively, and 1, p W 1, p (
), W0 (
) are the standard Sobolev spaces, with 1 < p < ∞. The approximation numbers am (E C (
)), m ∈ N, of E C (
) are defined as 1, p

p

am (E C (
)) := inf{E C (
) − P|WC (
) → L C (
) : rankP < m}

(1.1)

and for ε > 0 we set νC (ε,
) := max{m : am (E C (
))
ε}.

(1.2)

The objective of this paper is to determine, for a wide class of domains
(including ones with highly irregular, even fractal, boundaries), upper and lower bounds for νC (ε,
) as ε → 0, which can readily be estimated in examples. Roughly speaking, the core of our
is the union of rectanglular blocks or other regular polyhedra, and the remainder is made up of sets of small diameter and irregular sets which are generalized ridged domains (GRDs); see Definition 1.1 below. The latter portions include regions near parts of the boundary of
which may be horns, cusps, spirals or † ‡

E-mail: [email protected] E-mail: [email protected]

Quart. J. Math. 55 (2004), 267–302; doi: 10.1093/qmath/hag050 c Oxford University Press 2004; all rights reserved Quart. J. Math. Vol. 55 Part 3 

268

W. D . EVANS

et al.

fractal curves. Any GRD
which is not unduly pathological falls within this category. The lower bound for νC (ε,
) is determined in terms of the number µ0 (ε,
) := max{dim S : α(S)  1/ε},

(1.3)

1, p

where S is a subset of W0 (
) of finite dimension dim S and
 ∇ f  L p (
) . α(S) := sup  f  L p (
) f ∈S When p = 2, νC (ε,
) = N (ε −2 , − N ,
), the counting function for the positive eigenvalues of the Neumann Laplacian − N ,
, and µ0 (ε,
) = N (ε −2 , − D,
), where − D,
is the Dirichlet Laplacian on
; see [6, Lemma 5.2]. Note that in our present notation, ν
(ε), µ0
(ε) in [6] are denoted by νC (ε,
), µ0 (ε,
) respectively. In the estimates we derive, there are ‘main’ terms involving (1.2) and (1.3) on the unit cube (0, 1)n and error terms determined by the nature of the boundary of
. When p = 2, these estimates yield asymptotic formulae for N (ε −2 , − N ,
) and N (ε−2 , − D,
) with error terms. In [6, section 5], an analogue of Dirichlet–Neumann bracketing is developed for νC (ε,
), µ0 (ε,
) and related quantities, for 1 < p < ∞. This forms the basis of the method used in this paper. Another significant feature of the analysis is the role of Hardy-type operators on trees associated with those portions of
which are generalized ridged domains. This allows recent work on the approximation numbers of such operators to be applied. We refer to [4, 6] for a detailed description of GRDs and their properties, and also to a discussion of the analysis on trees. Here we just give a brief summary. Let  be a tree, that is, a connected graph without loops or cycles, where the edges are non-degenerate closed line segments whose endpoints are the vertices. The distance, dist(x, y), between x, y ∈  is defined to be the length of the unique polygonal path which joins x and y, and this defines a metric topology on . When endowed with the natural one-dimensional Lebesgue measure,  is a σ -finite measure space. The path joining two points x, y ∈  may be parametrized by s(t) := dist(x, t), and for g ∈ L 1loc () we have   y

dist(x,y)

g(t)dt :=

x

g(t (s))ds;

0

see [6, section 2] for further details. D EFINITION 1.1 A domain
is called a generalized ridged domain (GRD) if there exist real-valued functions u, ρ, τ , a tree  and positive constants α, β, γ , ζ such that the following conditions are satisfied: • u :  →
, ρ :  → R+ ≡ (0, ∞) are Lipschitz; • τ :
→  is surjective and for each x ∈
, there exists a neighbourhood V (x) such that for all y ∈ V (x), |τ (x) − τ (y)|  γ |x − y|, where | · | denotes the metric on ; thus τ is uniformly locally Lipschitz; • |x − u ◦ τ (x)|  αρ ◦ τ (x) for all x ∈
; • |u (t)| + |ρ (t)|  β for all t ∈ ; • with Bt := B(u(t), ρ(t)) and C(x) := {y : sy + (1 − s)x ∈
for all s ∈ [0, 1]}, we have that for all x ∈
, C(x) ∩ Bτ (x) contains a ball B(x) such that |B(x)|/|Bτ (x) |
ζ > 0. The curve t  → u(t) :  →
is called the generalized ridge of
.

ON THE APPROXIMATION NUMBERS OF SOBOLEV EMBEDDINGS

269

The map τ in Definition 1.1 defines a positive Borel measure µ on  such that for any open subset 0 of , µ(0 ) = |τ −1 (0 )|, and   F(t)dµ(t) = (F ◦ τ )(x)d x; 




this is initially defined on compactly supported functions on  but extends by continuity to give an isometry F  → F ◦ τ from L p (, dµ) to L p (
) when L p (, dµ) is endowed with the norm  1/ p p F L p (,dµ) := |F(t)| dµ . 

When µ is replaced by the Lebesgue measure on  we use the notation L p (). The paper is organized as follows. Section 2 contains an analysis of the one-dimensional problem when
is an open interval. In this case the norms of E C (
) and E 0 (
) are equal and are determined precisely (Theorem 2.1), and bounds for νC (ε,
) and µ0 (ε,
) given which yield the same asymptotic limit (Theorem 2.3). Similar results have also been independently obtained recently in [2, 7]; see also Remark 7.1 below. In order to apply results from [6], relationships 1, p p between E C (
) and embeddings E g (
) between subspaces Wg and L g (
) of W 1, p (
) and 1, p p L p (
) respectively (related to the spaces W M (
), L M (
) of [6]) are needed, and these are presented in section 3. This requires a comparison of various norms on quotient spaces; we note in p particular Theorem 3.2 which asserts that the quotient norm on L C (
) is equal to        p
f gd x   min  f − : g ∈ L (
), gd x = 0 .  

gd x L p (
) Similar results are obtained for spaces defined on trees. The main theorems, in which the upper and lower bounds for νC (ε,
) are established, are given in sections 4 and 5. Examples are discussed in section 6. 1, p 1, p p The norms on L p (), L p (
), W0 (
), WC (
) and L C (
) are denoted respectively by
 1/ p  f  p, = | f (t)| p dt ,
  f  p,
=
  f 1, p,
=
 [ f ]1, p,
=












