Mathematics
DISSERTATION Karlstad University Studies
2010:16
Robert Algervik
Embedding Theorems
for Mixed Norm Spaces
Karlstad University Studies
2010:16
Robert Algervik
Embedding Theorems
for Mixed Norm Spaces
DISSERTATION
Karlstad University Studies 2010:16 ISSN 1403-8099
ISBN 978-91-7063-306-5 © The Author
Distribution: Karlstad University
Faculty of Technology and Science Mathematics
651 88 Karlstad Sweden
+46 54 700 10 00 www.kau.se
Embedding Theorems for Mixed Norm Spaces and Applications Robert Algervik
Department of Mathematics Karlstad University
Abstract This thesis is devoted to the study of mixed norm spaces that arise in connection with embeddings of Sobolev and Besov type spaces. We study different structural, integrability, and smoothness properties of functions satisfying certain mixed norm conditions. Conditions of this type are determined by the behaviour of linear sections of functions. The work in this direction originates in a paper due to Gagliardo (1958), and was further developed by Fournier (1988), by Blei and Fournier (1989), and by Kolyada (2005).
Here we continue these studies. We obtain some refinements of known embeddings for certain mixed norm spaces introduced by Gagliardo, and we study general properties of these spaces. In connection with these results, we consider a scale of intermediate mixed norm spaces, and prove intrinsic embeddings in this scale.
We also consider more general, fully anisotropic, mixed norm spaces. Our main theorem states an embedding of these spaces to Lorentz spaces. Applying this result, we obtain sharp embedding theorems for anisotropic Sobolev-Besov spaces, and anisotropic fractional Sobolev spaces. The meth-ods used are based on non-increasing rearrangements, and on estimates of sections of functions and sections of sets. We also study limiting relations between embeddings of spaces of different type. More exactly, mixed norm estimates enable us to get embedding constants with sharp asymptotic be-haviour. This gives an extension of the results obtained for isotropic Besov spaces by Bourgain, Brezis, and Mironescu, and for anisotropic Besov spaces by Kolyada.
We study also some basic properties (in particular the approximation properties) of special weak type spaces that play an important role in the construction of mixed norm spaces, and in the description of Sobolev type embeddings.
In the last chapter, we study mixed norm spaces consisting of functions that have smooth sections. We prove embeddings of these spaces to Lorentz spaces. From this result, known properties of Sobolev-Liouville spaces fol-low.
Acknowledgements I thank my supervisor Professor Viktor I. Kolyada for guiding me in this area of mathematics. I am especially grateful to him for his patience, and for always taking the time to answer my questions. I also thank my family and my friends and colleagues for their encouragement and support.
This work was financially supported by the Graduate School in Mathe-matics and Computing (FMB).
Contents
1. Introduction 1
1.1. Antecedent results 1
1.2. Main objectives 5
1.3. Summary 5
2. Definitions and auxiliary propositions 11
2.1. Hardy’s inequalities 11
2.2. The non-increasing rearrangement 12
2.3. Lorentz spaces 17
2.4. Iterative rearrangements 20
3. Some geometric results 25
4. The spaces Λσ 31
4.1. Some general properties of the spaces Λσ 31
4.2. Approximation in Λσ 34
5. Mixed norm spaces 47
5.1. Some lemmas 47
5.2. The main theorem 56
6. Applications 65
6.1. Anisotropic Sobolev-Liouville spaces 65
6.2. Limiting embeddings and anisotropic Sobolev-Besov spaces 73
7. On Fournier’s theorem 79
7.1. Iterative rearrangement inequalities 79
7.2. Intermediate embeddings 84
7.3. Sobolev spaces 92
7.4. Limiting relations 93
7.5. On relations between the spaces Vp 98
8. Functions with smooth sections 103
8.1. Embeddings to Lorentz spaces 104
8.2. Embeddings to classes of smooth functions 110
8.3. Sobolev-Liouville spaces 125
1. Introduction
This thesis relates to one of the fundamental directions in the theory of function spaces - embeddings and inequalities. We study different struc-tural, integrability, and smoothness properties of functions satisfying certain mixed norm conditions. Conditions of this type are determined by the be-haviour of linear sections of functions. Initially they arose in the works of Gagliardo (1958) and Fournier (1988) in connection with embeddings of Sobolev spaces.
1.1. Antecedent results. A function f ∈ Lp(Rn), 1 ≤ p < ∞, is said to belong to the Sobolev space Wp1(Rn) if f has usual (weak) derivatives Dkf ∈ Lp(Rn) for all 1 ≤ k ≤ n. In 1938 Sobolev proved the following, now
classical, theorem.
Theorem 1.1. Let n ≥ 2, 1 < p < n, and q = np/(n − p). If f ∈ Wp1(Rn) then f ∈ Lq(Rn) and kf kq≤ c n X k=1 kDkf kp. (1.1)
It was first in 1958 that this theorem was extended to the case p = 1. This was done independently by Gagliardo and Nirenberg. The next lemma was the central part of Gagliardo’s approach (see [17]). We use the notation ˆ
xk for the vector in Rn−1 obtained from a given vector x ∈ Rn by removing
its kth coordinate.
Lemma 1.2. Let n ≥ 2. Assume that the functions gk ∈ L1(Rn−1), k =
1, ..., n, are non-negative. Then Z Rn n Y k=1 gk(ˆxk)1/(n−1)dx ≤ Yn k=1 Z Rn−1 gk(ˆxk)dˆxk 1/(n−1) . Let f ∈ W11(Rn). For almost every x ∈ Rn,
|f (x)| ≤ 1 2
Z
R
|Dkf (x)|dxk≡ gk(ˆxk), k = 1, ..., n. (1.2)
As usual, we let p0denote the H¨older conjugate of p, i.e. we set p0 = p/(p−1) (1 ≤ p ≤ ∞). Applying Lemma 1.2, we obtain
kf kn0 ≤ 1 2 Yn k=1 kDkf k1 1/n .
This implies Theorem 1.1 for p = 1. However, one can obtain a stronger statement from Lemma 1.2. Let
Vk ≡ L1xˆk(R
n−1)[L∞
be the space with the mixed norm
kf kVk ≡ kΨkkL1(Rn−1),
where
Ψk(ˆxk) = ess supxk∈R|f (x)|.
We say that the L1-norm is the “exterior” norm of Vk and the L∞-norm is
the “interior” norm. We shall also denote V =
n
\
k=1
Vk. (1.3)
Applying Lemma 1.2 to the functions Ψk gives the following theorem.
Theorem 1.3. Let n ≥ 2. If f ∈ ∩n k=1Vk, then f ∈ Ln 0 (Rn) and kf kn0 ≤ Yn k=1 kf kVk 1/n .
For f ∈ W11(Rn), inequality (1.2) gives kf kVk ≤
1
2kDkf k1. (1.4)
This estimate and Theorem 1.3 imply inequality (1.1) for p = 1.
For a measurable set E ⊂ Rm, we denote by mesmE the Lebesgue measure
of E in Rm.
Let S0(Rn) be the class of all measurable almost everywhere finite
func-tions f on Rnsuch that the distribution function λf(y) = mesn{x ∈ Rn: |f (x)| > y}
is finite for all y > 0. Let f∗ denote the non-increasing rearrangement of a function f ∈ S0(Rn) (the definition is given in Section 2.2). If 0 < q, p <
∞, then the Lorentz space Lq,p(Rn) is defined as the class of all functions
f ∈ S0(Rn) such that kf kq,p= Z ∞ 0 t1/qf∗(t)pdt t 1/p < ∞.
For any fixed q, the Lorentz spaces increase as the secondary index p in-creases (see Section 2.3 below).
It is well known that the left-hand side in (1.1) can be replaced by the stronger Lorentz norm (see [14], [39], [40], and [42]). That is, the following theorem holds.
Theorem 1.4. Let n ≥ 2 and 1 ≤ p < n. Set q = np/(n − p). If f ∈ Wp1(Rn), then f ∈ Lq,p(Rn) and kf kq,p≤ c n X k=1 kDkf kp. (1.5)
In [16], Fournier proved this theorem for p = 1, using the following re-finement of Theorem 1.3.
Theorem 1.5. Let n ≥ 2. If f ∈ ∩nk=1Vk, then f ∈ Ln
0,1 (Rn) and kf kn0,1≤ n0 Yn k=1 kf kVk1/n. (1.6)
Observe that for the characteristic function of the unit cube in Rn we have equality in (1.6). Thus, the constant n0 is optimal.
Some extensions of Theorem 1.5 were obtained in the paper [8] due to Blei and Fournier. In particular, it was proved that for any 1 < r ≤ ∞
kf kq,1 ≤ c n X k=1 kf k Vk(r), (1.7) where q = nr/(nr − r + 1) and Vk(r) = L1xˆk(Rn−1)[Lrxk(R)] (k = 1, . . . , n).
It was shown in [16], [36] that the preceding results give a sharpening of some inequalities for bilinear forms proved by Hardy and Littlewood.
In view of (1.4), Theorem 1.5 immediately implies Theorem 1.5 for p = 1. Fournier [16, p. 66] observed that it was not clear how the methods based on mixed norm estimates could be applied to obtain (1.5) also for 1 < p < n. This problem was studied by Kolyada in [29]. He introduced a scale of more general mixed norm spaces in which the interior norms are defined by conditions on the rearrangements with respect to specific variables. These conditions are expressed in terms of the “weak” spaces Λσ.
Let σ ∈ R. Denote by Λσ(R) the class of all functions f ∈ S0(R) such
that
kf kΛσ = sup
t>0
tσ(f∗(t) − f∗(2t)) < ∞. (1.8) If 0 < σ < ∞ and r = 1/σ, then Λσ = Lr,∞ (where Lr,∞ is the Marcinkiewicz space weak-Lr). If σ = 0, then Λσ coincides with the space weak-L∞introduced in [4]. If σ < 0, then (1.8) is a weak version of Lipschitz condition for the rearrangement (see Section 4).
Theorem 1.6. Let n ≥ 2. Assume that 1 ≤ p < ∞ and that αk, k = 1, ..., n,
are positive numbers such that α ≡ n n X k=1 1 αk −1 < n p. (1.9) Set σk= 1 p − αk, Vk≡ L p ˆ xk(R n−1)[Λσk xk(R)],
and q = np/(n − αp). Suppose that f ∈ S0(Rn) and f ∈ ∩nk=1Vk. Then
f ∈ Lq,p(Rn) and kf kq,p ≤ c n Y k=1 kf kα/(nαk) Vk , (1.10) where c = cn Yn k=1 (nαk− α)α/(nαk) −1/p (1.11) and cn depends only on n.
Observe that Theorem 1.6 remains true for α = n/p, q = ∞ (the space L∞,p is defined in Section 2.3).
It follows from Theorem 1.6, that the interior Lr-norm on the right-hand side of (1.7) can be replaced by the weaker Lr,∞-norm for 1 < r < ∞, and by the norm in weak-L∞ (see (4.4) below) for r = ∞.
It was proved in [29] (see Lemma 6.4 below) that if a function f ∈ Lp(Rn) has a usual (weak) derivative Dkf ∈ Lp(Rn), then
kf k
Lpˆxk[Λ1/p−1xk ]≤ ckDkf kp, 1 ≤ p < ∞.
Hence, there holds the embedding Wp1(Rn) ⊂ n \ k=1 Lpxˆ k(R n−1)[Λ1/p−1 xk (R)]. (1.12)
We now obtain Theorem 1.4 in two steps. The first (and simplest) step is (1.12) and the second step is Theorem 1.6 with α1= · · · = αn= 1. Observe
that no smoothness condition is imposed on the functions in the second step.