1/ p | f (x)| p d x

, 1/ p

|∇ f (x)| p d x

, 1/ p

|∇ f (x)| p d x

,

[ f ] p,
= inf{ f − c p,
: c ∈ C}. 1, p

p

Note that elements of the quotient spaces WC (
), L C (
) are equivalence classes [·], members of the same equivalence class differing by a constant. If the Poincar´e inequality is valid on
, that is,      f − 1 (1.4) f (x)d x     K ∇ f  p,
|
| p,


270

W. D . EVANS

et al. 1, p

for some constant K , then the norm defined above on WC (
) is equivalent to the standard quotient space norm, namely inf  f − cW 1, p (
) ,

c∈C

where

 p p 1/ p  f W 1, p (
) :=  f  p,
+ ∇ f  p,
, 1, p

and WC (
) is a Banach space. It is proved in [4, Theorem 2.6] that (1.4) is equivalent to inf{E(
) − P|W 1, p (
) → L p (
) : rankP < ∞} < 1,

(1.5)

where E(
) is the embedding W 1, p (
) → L p (
). We shall assume throughout that the Poincar´e 1, p inequality (1.4) holds on
, or equivalently, that WC (
) is a Banach space and E C (
) is bounded. In fact, we are mainly interested in the case when E(
) is compact, which is so if and only if the quantity on the left-hand side of (1.5) is 0, and hence (1.4) is satisfied. The norm of a bounded operator T : X → Y will be denoted by T |X → Y , but simplified to T  when the spaces involved are clear. Various positive constants are denoted by the same letter K . 2. The case
= (a, b) Let c ∈
= (a, b), and let G c be the Hardy operator  x (G c f )(x) := f (t)dt,

f ∈ L p (
).

(2.1)

c

Then G c : L p (
) → L p (
) is bounded with norm G c   |
| since  






x c

p 

p f (t)dt

d x   f  p,
(max |x − c|) p−1 d x  |
|

p


x∈
p  f  p,
.

(2.2)

We also have by scaling that G a  = G b  = |
|G 0 |L p (0, 1) → L p (0, 1) = |
|α p ,

(2.3)

where

α p = p 1/ p ( p )1/ p sin(π/ p)/π

(2.4)

for p = p/( p − 1); see [1] and Remark 7.1 below. T HEOREM 2.1 If c = 12 (a + b), E 0 (
) = E C (
) = G c |L p (
) → L p (
) =

αp |
|. 2

(2.5)

ON THE APPROXIMATION NUMBERS OF SOBOLEV EMBEDDINGS

Proof. Given ε > 0, there exists φ ∈ L p (c, b) such that  G c φ p,(c,b)
G c |L p (c, b) → L p (c, b) − ε φ p,(c,b)
 αp = |
| − ε φ p,(c,b) 2

271

(2.6)

by (2.3). Define the function

ψ(t) =

φ(2c − t), a < t  c, φ(t), c < t < b,

which is symmetrical about the mid-point c. Then (x) = (G c ψ)(x) is anti-symmetrical about c and we have ψ p,
= 21/ p φ p,(c,b) ,  p,
= 21/ p G c φ p,(c,b) . These equations and (2.6) yield
G c ψ p,



 αp |
| − ε ψ p,
2

(2.7)

and hence G c |L p (
) → L p (
)


αp |
|. 2

(2.8)

We also have for the above  that  L p (
) =  p,
. C

(2.9)

To see this, first note that, for any f ∈ L p (
), there exists a unique constant c f such that  f  L p (
) = inf  f − k p,
=  f − c f  p,
; C

k∈C

see [8, Lemma 3.6]—this is a consequence of the uniform convexity of L p . If k0 = c , we have, p p since (c + t) = −(c − t) for 0  t  12 (b − a), that  − k0  p,
=  + k0  p,
and this implies that k0 = 0 and (2.9) holds. It therefore follows from (2.7) that
 αp  L p (
)
|
| − ε   p,
C 2 and so E C (
)
For f ∈ W 1, p (
), (G c f )(x) =



x c

αp |
|. 2

f (t)dt = f (x) − f (c).

(2.10)

272 Hence,

W. D . EVANS

et al.

 f  L p (
)  G c f  p,
 G c |L p (
) → L p (
) f  p,
C

gives E C (
)  G c |L p (
) → L p (
). Furthermore, by (2.3)
p G c f  p,


=

p G c f  p,(a,c)

p + G c f  p,(c,b)



p

αp |
| 2

and so E C (
)  G c |L p (
) → L p (
) 

p

 f  p,


αp |
|. 2

Thus (2.8) and (2.10) give (2.5) for E C (
) and G c . The proof for E 0 (
) is similar, with the above  replaced by

(G a φ)(x), a  x  c, (x) = −(G b φ)(x), c < x  b. R EMARK 2.2 It is proved in [2, 7] (see also Remark 7.1 below) that E 0 (0, 1) =

φ p,(0,1) αp 1 = p 1/ p ( p )1/ p sin(π/ p)/π = φ  p,(0,1) 2 2

with  φ(t) = sin p (π p t),

1

πp = 2

(1 − t p )−1/ p dt,

0

where sin p t is the inverse of the function elsewhere on R by periodicity.

1 2

p

 2t/ p 0

(1 − s p )−1/ p ds in [0, π p /2] and is defined

In the next theorem we use the following notation: νC (ε,
) := max{m : am (E C (
))
ε}, µC (ε,
) := max{dim S : α(S)  1/ε}, where α(S) =

 f  p,
 f  L p (
) 1, p f ∈S⊂W (
) sup C

C

and am (E C (
)) is the mth approximation number of E C (
). Also, ν0 (ε,
), µ0 (ε,
) are the 1, p 1, p corresponding numbers when WC (
) is replaced by W0 (
), and thus E C (
) by E 0 (
). T HEOREM 2.3 For 1 < p < ∞ and n ∈ N we have an (E 0 (
)) = an (E C (
)) =

αp |
|, 2n

(2.11)

ON THE APPROXIMATION NUMBERS OF SOBOLEV EMBEDDINGS


µ0


  αp αp |
|,
= ν0 |
|,
2n 2n
 αp = νC |
|,
= n. 2n

273

(2.12) (2.13)

Hence, lim εµ0 (ε,
) = lim εν0 (ε,
) = lim ενC (ε,
) =

ε→0

ε→0

ε→0

αp |
|. 2

(2.14)

Proof.In view of evident scaling properties, it is sufficient to give the proof for
= (0, 1). Let n
= i=0
i , where
0 = (0, 1/2n),
i = [ai , ai+1 ), i = 1, . . . , n − 1, where ai+1 − ai = 1/n n−1 and
n = [1 − 1/2n, 1). Set P f (x) = i=1 f (ci )χi (x), where ci = 12 (ai+1 + ai ) and χi is the characteristic function of
i . Then, by (2.5), for f ∈ C01 (
) and with c0 = 0, cn = 1, p