In [29], Theorem 1.6 was also applied to obtain optimal constants in embeddings of anisotropic Besov spaces.
1.2. Main objectives. As we can see, the use of mixed norm estimates clarifies the role of smoothness conditions in the embedding theorems for Sobolev and Besov type spaces. Moreover it was shown in [29], and it will be seen below in Section 6.2 and 6.1, that such estimates provide a flexible method which can be applied to the study of different types of function spaces.
The general objective of this thesis is the further study of mixed norm spaces. More concretely, we shall study the following:
• some general properties of the Fournier-Gagliardo space V (see (1.3)) and the spaces Λσ;
• refinements of embeddings of the Fournier-Gagliardo space V ; • a scale of intermediate mixed norm spaces, and intrinsic embeddings
in this scale;
• an extension of Theorem 1.6 to more general mixed norm spaces; • embeddings of mixed norm spaces of functions with smoothness
con-ditions on linear sections.
1.3. Summary. In what follows we give a brief description of the main content of this thesis.
Section 2 contains general definitions and known results. In particular, we define and consider some basic properties of the non-increasing rearrange-ment, the Lorentz spaces, and the iterative rearrangement.
In Section 3 we obtain some properties of the Fournier-Gagliardo space V , defined in (1.3). The most interesting result is the observation that kf kVk
has a clear geometric interpretation: it is the n-dimensional measure of the essential projection of the set
{(x, y) ∈ Rn× [0, ∞) : 0 ≤ y ≤ |f (x)|},
onto the hyperplane xk= 0 (see Theorem 3.3 below). We also show that V
is not invariant under rotation.
In Section 4 we study the space Λσ, defined in (1.8). As follows from the above, and as we will see in Section 6 below, the spaces Λσ have a relevant role in the description of Sobolev-type embeddings. This motivates us to study the basic properties of these spaces. As it was mentioned above, we show that Λσ relates to known spaces, in particular the Marcinkiewicz spaces. We also show that Λσ ⊂ L∞(R), for σ < 0.
Results for approximation of functions in Λσ are given in Section 4.2. We will see that approximation in the “norm” on Λσ behaves badly. However, we have obtained some positive results on approximation of functions f in this space. Our main result for the space Λσ is the following theorem.
Let C0(R) denote the class of all continuous functions with bounded
Theorem 1.7. Let f ∈ Λσ (σ ∈ R). Then there exists a sequence {fk},
fk∈ C0(R), such that {fk} converges to f in measure and kfkkΛσ → kf kΛσ.
This is in fact a special case of the more general result obtained in Theo-rem 4.10. Observe that this theoTheo-rem is similar to known results for approx-imation in variation (see [49], [24, Section 9.1]).
In Section 5 we prove our main result, Theorem 1.8 below. It is an extension of Theorem 1.6. This section also includes some relevant lemmas. In Theorem 1.4 all derivatives Dkf belong to the same space Lp(Rn).
Nevertheless, it is quite reasonable to suppose that the functions Dkf ,
k = 1, . . . , n, belong to different spaces Lpk(Rn). Such conditions
natu-rally appear in embedding theory as well as in applications. Furthermore, many authors have studied Sobolev and Besov spaces whose construction involves, instead of Lp-norms, norms in more general spaces - first of all, in the Lorentz spaces.
Therefore it is natural to study mixed norm spaces which are anisotropic not only with respect to interior norms, but also with respect to exterior norms. The main problem considered in this work is to extend Theorem 1.6 to these, more general, mixed norm spaces. This extension is given by Theorem 5.4, and it states in particular the following.
Theorem 1.8. Let n ≥ 2, 1 ≤ p1, . . . , pn, s1, . . . , sn< ∞, and α1, . . . , αn>
0. Put α = n n X k=1 1 αk −1 , p = n α Xn k=1 1 αkpk −1 , and s = n α Xn k=1 1 αksk −1 . Assume that p < n/α and put q = np/(n − αp). Set
σk=
1 pk
− αk, and Vk = Lxpˆkk,sk(Rn−1)[Λσxkk(R)],
and assume that
1 p−
α
n− σk> 0, for k = 1, . . . , n. Suppose that
f ∈ S0(Rn) and f ∈ n \ k=1 Vk. Then f ∈ Lq,s(Rn) and kf kq,s≤ c n Y k=1 kf kα/(nαk) Vk , (1.13)
We have obtained the constant in (1.13) explicitly. This explicit value is used in Section 6, where we consider applications of Theorem 1.8.
As we will show, Theorem 1.8 holds in the case p = n/α as well.
The proof of Theorem 1.8 is based on the approach given in the works of Kolyada [29] and Kolyada and P´erez [32].
In Section 6 we apply Theorem 1.8 to obtain sharp embedding theo-rems for anisotropic Sobolev-Liouville spaces and anisotropic Sobolev-Besov spaces. We also study limiting relations between embeddings of spaces of different type. More exactly, mixed norm estimates enable us to get embed-ding constants with sharp asymptotic behaviour. This gives an extension of the results obtained for isotropic Besov spaces Bpα by Bourgain, Brezis, and Mironescu [11], and for Besov spaces Bα1,...,αn
p by Kolyada [29].
As follows from the exposition given above, the Fournier-Gagliardo space V =
n
\
k=1
L1xˆk(Rn−1)[L∞xk(R)]
(from (1.3)) appears naturally in connection with embeddings of the space W11(Rn). In this work we continue the study of embeddings of the space V . In Section 7.1 we prove that V is embedded to the modified Lorentz space defined in terms of iterative rearrangements (see Section 2.4). In particular, V is embedded into the space Ln0,1(Rn), which is strictly smaller
that Ln0,1(Rn). Thus our result is a refinement of Fournier’s inequality (1.6). We observe that this result was inspired by embeddings of Sobolev spaces into the Lorentz spaces Lq,p(Rn) proved in [28].
It is natural to study the intrinsic relations between different mixed norm spaces. In Section 7.2 we introduce a one parameter family of spaces, con-taining the space V . This scale is formed by the spaces
Vp = n \ k=1 Lp,1xˆ k(R n−1)[Lrp,1 xk (R)],
1 ≤ p ≤ (n − 1)0, rp = p0/(n − 1). For p = 1 we have V1 = V and the
norms coincide, so the space V is included in the scale Vp. Further, for 1 < p ≤ (n − 1)0, we prove that V ⊂ Vp, and
kf kVp j ≤ ckf k 1/r0p Vj Y k6=j kf k1/pV 0 k , j = 1, . . . , n, (1.14)
for 1 < p ≤ (n − 1)0, rp = p0/(n − 1), f ∈ V . We obtain also some results
concerning the optimality of the estimates for the Vp-norms (Remark 7.16 and Theorem 7.28 below).
Embeddings of the space Vp are closely connected with Theorem 1.8. Namely, using this theorem, we prove
kf kn0,1≤ c n Y k=1 kf k1/n Vkp
(1 ≤ p < n0) for f ∈ Vp (when p = 1, this is inequality (1.6)).
In Section 7.3 we apply our results for the space Vp to obtain embedding
theorems for Sobolev spaces. We prove the inequality
n Y k=1 kf kVp k ≤ c n Y k=1 kDkf k1
(1 < p ≤ (n − 1)0) for all f ∈ W11(Rn). In particular this result states that W11(R2) ⊂ Vk2 = L2,1xˆ
k(R)[L
2,1
xk(R)], k = 1, 2.
This inclusion does not follow from the strong type Sobolev inequality (1.5), but it can be derived from known results for iterative rearrangements (Re-mark 7.23 below).
In Section 7.4, we consider some limiting relations for the Vp-norm. Re-call that inequality (1.14) holds for 1 < p ≤ (n − 1)0, and that V1 = V . In Theorem 7.26 below, we clarify the limiting behaviour of kf kVp
k, as p → 1+.
In Section 7.5 we obtain a result concerning the relations between spaces Vp with different values of p. In particular we see that these spaces do not form a monotone scale.
As is mentioned above, Theorem 1.6 (and its extension - Theorem 1.8) is closely related to embeddings of Sobolev spaces with smoothness not greater than 1. The objective of Section 8 is to obtain an extension of Theorem 1.6 related to Sobolev spaces of higher smoothness. In this case, the mixed norms from Theorem 1.6 are not suitable, since the definition of k · kΛσ
involves the rearrangement.
Instead we will consider mixed norm spaces with interior norms defined in terms of the modulus of continuity ωr(·, t) (see Section 8) of the sections of the function. For simplicity, we study only isotropic mixed norm spaces. Suppose λ > 0, and let r = r(λ) be the smallest integer such that λ < r. Let Cλ(R) be the space of all measurable functions ϕ on R such that
kϕk∗Cλ = sup
t>0
ωr(ϕ; t) tλ < ∞.
These seminorms will be used as interior norms instead of k·kΛσ. The mixed
norm spaces thus obtained, are denoted Ukp,λ = Lpxˆ
k(R
n−1)[Cλ
for 1 ≤ p < ∞ and λ > 0. We also put Up,λ= n \ k=1 Ukp,λ.
Observe that if 0 < α < 1, then kϕkΛ−α ≤ kϕk∗Cα (see Proposition 4.3
below).
Note that using the modulus of continuity of orders higher than r(λ) in the definition of k · k∗Cλ, would yield equivalent seminorms (Remark 8.16
below). We emphasize also that if λ ∈ N, then r(λ) = λ+1. If we considered instead the modulus of continuity of order λ, we would get a stronger (semi) norm. However, in our estimates, these norms will appear on the right hand side, and thus it is better to use weaker norms.
In Section 8.1, we consider the case when 0 < λ < (n − 1)/p, and we prove:
Theorem 1.9. Let 1 ≤ p < ∞, 0 < λ < (n − 1)/p, and q = np/(n − 1 − λp). Suppose that f ∈ S0(Rn), f ∈ Up,λ, and that
f∗(t) = O(t−1/q), t → ∞. (1.15) Then f ∈ Lq,p(Rn), and kf kq,p≤ c n X k=1 kf k Ukp,λ
where c depends only on p, λ, and n.
In the special case 1 ≤ p < ∞ and 0 < λ < min(1, (n − 1)/p), this theorem can be derived from Proposition 4.3 and Theorem 1.6. Moreover, the assumption (1.15) in Theorem 1.9 can be omitted for these values of p and λ (see Remark 8.6 below).
In Section 8.2, we first give some simple lemmas on the properties of the Steklov averages defined in (8.30). We then prove the following teorem. Theorem 1.10. Let 1 ≤ p ≤ ∞, and (n − 1)/p < λ < ∞. Suppose that f ∈ S0(Rn) and f ∈ Up,λ. Then there exists a bounded and uniformly
continuous function g ∈ Up,λ, such that f = g a.e. and kgkUp,λ k
= kf kUp,λ k
, k = 1, . . . , n. Moreover, if β ≡ λ − (n − 1)/p and s > β, s ∈ N, then
ωs(g; δ) ≤ cδβ n X k=1 kf kUp,λ k
We emphasize that this theorem does not hold for s = β, β ∈ N (see Remark 8.14).
In Section 8.3 we first prove some basic properties of functions with frac-tional derivatives. Such generalized derivatives, and the Sobolev-Liouville spaces, are defined in Section 6. The main result in Section 8.3 is the fol-lowing theorem.