E 0 (
) f − P f  p,
=

n 

G ci f  p,
i 




p

i=0

p  αp  f  p,
. 2n

1, p

Hence, since C01 (
) is dense in W0 (
), we have 1, p

E 0 − P|W0 (
) → L p (
) 

αp . 2n

(2.15)

Since rank P  n − 1, it follows that αp 2n

(2.16)

 αp ,
 n. 2n

(2.17)

an (E 0 (
))  and so


ν0

n Next, we choose the partition
= i=1
i , where |
i | = |
|/n for i = 1, . . . , n. Let M 1, p 1, p be the subspace of W0 (
) spanned by {φ1 , . . . , φn }, where, for ε > 0, the φi ∈ W0 (
i ) are translations of one another, and φi  p,
i 2n = ; E 0 (
i )φi  p,
i αp see Remark 2.2. Then dim M = n and for non-zero φ ∈ M, φ  p,
2n = . E 0 (
)φ p,
αp  It follows that µ0 α p /2n,

n. Since µ0 (ε,
)  ν0 (ε,
) is always true by [6, Lemma 5.1, (5.6)], we have proved that  

αp αp µ0 ,
= νo ,
= n 2n 2n

274

W. D . EVANS

et al.

and therefore an (E 0 (
)) = α p /2n. n To prove the result for E C (
), we again choose the partition
= i=1
i , where |
i | = 1/n  1, p n for i = 1, . . . , n. For g ∈ [ f ], f ∈ WC (
), set Pg(x) = i=1 f (ci )χi (x), where ci is the 1, p

p

mid-point of
i . As a map from WC (
) into L C (
), P has rank at most n − 1 as is seen on n [ f (ci ) − f (c1 )]χi . Furthermore, writing Pg = f (c1 ) + i=2 p 
 n  αp p p  f − Pg p,
= G ci f  p,
i   f  p,
. 2n i=1 Hence αp 1, p p E C (
) − P|WC (
) → L C (
)  2n and so, since rank P  n − 1, an (E C (
)) 

αp . 2n

From Theorem 2.1, for i = 1, . . . , n,

  αp 1, p p E C (
i )|WC (
i ) → L C (
i ) = sup inf  f − c p,
i ;  f  p,
i = 1 = . 2n c∈C

Thus, if δ < α p /2n, there exists ψi ∈ W 1, p (
i ) such that ψi  p,
i > 0 and [ψi ] L p (
i ) ≡ inf ψi − c p,
i > δψi  p,
i . c∈C

C

Define ψi to be zero outside
i and such that, by a suitable selection of constant values outside 1, p p
i , ψi ∈ W 1, p (
). Now let P : WC (
) → L C (
) have rank less than n. It follows that there exist constants λi , i = 1, . . . , n, not all zero, such that n 

[ψ] =

λi [ψi ],

P[ψ] = 0.

i=1

Then p

p

(E C (
) − P)[ψ] L p (
) = [ψ] L p (
) = C

C


δp

n 

n  i=1

p

λi [ψi ] L p (
) C

i

λi ψi  p,
i = δ p ψ  p,
. p

p

i=1

Since P of rank less than n and δ < α p /2n are arbitrary, we have proved that an (E C (
))


αp , 2n

and hence the proof is complete. The results for ν0 and νC in Theorem 2.3 have been obtained independently in [2, 7].

ON THE APPROXIMATION NUMBERS OF SOBOLEV EMBEDDINGS

275

3. Quotient spaces, norms and approximation numbers The domain
is now a subset of Rn , n
1, of finite volume |
|. The quotient space L C (
) ≡ L p (
)/C has norm  f  L p (
) ≡ [ f ] p,
:= inf  f − c p,
c∈C

C

(3.1)

and functions are equivalent if and only if they differ by a constant almost everywhere. Another relevant space in subsequent analysis is    p p
f gd x  L g (
) := f ∈ L (
) :
g ( f ) := =0 , (3.2)
gd x  p where g ∈ L p (
), 1/ p + 1/ p = 1, and
gd x = 0. Note that L 1 (
) is the space denoted by p p L M (
) in [6, section 3]. Clearly L g (
) is a closed subspace of L p (
) and f  → f −
g ( f ) is a p projection of L p (
) onto L g (
). We shall write  f  L gp (
) =  f −
g ( f ) p,
,

f ∈ L p (
),

and hence  f  L gp (
) =  f  p,
,

(3.3)

p

f ∈ L g (
). p

L EMMA 3.1 Let V
denote the canonical map V
: L p (
) → L C (
), given by f  → [ f ]. Then, for all f ∈ L p (
), V
f  L p (
)   f  L gp (
)  c( p, g)V
f  L p (
) , C

C

where c( p, g) = 1 +

(3.4)

|
|1/ p g p ,
 . |
gd x|

When p = 2, and g = 1 V
f  L 2 (
) =  f  L 2 (
) =  f −
1 ( f )2,
. C

1

(3.5)

Proof. The first inequality in (3.4) is obvious, while the second is a consequence of 1 |
g ( f )|   g p ,
 f  p,
|
g(x)d x| which follows from H¨older’s inequality. The identity (3.5) is well known. We now set 1, p

Wg (
) := { f : f ∈ W 1, p (
),
g ( f ) = 0} with norm  f W 1, p (
) := ∇ f  p,
g

(3.6)

276

W. D . EVANS

et al.

and 1, p

p

E g (
) : Wg (
) → L g (
); 1, p

1, p

1, p

W1 (
) is the space W M (
) in [6]. Note that Wg (
) is a closed subspace of W 1, p (
) if the Poincar´e inequality (1.4) is satisfied. We have the direct sum decomposition ˙ W 1, p (
) = Wg (
)+C 1, p

(3.7)

determined by f = [ f −
g ( f )] +
g ( f )  1, p since
g
g ( f ) =
g ( f ), and the map W : [ f ]  → f −
g ( f ) is an isometry between WC (
) 1, p

p

p

and Wg (
). The map W is also a topological isomorphism of L C (
) onto L g (
), by Lemma 3.1, satisfying [ f ] L p (
)  W [ f ] L gp (
) C

(3.8)

=  f  L gp (
)  c( p, g)[ f ] L p (
) .