Theorem 1.11. Let 1 ≤ p < ∞, α > 1/p, and λ = α − 1/p. Suppose that f ∈ Lα
p(Rn). Then there exists a function f0 ∈ Up,λ which is equivalent to
f and satisfies
kf0k
Ujp,λ ≤ ckD α
jf0kp, j = 1, . . . , n, (1.16)
where c depends only on n, p, and α.
For the Sobolev-Liouville spaces, inequality (1.16) plays the same role as (1.4) for W11(Rn).
We apply this theorem to illustrate that the properties of the mixed norm spaces Up,λare consistent with known results for the Sobolev-Liouville spaces Lαp(Rn) (see Theorem 8.22 and Theorem 8.23 below).
2. Definitions and auxiliary propositions
This section contains definitions and known results. Section 2.1 contains Hardy type inequalities that we need. In Section 2.2 we define the non-increasing rearrangement of a function and give some of its basic properties. This definition was first given by Hardy and Littlewood [19]. Estimates in terms of rearrangements will be important in the following sections. In Section 2.3 we introduce the Lorentz spaces. In the last section, Section 2.4, we consider iterative rearrangements.
2.1. Hardy’s inequalities. The next theorem was proved by Hardy (see e.g. [5, p. 124]).
Theorem 2.1. Let α > 0 and 1 ≤ p < ∞. If f is a non-negative measurable function on R+≡ (0, ∞) then Z ∞ 0 tα−1 Z ∞ t f (u)dupdt1/p ≤ p α Z ∞ 0 tp+α−1f (t)pdt1/p (2.1) and Z ∞ 0 t−α−1 Z t 0 f (u)du p dt 1/p ≤ p α Z ∞ 0 tp−α−1f (t)pdt 1/p . (2.2) If, as in the above theorem, f is a non-negative measurable function on R+ and α > 0, there hold the obvious inequalities
sup t>0 tα Z ∞ t f (u)du ≤ 1 αsupt>0 t1+αf (t) (2.3) and sup t>0 t−α Z t 0 f (u)du ≤ 1 αsupt>0 t1−αf (t). (2.4)
For u, v > 0, we let Γ(u) and B(u, v) denote the usual Gamma- and Beta-functions, respectively. Recall that
Γ(u) = Z ∞ 0 e−ttu−1dt (2.5) and B(u, v) = Γ(u)Γ(v) Γ(u + v) Z 1 0 tu−1(1 − t)v−1dt.
The next inequality is similar to Hardy’s inequality (2.1), but for the case 0 < p < 1 and for non-increasing functions. It was obtained by Bergh, Burenkov, and Persson (see [6, Corollary 3.7]). We stress that the constant in (2.6) is the best possible.
Theorem 2.2. Let f be a non-negative non-increasing function on R+.
Suppose that α > 0 and 0 < p < 1. Then Z ∞ 0 tα−1 Z ∞ t f (u)du p dt ≤ pB(p, α) Z ∞ 0 tα+p−1f (t)pdt. (2.6) Remark 2.3. A different proof of inequality (2.6) was given in [29, Lemma 2.5], but with the worse constant:
c = e 1 + p α .
2.2. The non-increasing rearrangement. Let X ⊂ Rnbe a measurable set and let f be a measurable function on X. For y ≥ 0 we define the distribution function of f by
λf(y) = mesn{x ∈ X : |f (x)| > y}.
Observe that λf may take the value ∞. If mesnX = ∞, we let S0(X) denote
the class of all measurable almost everywhere finite functions f on X, for which λf(y) < ∞ for all y > 0. Two functions f ∈ S0(X), X ⊂ Rn, and
g ∈ S0(Y ), Y ⊂ Rm, are said to be equimeasurable if
λf(y) = λg(y), y ≥ 0.
A non-negative and non-increasing function f∗ on R+ which is
equimeasur-able with f is said to be a non-increasing rearrangement of the function f ∈ S0(X). We will also assume that f∗ is left-continuous on R+. Under
this condition f∗ is defined uniquely by (see [33, p. 59]),
f∗(t) = inf{y > 0 : λf(y) < t}. (2.7)
We will refer to f∗ as the rearrangement of f . We say that a function is rearrangable on X if it belongs to S0(X). Note that if two rearrangable
functions are equimeasurable, then their rearrangements coincide. Further, for f ∈ S0(X), there holds the identity (see [12, p. 32])
f∗(t) = sup E inf x∈E|f (x)| , (2.8)
where the supremum is taken over all measurable sets E ⊂ X having mea-sure t. Notice that it is sufficient to take supremum only over all Fσ-sets of
measure t in (2.8). Indeed, let f ∈ S0(X) and denote
g(t) = sup A inf x∈A|f (x)| ,
where the supremum is taken over all Fσ-sets A ⊂ X with mesnA = t.
Obviously g(t) ≤ f∗(t). Further, for every measurable set E ⊂ X with mesnE = t, there exists an Fσ-subset A with mesnA = t such that
inf
To verify the last property, observe that E contains a sequence {xk} such that 0 ≤ |f (xk)| − inf x∈E|f (x)| ≤ 1 k.
We can assume that also A contains this sequence, and then (2.9) follows. From (2.9), we see that
inf
x∈E|f (x)| ≤ g(t),
for every measurable set E ⊂ X with mesnE = t, and thus f∗(t) ≤ g(t).
This proves that f∗(t) = g(t). In what follows, we will have X = Rn, X = Rn+, or X = Rn× Rm+ (n, m ≥ 1).
Let f ∈ S0(Rn). We now give some basic properties of the rearrangement
that will come to use in what follows. Put
At= {x : |f (x)| > f∗(t)},
t > 0. By the definition of f∗ it holds that
mesnAt≤ t. (2.10)
It is also a consequence of the definition of f∗ that the measure of the set Bt= {x : |f (x)| ≥ f∗(t)} satisfies
mesnBt≥ t. (2.11)
For each f ∈ S0(Rn) and every scalar a ∈ R it is immediate that af ∈
S0(Rn) (the distribution function of af is y 7→ λf(y/a), so it is finite). It
follows directly from (2.8) that
(af )∗(t) = |a|f∗(t), (2.12)
for all t > 0.