(3.9)

C

1, p

p

Note that if f ∈ Wg (
) and h ∈ [ f ], then h = f +
g (h) and similarly between L C (
) and p L g (
), with W −1 f = [ f ]. In the analysis to follow, it is helpful to distinguish between the maps 1, p

1, p

W1 : [ f ]  → f −
g ( f ) : WC (
) → Wg (
), p p W0 : [ f ]  → f −
g ( f ) : L C (
) → L g (
).

(3.10) (3.11)

From the preceding remarks, both maps are surjective, W1 is an isometry and 1  W0   c( p, g).

(3.12)

E g (
) = W0 E C (
)W1−1

(3.13)

W0−1 E g (
)W1 .

(3.14)

It is readily seen that

E C (
) = T HEOREM 3.2 For all f ∈ L p (
),

V
f  L p (
) = min  f −
g ( f ) p,
, g∈S

C

where

   S := g : g ∈ L p (
), g(x)d x = 0 .


When p = 2 we have V
f  L 2 (
) =  f −
1 ( f )2,
. C

(3.15)

277

ON THE APPROXIMATION NUMBERS OF SOBOLEV EMBEDDINGS

Proof. If f 0 ∈ L p (
) and V
f 0 = 0, then f 0 = c, a constant a.e., and, for any g ∈ S, V
f 0  L p (
) = c −
g (c) p,
= 0. C

Hence, from (3.4), V
f 0  L p (
) = min  f 0 −
g ( f 0 ) p,
. S

C

Suppose V
f 

p L C (
)

= 0. Then, there exists a unique constant c = c( f ) such that V
f  L p (
) =  f − c p,
C

(cf. [9, Lemma 3.6]). Since f − c = 0, the one-dimensional vector space {λ( f − c) : λ ∈ C} does not contain the constant function 1. Therefore, by the Hahn–Banach theorem, there exists ∗ l ∈ L p (
) , the dual of L p (
), such that l( f − c) = 0,

l(1) = 1.



Hence, there exists g0 ∈ L p (
) such that  [ f (x) − c]g0 (x)d x = 0,


Hence,
g0 ( f ) = c and




g0 (x)d x = 1.

V
f  L p (
) =  f −
g0 ( f ) p,
. C

Since V
f  L p (
)   f −
g ( f ) p,
C

for any g ∈ S, (3.15) follows. The case p = 2, g = 1 was noted in (3.5). The approximation numbers of E g (
) and E C (
) both feature in the analysis to follow. They are related by a lemma. L EMMA 3.3 For any g ∈ S, am (E C (
))  am (E g (
))  c( p, g)am (E C (
)),

(3.16)

where c( p, g) is given in Lemma 3.1; when g = 1, c(2, 1) = 1 and c( p, 1) = 2 for p = 2. Therefore
 ε νC (ε,
)  νg (ε,
)  νC ,
. (3.17) c( p, g) 1, p

p

Proof. Let P : WC (
) → L C (
) be of rank less than m. Then, from (3.13) and (3.14), E g (
) − W0 P W1−1 |Wg (
) → L g (
) = W0 (E C − P)W1−1 |Wg (
) → L g (
) 1, p

p

1, p

1, p

p

p

 c( p, g)E C − P|WC (
) → L C (
). Since rank(W0 P W1−1 )  rankP < m, the second inequality in (3.16) follows. The first inequality is proved similarly. The inequalities (3.17) are immediate consequences of (3.15).

278

W. D . EVANS

et al.

Similar results to the preceding ones in this section hold also on the tree . The analogous spaces defined on  are the following: p

L C (, dµ) : u L p (,dµ) := inf u − c L p (,dµ) ; C C c∈C    p p  uhdµ  L h (, dµ) : = u ∈ L (, dµ) : ωh (u) := =0 ;  hdµ u L p (,dµ) : = u − ωh (u) L p (,dµ) , h



where h ∈ L p (, dµ) and

 

(3.18)

(3.19)

hdµ = 0;

L C (, dµ) : u L 1, p (,dµ) = u  L p (,dµ) ; C  1, p L h (, dµ) := u ∈ L 1, p (, dµ) : ωh (u) = 0 ; u L 1, p (,dµ) := u  L p (,dµ) . 1, p

(3.20)

(3.21)

h

We denote the associated embeddings by 1, p

p

IC () : L C (, dµ) → L C (, dµ), 1, p p Ih () : L h (, dµ) → L h (, dµ).

(3.22) (3.23)

The maps 1, p

1, p

T1 : [u]  → u − ωh (u) : L C (, dµ) → L h (, dµ), p p T0 : [u]  → u − ωh (u) : L C (, dµ) → L h (, dµ)

(3.24) (3.25)

are respectively an isometry and an isomorphism, with 1  T0   d( p, h) := 1 +

µ()1/ p h L p (,dµ)  . |  hdµ|

(3.26)

The analogue of Lemma 3.3 is as follows. L EMMA 3.4 am (IC ())  am (Ih ())  d( p, h)am (IC ())

(3.27)

and νC (ε, )  νh (ε, )  νC (

ε , ), d( p, h)

where d( p, h) is given in (3.26); in particular d(2, 1) = 1 and d( p, 1) = 2 for p = 2.

(3.28)

ON THE APPROXIMATION NUMBERS OF SOBOLEV EMBEDDINGS

279

4. An upper bound for ν1 (ε,
) Let
be a generalized ridged domain and, for ε > 0, set 
(ε) := x ∈
: ρ ◦ τ (x)
ε . Then
\
(ε) =

q(ε) 


j (ε),

j=1

where the
j (ε) are the connected components of
\
(ε). The following is proved in [5, section 3]. Let Fε be a tesselation of Rn by cubes Q of side s(ε), where s(ε) :=

κζ 1/n ε, 2(α + 1)

and 4βγ κ < n; the tesselation can be either by closed cubes with disjoint interiors or disjoint halfopen cubes. For each x ∈
(ε), there exists Q ∈ Fε such that the cone C(x, Q) := {t x + (1 − t)y : y ∈ Q, 0  t  1} ⊂
. Denoting by VQ the union of all the cones C(x, Q) for a given Q, and setting  Q(ε) := Q ∈ Fε : VQ = ∅ ,  Q0 (ε) := Q ∈ Q(ε) : Q ∈
∩
(ε) , it follows that
(ε) ⊆



h(ε) := Q(ε),

Q(ε)

r (ε) := Q0 (ε),

VQ , and hence we have


=





q(ε) 
   Q ∪ WQ ∪
j (ε) ,

Q0 (ε)

Q(ε)

j=1

where W Q is a measurable subset of VQ . Furthermore, it is proved in [5, Lemma 3.4] that for Q ∈ Fε and f ∈ C 1 (
) ∩ W 1, p (
)  f − f Q W Q  c p 2n−1 nζ −1 (α + 1)n κε∇ f W Q with a constant c p which depends only on p and n. Hence for sufficiently small κ we have  f − f Q W Q < ε∇ f W Q . If for each x ∈
j (ε), the cone C(x) ∩ Bτ (x) in Definition 1.1 contains a ball B(x) which lies in
j (ε) then
j (ε) has all the important features of a GRD. If each
j (ε) has this property and q(ε) < ∞, then
can be decomposed in accordance with (4.1) in the first of our two main theorems. T HEOREM 4.1 Given ε > 0, let q(ε), h(ε) ∈ N be such that
=


q(ε)  i=0


q(ε)+h(ε)  
i (ε) ∪
i (ε) ∪ N , i=q(ε)+1

where the
i (ε) are disjoint open sets, N is a null set and

(4.1)

280

W. D . EVANS

et al.

r (ε) •
0 (ε) = j=1 Q j (ε) ∪ N , where the Q j (ε) are disjoint rectangular blocks with edge lengths l1 (ε, j)  l2 (ε, j)  · · ·  ln (ε, j); •
i (ε), i = 1, . . . , q(ε) are generalized ridged domains with associated trees i (ε), for which the constants α and ζ in Definition 1.1 are such that α, ζ −1 are bounded independently of ε and sup ( ◦ τ )(x) < ε;

x∈
i (ε)

•
i (ε), i = q(ε) + 1, . . . , q(ε) + h(ε) have the Poincar´e property [ f ] p,
i (ε) = inf  f − c p,
i (ε)  ε∇ f  p,
i (ε) , c∈C

f ∈ W 1, p (
i (ε)).

For any δ ∈ (0, 1) and k = 0, 1, . . . , n − 1, define  L k := j : j ∈ {1, . . . , r (ε)}, lk (ε, j)  εδ −1 < lk+1 (ε, j) ,

p j (ε, δ, 0) :=

(4.2)

n  

n   ε−1 δls (ε, j) + 1 − ε −1 δls (ε, j) ,

s=1

s=1 n 

p j (ε, δ, k) := [ε −1 δ]n−k

ls (ε, j), 1  k  n − 2,

s=k+1

p j (ε, δ, n − 1) := 1,

(4.3)

where l0 (ε, j) = 0. Then ε n [ν1 (ε,
) + 1]  δ n {ν1 (δ, U ) + 1}



|Q j (ε)|

j∈L 0

+K εn δ −n

n−1  

p j (ε, δ, k)

k=0 j∈L k

+εn

q(ε) 

 {ν1 ci ε, i (ε) } + ε n {q(ε) + h(ε)},

(4.4)

i=1

where U is the unit cube (0, 1)n and K , ci are constants independent of ε and δ. Moreover,  inf δ n [ν1 (δ, U ) + 1] = lim t n ν1 (t, U ) =: L U . δ>0

t→0

(4.5)

Proof. We are motivated by ideas from [5] and use results which are proved in [6, section 5] for E 1 (which is denoted in [6] by E M ). From [6, Lemma 5.3] ν1 (ε,
) + 1  ν1 (ε,
0 (ε)) + 1 +

q(ε)  i=1

= I1 + I2 + I3 ,

{ν1 (ε,
i (ε)) + 1} +

q(ε)+h(ε) 

{ν1 (ε,
i (ε)) + 1}

i=q(ε)+1

(4.6)

281

ON THE APPROXIMATION NUMBERS OF SOBOLEV EMBEDDINGS

say. Let Q be a rectangular block with edges l1  l2  · · ·  ln . Suppose first that 



1  ε−1 δl1  ε −1 δl2  · · ·  ε −1 δln ,

(4.7)

and set Mr = ε −1 δlr . Then, Mr
1 and, for some θr ∈ [0, 1), we have ε−1 δlr = Mr + θr .

(4.8)

ε−1 Q

δ −1 ,

We divide into M1 M2 · · · Mn n-dimensional cubes of side µ = and a remainder S made  up of ns=1 (Ms + 1) − ns=1 Ms rectangular blocks of sides η1 µ, · · · , ηn µ, where ηr = 1 or θr . Further, subdivide each of the rectangular blocks in S into k(µ), say, rectangular blocks T which are sufficiently small that ν1 (1, T ) = 0; note that k(µ) = O(µn ). It follows from [6, Lemmas 5.3 and 5.6] that ν1 (ε, Q) + 1 = ν1 (1, ε −1 Q) + 1


 n

 M1 · · · Mn {ν1 (1, µU ) + 1} +

(Ms + 1) −

s=1

 ε−n δ n |Q|[ν1 (1, µU ) + 1] + K δ −n

 n

n 

 Ms k(µ)

s=1

(ε−1 δls + 1) −

s=1

n 

 (ε−1 δls ) . (4.9)

s=1

Next, suppose that ε−1 δlk < 1  ε −1 δlk+1

(4.10)

for some k ∈ (0, n − 1). We then have  that M1 = · · · = Mk = 0 and 1  Mk+1  · · ·  Mn . This time we divide ε−1 Q into at most ns=k+1 (Ms + 1) rectangular blocks whose sides are of length at most µ. As in the previous case, each of these rectangular blocks is divided into O(µn ) blocks T which are sufficiently small that ν1 (1, T ) = 0. We now have ν1 (ε, Q) + 1 = ν1 (1, ε −1 Q) + 1 n  (Ms + 1)µn K s=k+1

 K δ −n

n 

(ε −1 δls ).

(4.11)

s=k+1

Finally, suppose that ε−1 δln < 1.

(4.12)

We again divide ε−1 Q into O(µn ) blocks T for which ν1 (1, T ) = 0 and obtain ν1 (ε, Q) + 1 = ν1 (1, ε −1 Q) + 1  K δ −n .

(4.13)

From (4.9), (4.11) and (4.13) we have εn I1 = ε n {ν1 (ε,
0 (ε)) + 1} 

r (ε) 

ε n {ν1 (ε, Q j (ε)) + 1}

j=1

 δ n [ν1 (δ, U ) + 1]

 j∈L 0

|Q j | + K ε n δ −n

n−1   k=0 j∈L k

p j (ε, δ, k).

(4.14)

282

W. D . EVANS

et al.