For f, g ∈ S0(Rn) and t, s > 0 it holds that (see [33, p. 67])
(f + g)∗(t + s) ≤ f∗(t) + g∗(s). (2.13) Let f ∈ S0(Rn) and fix ε > 0. Since f∗ and f are equimeasurable, we
have
mes1{t > 0 : f∗(t) > ε} = λf(ε) < ∞.
Since f∗ is non-increasing it follows that f∗(t) ≤ ε for all t > λf(ε). Thus,
lim t→∞f ∗ (t) = 0. (2.14) We also have lim t→0+f ∗(t) = kf k ∞. (2.15)
Indeed, let y0 denote this limit. By (2.10) it holds that
for all t > 0. Thus mesn{x : |f (x)| > y0} = 0, so that kf k∞ ≤ y0.
Furthermore, (2.11) implies that kf k∞ ≥ f∗(t) for all t > 0, and therefore
kf k∞≥ y0.
We also mention the following result [33, p. 67]
Proposition 2.4. If the sequence {fk} ⊂ S0(Rn) converges in measure to
the function f ∈ S0(Rn), then fk∗ → f∗ at every point of continuity of f∗.
Let C(Rn) denote the class of all bounded continuous functions on Rn. Lemma 2.5 and Lemma 2.6 below, state known elementary properties of the rearrangement of a continuous function.
Lemma 2.5. Let f ∈ S0(Rn) ∩ C(Rn). Then, for every t0 > 0 there exists
a point x0∈ Rn such that f∗(t0) = |f (x0)|.
Proof. Fix t0 > 0. It is immediate from the definition of f∗ that 0 ≤
f∗(t0) ≤ kf k∞. First we assume that f∗(t0) = 0. Suppose |f (x)| > 0 for all
x ∈ Rn. Let E ⊂ Rnbe a compact set having measure t0. Since f ∈ C(Rn)
there exists x1 ∈ E where
f∗(t0) ≥ inf
x∈E|f (x)| = |f (x1)| > 0,
which is a contradiction.
Next we suppose that f∗(t0) = kf k∞. According to (2.11), it holds that
mesn{x : |f (x)| = kf k∞} = mesn{x : |f (x)| ≥ f∗(t0)} ≥ t0> 0,
so there exists x0 ∈ Rn where |f (x0)| = kf k∞= f∗(t0).
The remaining case is when 0 < f∗(t0) < kf k∞. Since f ∈ S0(Rn), we
can not have |f (x)| > f∗(t0) > 0 for all x ∈ Rn. So there exists x0 ∈ Rn
such that
0 ≤ |f (x0)| ≤ f∗(t0). (2.16)
Clearly there also exists a point x00∈ Rnwhere
f∗(t0) ≤ |f (x00)| ≤ kf k∞. (2.17)
Since f has the intermediate value property, it follows from (2.16) and (2.17) that there exists some x0 along the line segment from x0 to x00 for which
|f (x0)| = f∗(t0).
Lemma 2.6. Let f ∈ S0(Rn) ∩ C(Rn). Then f∗ is continuous on R+.
Proof. Fix t0 > 0. Assume that f∗ is discontinuous at t0. Since f∗ is
left-continuous and non-increasing, it follows that y0≡ lim
t→t0+
So, f∗ takes no values in (y0, f∗(t0)). Let τ ∈ (y0, f∗(t0)) and suppose
|f (x0)| = τ for some x0 ∈ Rn. Since f is continuous, there exists some
δ > 0 such that if x1 ∈ Rn and |x0− x1| < δ then
τ − |f (x1)| = |f (x0)| − |f (x1)| < f∗(t0) − y0. Therefore mesn{x : |f (x)| ∈ (y0, f∗(t0))} > 0.
But, f and f∗ are equimeasurable so
mesn{x : |f (x)| ∈ (y0, f∗(t0))} = mes1{s > 0 : f∗(s) ∈ (y0, f∗(t0))} = 0,
which is a contradiction. Thus, if f∗ is discontinuous at t0, then |f | takes
no values in the interval (y0, f∗(t0)). By (2.11)
mesn{x : |f (x)| ≥ f∗(t0)} ≥ t0 > 0.
Again by (2.11) and the equimeasurability of f and f∗, mesn{x : f∗(t0+ 1) ≤ |f (x)| ≤ y0} =
= mesn{x : |f (x)| ≥ f∗(t0+ 1)} − mes1{s > 0 : f∗(s) > y0} ≥ 1,
so |f | takes values greater than f∗(t0) and values less than y0. Since f has
the intermediate value property, it follows that the whole interval (y0, f∗(t0))
is in the range of |f |. Thus, the assumption that f∗is discontinuous at some
point t0 leads to a contradiction.
Let f be continuous on a set E ⊂ Rn. The total modulus of continuity of f is the function δ 7→ ω(f ; δ), which is defined for all δ > 0 by
ω(f ; δ) = sup{|f (x) − f (y)| : x, y ∈ E, |x − y| ≤ δ}. (2.18) The supremum is over all x and y in the domain E of f such that |x−y| < δ. For all α > 0 it holds that (see [38, p. 148])
ω(f ; αδ) ≤ (α + 1)ω(f ; δ). (2.19)
The inequality stated by the next proposition is known, but we give a simpler proof of it. Similar estimates can be found e.g. in [20], [37] and [25]. Proposition 2.7. Let f ∈ S0(Rn) ∩ C(Rn). Then
ω(f∗; δ) ≤ c ω(f ; δ1/n), (2.20)
for all δ > 0, where c = 2vn−1/n+ 1 and vn is the measure of the unit ball in
Proof. By the triangle inequality we have ω(|f |; δ) ≤ ω(f ; δ), so we may assume that f ≥ 0. Fix 0 < t0 < t00 and estimate f∗(t0) − f∗(t00). We can assume that f∗(t00) < f∗(t0). Let
A0= {x : f (x) = f∗(t0)} and A00= {x : f (x) = f∗(t00)}.