Let
be any GRD with associated tree  , and, in the notation of [4, 6], set (T F)(x) := (F ◦ τ )(x), x ∈
,  1 (M f )(t) := f (x)d x, |Bt | Bt

(4.15) (4.16)

where Bt = B(u(t), ρ(t)), t ∈  , and set ψ(s) :=

ds , u(s) := ψ(s)1/ p , v(s) := ψ(s)−1/ p , dµ

(4.17)

where ds denotes Lebesgue measure on  . The maps T1 : [F]  → [T F] : L C ( , dµ) → WC (
), p p T0 : [F]  → [T F] : L C ( , dµ) → L C (
), 1, p

1, p

M1 : [ f ]  → [M f ] : WC (
) → L C ( , dµ), p p M0 : [ f ]  → [M f ] : L C (
) → L C ( , dµ) 1, p

1, p

(4.18)

are well defined, and we have a commuting diagram Fig. 1. The following identities have important implications for the proof: M0 E C = I C M1 , E C T1 = T0 IC ;

(4.19) (4.20)

we suppress the dependence on
and  to simplify the notation. From E C = E C − T0 M0 E C + T0 M0 E C we have by [3, Proposition II.2.2] am (E C )  E C − T0 M0 E C  + am (T0 M0 E C ) and, on using (4.19), am (T0 M0 E C ) = am (T0 IC M1 )  T0 am (IC M1 )  am (IC )M1 .

(4.21)

Furthermore, from [6, Lemma 3.4], E C − T0 M0 E C   c(
) sup(ρ ◦ τ )(x) =: c(
)k(
),


we have that

c(
)  ζ −1 (α + 1)n c p {α + 1 + 2n+1 c p },

where ζ, α are the constants in Definition 1.1, and c p is the norm of the maximal function as a

ON THE APPROXIMATION NUMBERS OF SOBOLEV EMBEDDINGS

WC (
)  1, p

M1

-

283

L C ( , dµ) 1, p

T1 EC

IC ?

M0

L C (
)  p

-

? L C ( , dµ) p

T0 Fig. 1

map on L p (Rn ). In view of the hypothesis, when
=
i (ε), i = 1, 2, . . . , q(ε), the associated constants c(
) are bounded independently of ε. Hence, am (E C )  ck(
) + M1 am (IC ).

(4.22)

It follows for
=
i (ε), i = 1, 2, . . . , q(ε), that am (IC ( )) < ε and hence

⇒

(i)

am (E C (
)) < [c + M1 ]ε

 (i) νC (ε, i )
νC [c(i) + M1 ]ε,
i (ε) ,

i = 1, 2, . . . , q(ε),

(4.23)

(4.24)

(i)

where M1 is the map M1 in (4.18) when
,  are
i (ε), i (ε) respectively; from [4, Lemma 4.2; (i) 6, Lemma 3.3], M1   c p (α + 1)n which is bounded independently of ε by hypothesis. In view of Lemma 3.3 and Lemma 3.4, (4.24) yields  (i) ν1 (ε, i (ε))
ν1 c( p, 1)[c(i) + M1 ]ε,
i (ε) , (4.25) and in (4.6), ε n I2 

q(ε) 

ε n {ν1 (ci ε, i (ε)) + 1},

(4.26)

i=1

 (i) −1 where ci = c( p, 1)[c(i) + M1 ] . For the
i (ε) in I3 , we have that 1, p

p

E C |WC (
i (ε)) → L C (
i (ε))  ε and hence ν1 (ε,
i (ε)) = 0,

i = q(ε) + 1, · · · , q(ε) + h(ε).

The existence of the limit L U and the identity (4.5) are proved in [6, Theorem 5.7]. The theorem is therefore proved.

284

W. D . EVANS

et al.

R EMARK 4.2 In the hypothesis of Theorem 4.1,
is predominantly composed of the set
0 (ε) which is a union of rectangular blocks, the remainder being made up of q(ε) GRDs of ‘small width’ and h(ε) sets satisfying the Poincar´e inequality and having ‘small diameter’. The GRDs are irregular portions of
(like cusps or spirals) left over after
0 (ε) has been removed, while the Poincar´e domains are less singular. If we choose δ = δ(ε), say, in (4.4), the first term on the right-hand side is the main term as ε → 0, except in pathological cases when the other terms might dominate and yield a non-standard asymptotic estimate even in the case p = 2 for the counting function of the eigenvalues of the Neumann Laplacian. In all cases an appropriate choice of δ is crucial. R EMARK 4.3 In the estimate (4.4), the limit L U and the error δ n [ν1 (δ, U ) + 1] − L U are not known for p = 2. When p = 2, ν1 (δ, U ) = νC (δ, U ) = N (δ −n , − N ,U ) = (2π )−n ωn δ −n + O(δ −n+1 ) as δ → 0, where ωn is the volume of the unit ball in Rn ; see [6, (5.12)]. When n = 1, it follows from (2.14) that lim tνC (t, U ) =

t→0

αp 2

and hence, from (3.17), αp αp  L U  c( p, 1) , 2 2 where c(2, 1) = 1 and c( p, 1) = 2 for p = 2. R EMARK 4.4 In Theorem 4.2, the rectangular blocks Q j (ε), j = 1, 2, · · · , r (ε) in
0 (ε) can be replaced by any similar sets which fit together to form a tesselation of
0 (ε). This observation can greatly ease the application of the result. For instance, to apply our methods and results to the Koch snowflake, the choice of equilateral triangles, or regular hexagons, is appropriate. The proof of Theorem 4.1 goes through as before on noticing that [6, Theorem 5.7] continues to hold, mutatis mutandis, for a regular polyhedron Q, namely lim {ε 2 ν1 (ε, Q)} = inf ε 2 [ν1 (ε, Q) + 1].

ε→0

ε>0

(4.27)

Recent research has uncovered a great deal of information about the quantity ν1 (ε, ) which features on the right-hand side of (4.4). Before we can comment further, we need to establish the connection with the Hardy operator Hc : L p () → L p () on a tree  defined by  t (Hc F)(t) := v(t) F(s)u(s)ds, c, t ∈ . (4.28) c

For F ∈ L 1, p (, dµ) we have



t

F(t) − F(c) = c





t

F (s)ds =

F (s)v(s)u(s)ds

c

by (4.17). This can be written as F(t) − F(c) = (Rc D F)(t),

(4.29)

ON THE APPROXIMATION NUMBERS OF SOBOLEV EMBEDDINGS

285

where D : [F]  → F v : L C (, dµ) → L p () 1, p

(4.30)

and 

t

(Rc F)(t) =

F(s)u(s)ds;

Rc : L p () → L p (, dµ).

(4.31)

c

Note that D is surjective, and is isometric since D F p, = F  L p (,dµ) = F L p (,dµ) .