Since f ∈ S0(Rn) ∩ C(Rn), the sets A0 and A00are nonempty by Lemma 2.5.
Fix N ≥ 2. We will show that there exist points x0 ∈ A0 and x00 ∈ A00 such that |x0− x00| < 2N + 1 N − 1v −1/n n (t 00− t0 )1/n. (2.21)
Let d be the distance from A0 to A00, i.e.
d = inf{|x0− x00| : x0 ∈ A0, x00∈ A00}.
If d = 0 then |x0− x00| can be chosen arbitrarily small, in particular so small
that (2.21) is satisfied. Assume that d > 0. Then there exists x0 ∈ A0 and x00 ∈ A00 such that |x0− x00| < (1 + 1/N )d. Let these points be chosen so
that the function τ 7→ f (x0τ + (1 − τ )x00) only takes values in (f∗(t00), f∗(t0)) for τ ∈ (0, 1). Set λN = N/(N + 1) − 1/2 > 0. Let B be the ball in Rn
centered at p = (x0+ x00)/2 of radius λN|x0− x00|. Then B ∩ A0 = ∅. Indeed,
suppose there exist a point y0 ∈ B ∩ A0. Then |y0− x00| ≤ |y0−x 0+ x00 2 | + | x0+ x00 2 − x 00| < < (λN + 1 2)|x 0− x00| < (λ N + 1 2)(1 + 1 N)d = d, which is a contradiction. Similarly B ∩ A00= ∅.
Let
E = {x : f∗(t00) < f (x) < f∗(t0)}.
We will prove that B ⊂ E. By choice of x0 and x00 we know that f∗(t00) < f (p) < f∗(t0).
Suppose there exists a point q ∈ B where f (q) < f∗(t00). Since f has the intermediate value property there exists a point r along the line segment from p to q where f (r) = f∗(t00). Thus r ∈ B ∩ A00, which is a contradiction. In the same way the assumption that f (x) > f∗(t0) for some x ∈ B leads to a contradiction. This proves that B ⊂ E. By our observations (2.10) and (2.11) we then obtain
mesnB ≤ mesnE ≤ t00− t0.
This gives inequality (2.21). Now
f∗(t0) − f∗(t00) = f (x0) − f (x00) ≤ ω f ; 2N + 1 N − 1v −1/n n (t 00− t0 )1/n.
Since N is arbitrary, we obtain
f∗(t0) − f∗(t00) ≤ ω f ; 2vn−1/n(t00− t0)1/n. (2.22)
By (2.19), this implies (2.20).
Remark 2.8. Let n = 1. Then we have c = 2 in (2.20). However, in this case (2.22) gives
ω(f∗; δ) ≤ ω(f ; δ), (2.23)
that is, (2.20) holds with c = 1. The following shorter proof of inequality (2.23) is already known. Let 0 < t < t + h. Assume that f∗(t) > f∗(t + h). By Lemma 2.5, there exists x0, x00∈ R such that |f(x0)| = f∗(t), |f (x00)| =
f∗(t + h), and f∗(t + h) < |f (x)| < f∗(t) for all x between x0 and x00. It is clear that |x0− x00| ≤ h since otherwise we would have
mes1{x : f∗(t + h) < |f (x)| < f∗(t)} > h,
which is a contradiction (by (2.10) and (2.11), this set has measure at most h). Thus,
f∗(t) − f∗(t + h) = |f (x0)| − |f (x00)| ≤ ω(f ; h). This implies inequality (2.23).
2.3. Lorentz spaces. The Lorentz spaces Lq,pform a two parameter family of spaces that contains the Lebesgue spaces Lp. We give here the definition and some basic properties.
We observe first that the rearrangement preserves the Lp-norm. Indeed it holds that (see [44, p. 191-192])
Z Rn |f (x)|pdx = Z ∞ 0 [f∗(t)]pdt, (2.24)
for all 0 < p < ∞, and
kf k∞= kf∗k∞. (2.25)
It follows from Lemma 3.17 on page 201 in [44] that given f ∈ S0(Rn)
and t > 0, there exists a measurable set Et ⊂ Rn having measure t such
that Z Et |f (x)|dx = sup |E|=t Z E |f (x)|dx = Z t 0 f∗(u)du, (2.26)
where |E| denotes the measure of E and the supremum is over all measurable sets E ⊂ Rn having measure t.
In what follows we set
f∗∗(t) = 1 t
Z t
0
f∗(u)du. (2.27)
It follows from (2.26) that the operator f 7→ f∗∗ is subadditive, that is, (f + g)∗∗(t) ≤ f∗∗(t) + g∗∗(t). (2.28)
By the property (2.15) of the rearrangement, we have lim
t→0+f
∗∗(t) = kf k
∞. (2.29)
If f ∈ S0(Rn) and f ∈ L1(E), for each measurable set E ⊂ Rn with
mesnE < ∞, then f∗∗(t) < ∞ for all t > 0. Thus, in this case, we have,
applying (2.14),
lim
t→∞f ∗∗
(t) = 0. (2.30)
As was already mentioned in Section 1, when 0 < q, p < ∞, the space Lq,p(Rn) is defined as the class of all f ∈ S0(Rn) such that
kf kq,p ≡ Z ∞ 0 [t1/qf∗(t)]pdt t 1/p < ∞.
By (2.24) we have that Lp,p coincides with the space Lp, 0 < p < ∞. For
0 < q < ∞ we let Lq,∞(Rn) be the space of all f ∈ S0(Rn) for which
kf kq,∞≡ sup t>0
t1/qf∗(t) < ∞.