(4.32)

IC () = V Rc D = V v −1 Hc D,

(4.33)

C

Also, (4.29) gives

where V : L p (, dµ) → L C (, dµ) is the canonical map and v −1 is multiplication by 1/v; the map F  → v −1 F is an isometry between L p () and L p (, dµ). The identity (4.33) leads to the following estimate for the approximation numbers of IC () in terms of those for Hc . p

L EMMA 4.5 Let Hc be the Hardy operator defined in (4.28) with u, v given in (4.17), and so uv = 1. Then 1 2 am+1 (Hc )

 am (IC ())  am (Hc ),

m ∈ N0 ,

(4.34)

and hence ν(2ε, Hc )  νC (ε, )  ν(ε, Hc )

(4.35)

where ν(ε, Hc ) := max{m ∈ N0 : am (Hc )
ε}. Proof. From (4.33) and [3, Proposition II.2.2(iii)], we have am (IC ()  V v −1 Dam (Hc )  am (Hc ) since v −1 and D are isometries and V   1. To derive the first inequality in (4.34), define A : L p (, dµ) → C to be the one-dimensional operator  1 A f := f dµ. µ()  Denote the kernel of A by N and the restriction of V to N by VN . Then VN is one–one, since VN f = VN g implies that [ f − g] = 0 and hence f − g = c, a constant. But this means that p c = A( f − g) = 0 and so f = g. Furthermore VN has range L C (), for [ f ] = [ f − A f ] and f − A f ∈ N . Thus VN (I − A) f = [ f ] = V f,

(4.36)

286

W. D . EVANS

et al.

where I is the identity on L p (). This gives in (4.33) am (IC ()) = am (VN (I − A)Rc D), Rc = v −1 Hc  1, p p = inf VN (I − A)Rc D − P|L C (, dµ) → L C (, dµ) : rankP < m   p = inf VN Rc − A Rc − VN−1 P D −1 |L p () → L C (, dµ) : rankP < m 1, p

since D is an isometry of L C (, dµ) onto L p (). From (4.36), VN−1 [ f ] L p (,dµ)  2[ f ] L p (,dµ) and so

VN−1 |L C (, dµ) → L p (, dµ)  2; p

hence A Rc + VN−1 P D −1 : L p () → L p (, dµ) is bounded and of rank at most rankP +1 < m +1. It follows that  am (IC ())
VN−1 −1 inf Rc − Q|Q : L p () → L p (, dµ) : rankQ < m + 1 1 1
am+1 (Rc ) = am+1 (Hc ) 2 2 since v −1 : L p (, dµ) → L p () is an isometry. The lemma is therefore proved. R EMARK 4.6 In the case when  is an interval (c, ∞), Hc is bounded if Muckenhaupt’s A p condition holds, namely 


t

A p := sup

c 1. This case is of special interest as it is the most pathological. Its boundary has outer and inner Minkowski dimensions do , di respectively, given by do =

log 2 , log(1/c)

di = 1 −

log 2 1 + α α log(1/c)

(6.3)

and the upper Minkowski contents of
and its complement are finite; see [6, section 6.2.2]. Furthermore, 1 < di < do < 2. Let
m :=



{Q : Q ∈ i },


\
m =

i
m

2m 

Ai,m .

(6.4)

i=1

For each i, Ai,m is a GRD which is similar to
. To apply Theorem 4.1, we choose
0 (ε) =
m for an appropriate choice of m, with the remainder of
made up of the GRDs Ai,m . In Ai,m , the generalized ridge is the tree rooted at the point O in Fig. 4, and the map τ is the projection of PA onto OL (or what corresponds to SL in the first rectangle) and BC onto LR. The following facts are established in [6, section 6]. The symbol ≈ means that the quotient of the two sides exhibited is bounded between positive constants. • For x ∈ Q m and t on the part of the generalized ridge in Q m |∇τ (x)| ≈ 1, dµ ≈ αm , dt |ρ (t)|  1.

(6.5)

ON THE APPROXIMATION NUMBERS OF SOBOLEV EMBEDDINGS

293

• ν1 (ε, Ai,m ) = 0 if m
m(ε), where m(ε) =

log(1/ε) + O(1). log(1/c)

(6.6)

6.1. Upper bound We apply Theorem 4.1 with 


0 (ε) =

{Q : Q ∈ i },

i
m(ε)


i (ε) = Ai,m(ε) , i = 1, · · · q(ε).

(6.7)

Note that the number n i of rectangles in i satisfies n 0 = 1, n i = 2i−1 , i
1 , and hence in the notation of Theorem 4.1, r (ε) = 1 +

m(ε) 

2i−1 ,

q(ε) = 2m(ε) .

i=1

From Theorem 4.1 we have ε2 [ν1 (ε,
) + 1]  ε 2 {ν1 (ε,
0 (ε)) + 1} 1     δ 2 [ν1 (δ, U ) + 1] |Q j (ε)| + K ε 2 δ −2 p j (ε, δ, k) + ε 2 q(ε),(6.8) j∈L 0

k=0 j∈L k

where l1 (ε, j) = 2cα j and l2 (ε, j) = 2c j . Furthermore, j ∈ L 0 if and only if l1 (ε, j)
ε/δ and hence j  j0 , where j0 = j0 (ε) :=

log(2δ/ε) . α log(1/c)

(6.9)

Thus from (6.8) ε2 [ν1 (ε,
) + 1]  δ 2 [ν1 (δ, U ) + 1] +K ε2 δ −2





n j |Q j |

j
j0

n j p j (ε, δ, 0) +

j
j0



 n j + ε 2 q(ε).

j> j0

In (6.10), on noting that for j ∈ L 0 , εδ −1 < cα j /2 < 1, it readily follows that  j
j0

n j |Q j |  |
0 (ε)|,

(6.10)

294

W. D . EVANS



n j p j (ε, δ, 0) =

j
j0

j0 

et al.

 n j 4δε −1 (cα j + c j ) + 1

j=0

=O(δε−1 

=O δ

j0 

(2c) j )

j=0 1+1/α(d0 −1) −1−((d0 −1)/α)

ε



=O(δε−d0 ), 

nj 

j> j0

m(ε) 

2 j−1

j=1

2m(ε) = O(ε −d0 ).

(6.11)

Hence, we have ε2 [ν1 (ε,
) + 1]  δ 2 [ν1 (δ, U ) + 1]|
0 (ε)| + O(δ −2 ε 2−d0 ).

(6.12)

In particular, this gives lim sup{ε 2 ν1 (,
)}  L U |
|. ε→0

6.2. Lower bound We now take
0 (ε) =



(6.13)

{Q : Q ∈ i },

i
m−1


i (ε) = Q ∈ m ,

i = 1, 2, · · · , q(ε),

(6.14)

where m = m(ε). Thus the GRDs
i (ε) are the rectangles of side 2cαm × 2cm in m , and q(ε) = 2m . We therefore have, using | · | to denote Lebesgue measure, |i (ε)| = 2(cαm + cm ). Also, in (5.3), |Jt | ≈ αm = cαm and φ(s) ≈

 {x:τ (x)=s∈Jt }

dθ(x)
2cα(m+1) .

Since dµ/dt ≈ cαm from (6.5), it is readily seen that
  p/ p  ε− p φ − p / p ds dµ(t) = O(ε 1+ p(α/ p −1) ). i (ε)

Jt

Hence (5.3) is satisfied since 1 + p(α/ p − 1) > 1 + p(1/ p − 1) = 0, and, in view of (5.5), we have for the GRDs
i (ε) ε2

q(ε)  i=1

µ0 (ε,
i (ε))
K ε2m (cαm + cm )
K ε(2c)m ≈ ε 2−d0 .

295

ON THE APPROXIMATION NUMBERS OF SOBOLEV EMBEDDINGS

Consequently Theorem 5.1 yields 

ε µ0 (ε,
)
δ µ0 (δ, U ) 2

2

n j |Q j | − K εδ

−1

j
j0



 n j l2 (ε, j)

j
j0

+K ε2−d0 .

(6.15)

In (6.15) 

n j |Q j | = |
| − O

j
j0



[2c1+α ] j = |
| − O

j> j0



1−((d0 −1)/α)  ε , δ

and εδ

−1





n j l2 (ε, j) = O εδ

j
j0

−1



(2c)

j
j0

j





1−((d0 −1)/α)  ε . =O δ

Since δ 2 µ0 (δ, U ) is bounded, for δ ∈ (0, 1) by (5.2), Theorem 5.1 therefore gives

1−((d0 −1)/α)  ε ε µ0 (ε,
)
δ µ0 (δ, U )|
| − O + K ε 2−d0 . δ 2

2

(6.16)

Note that 1 − ((d0 − 1)/α) > 2 − d0 > 0. On choosing δ = δ(ε), where δ(ε)1−((d0 −1)/α) = ε (1−1/α)(d0 −1)−θ for 0 < θ < (1 − 1/α)(d0 − 1), we have
1−((d0 −1)/α) ε = ε 2−d0 +θ . δ Hence, we have as ε → 0, ε2 µ0 (ε,
))
δ(ε)2 µ0 (δ(ε), U )|
| + K ε 2−d0 − O(ε 2−d0 +θ ),

(6.17)

where K and θ are positive constants. In [6, Theorem 6.3] it is proved that when p = 2, ε2 µ0 (ε,
)) −

1 |
| ≈ ε 2−d0 . 4π

(6.18)

7. Example 2 Our second example is the horn-shaped domain 
= x = (x1 , x2 ) : |x2 | < (x1 ), 0 < x1 < ∞ , where θ

(t) = e−t ,

θ > 2.

(7.1)

296

W. D . EVANS

et al.

It is shown in [4, section 6.3] that this is a GRD with generalized ridge (0, ∞), τ (x1 , x2 ) = x1 and dµ θ = (t) = e−t . dt Furthermore, the embedding W 1, p (
) → L p (
), and hence E 1 (
) by [4, Theorem 2.10], is compact; in fact θ > 1 is enough to ensure this, but taking θ > 2 simplifies the analysis to follow. For R > 0, set
R := {x ∈
: x1 > R}. Then we have the Poincar´e inequality  f − f
R  p,
R  D(
R )∇ f  p,
R , where by (4.22) and [5, section 6] there exists a constant K depending on
but not on R such that  D(
R )  K k(
R ) + N R , with k(
R ) = sup ρ ◦ τ,
R


 NR =

sup

R
X K 0 ε −1/(θ−1) .

(7.5)

ON THE APPROXIMATION NUMBERS OF SOBOLEV EMBEDDINGS

Let lε = ε 1−σ , where σ ∈ (0, 1), and set   K 0 ε−1/(θ−1) kε := 1 + ≈ ε σ −θ/(θ−1) , lε

297

(7.6)

Rε := kε lε ≈ ε −1/(θ−1) ,

(7.7)

where [·] denotes the integer part. Then D(
Rε ) < ε and hence ν1 (ε,
Rε ) = 0.

(7.8)

We now partition
as follows:
=

kε  

Q j ∪ T j+ ∪ T j− ∪
Rε ∪ N ,

(7.9)

j=1

where N is a null set and   Q j = ( j − 1)lε , jlε × − ( jlε ), ( jlε ) ,  T j+ = x = (x1 , x2 ) : ( j − 1)lε < x1 < jlε , ( jlε ) < x2 < (x1 ) ,  T j− = x = (x1 , x2 ) : ( j − 1)lε < x1 < jlε , −(x1 ) < x2 < −( jlε ) . The sets T j+ , T j− are GRDs with  j = (( j − 1)lε , jlε ), τ the vertical projection, u any Lipschitz map of  j into T j+ or T j− and dµ/dt ≈ |(t) − ( jlε )|. It is readily seen that the values of the constants α, β, γ , ζ in Definition 1.1 can be chosen to be independent of ε and j; they compare with the corresponding constants for
. From [6, Lemma 5.3] and (7.9) ν1 (ε,
) 

kε  

kε         ν1 (ε, Q j ) + 1 + ν1 (ε, T j+ ) + 1 + ν1 (ε, T j− ) + 1 + ν1 (ε,
Rε ) + 1

j=1

j=1

= I1 + I2 + I3 ,

(7.10)

say. We already know from (7.8) that I3 = 1. We use the same technique as in Example 1 to estimate I1 . For the shorter of the side lengths li (ε, j), i = 1, 2, of Q j we now have

θ if 2e−( jlε )
lε , lε l1 (ε, j) = θ θ 2e−( jlε ) if 2e−( jlε ) < lε .   1/θ  Hence, on setting n ε := lε−1 ln(2lε−1 ) ≈ ε −1+σ {ln(ε −1+σ )}1/θ , we have

if j  n ε , lε l1 (ε, j) = θ 2e−( jlε ) if j > n ε . We now repeat the argument used for (6.10) to obtain, with
0 :=

k ε

j=1

Q j,

kε      ε2 I1  δ 2 ν1 (δ, U ) + 1 |
0 | + K εδ −1 l1 (ε, j) + l2 (ε, j) + K ε 2 δ −2 kε . j=1

298

W. D . EVANS

et al.

Since kε 

l2 (ε, j) =

j=1


if σ