We also set L∞,∞(Rn) ≡ L∞(Rn). When 0 < p ≤ s ≤ ∞, 0 < q < ∞, there holds the inequality (see [44, Theorem 3.11, p. 192])
s q 1/s kf kq,s≤p q 1/p kf kq,p. (2.31)
Note that we get equality in (2.31) for f = χE, mesnE < ∞, so the constants
can not be improved.
The last range of the parameters for which we define the Lorentz space is when q = ∞, 0 < p < ∞. Then we let L∞,p(Rn) consist of all f ∈ S0(Rn)
such that (see [3], [35]) kf k∞,p≡ Z ∞ 0 h f∗∗(t) − f∗(t) ipdt t 1/p < ∞. Observe that L∞,1(Rn) = L∞(Rn), (2.32)
and the norms coincide. Indeed, note that d
dtf
∗∗(t) = −1
t(f
∗∗(t) − f∗(t)), a.e. t > 0.
Hence, for every ε > 0, Z 1/ε ε h f∗∗(t) − f∗(t) idt t = f ∗∗ (ε) − f∗∗(1/ε).
Let ε → 0+. Then the left-hand side tends to kf k∞,1, and the right-hand
If 1 ≤ q, p < ∞ and f ∈ Lq,p(Rn), then by (2.31)
f∗(t) = O(t−1/q), (2.33)
as t → 0+ and as t → ∞.
For any function f ∈ S0(Rn), we will use the notation
∆f(t) ≡ f∗(t) − f∗(2t),
for t > 0. This difference will play an important role in the sequel. We now define the modified Lorentz norm, denoted k · k∗q,p, which will be equivalent to the Lorentz norm of f but which is defined in terms of ∆f. This modified
Lorentz norm was introduced in [29]. When 1 ≤ q < ∞ we set
kf k∗q,p= Z ∞ 0 [t1/q∆f(t)]p dt t 1/p , 1 ≤ p < ∞ sup t>0 t1/q∆f(t), p = ∞.
Clearly, kf k∗q,p ≤ kf kq,p. To show that kf kq,p ≤ ckf k∗q,p for some constant
c, we use the inequality:
f∗(2t) ≤ 1 ln 2 Z ∞ t ∆f(u) du u . (2.34)
To verify that (2.34) holds, fix t > 0 and take N > 2t. Then Z N t ∆f(u) du u = Z 2t t f∗(u)du u − Z 2N N f∗(u)du u ≥ f ∗ (2t) ln 2 − f∗(N ). Now (2.34) follows if we let N tend to ∞ and use (2.14). By (2.34), Hardy’s inequality (2.1), and (2.3) we obtain that
kf kq,p ≤
21/qq ln 2 kf k
∗
q,p, 1 ≤ q < ∞, 1 ≤ p ≤ ∞. (2.35)
We define the modified Lorentz norm also when q = ∞ and 1 ≤ p < ∞. In this case we set
kf k∗∞,p≡ Z ∞ 0 (∆f(t))p dt t 1/p .
To prove the equivalence between k·k∞,pand k·k∗∞,pwe will use the following
inequalities 1 2∆f t 2 ≤ f ∗∗(t) − f∗(t) ≤ 2 t Z t 0 ∆f(u)du. (2.36)
The left inequality in (2.36) is immediate, f∗∗(t) − f∗(t) ≥ 1 t Z t/2 0 [f∗(u) − f∗(t)]du ≥ 1 2[f ∗(t/2) − f∗(t)].
To prove the right inequality in (2.36) we take 0 < ε < t/2 and observe that 2 Z t ε ∆f(u)du ≥ Z t ε f∗(u)du − Z 2t t f∗(u)du ≥ Z t ε f∗(u)du − tf∗(t). The left inequality in (2.36) immediately implies that kf k∗∞,p ≤ 2kf k∞,p.
By the right inequality in (2.36) and Hardy’s inequality (2.2) we have that
kf k∞,p≤ 2kf k∗∞,p. (2.37)
2.4. Iterative rearrangements. We will consider rearrangements with respect to specific variables. Let f ∈ S0(Rn) and 1 ≤ k ≤ n. Fix
ˆ
xk ∈ Rn−1, and consider the function fxˆk(xk) = f (ˆxk, xk). By Fubini’s
theorem, fˆxk ∈ S0(R) for almost all ˆxk ∈ R
n−1. We denote the
rearrange-ment of f with respect to xk by Rkf . That is, we set
Rkf (t, ˆxk) = fˆx∗k(t).
This function is defined almost everywhere on R+× Rn−1. Moreover, Rkf
is a measurable function equimeasurable with f (see [9]). Let Pn denote
the set of all permutations σ = (k1, . . . , kn) of the numbers 1, 2, . . . , n. For
all σ ∈ Pn and f ∈ S0(Rn), we define the Rσ-rearrangement of f as the
function
Rσf (t) = Rkn· · · Rk1f (t), t ∈ R
n +.
That is, we obtain Rσf from f by “rearranging” f successively with respect
to the variables xk1, . . . , xkn, starting with xk1. By the above, the function
Rσf is equimeasurable with f . As we observed in Section 2.2, this means
that their rearrangements coincide, that is Rσf ∈ S0(Rn+) and
(Rσf )∗(τ ) = f∗(τ ), τ > 0. (2.38)
Moreover, Rσf is non-negative on Rn+, and non-increasing in each variable.
In what follows we set
π(t) =
n
Y
k=1
tk, t ∈ Rn+.
There holds the inequality
Rσf (t) ≤ f∗(π(t)), t ∈ Rn+. (2.39)
Indeed, since Rσf is non-increasing in each variable, we have
(Rσf )∗(π(t)) ≥ inf s∈Qt
Rσf (s) ≥ Rσf (t), (2.40)
where Qt= (0, t1) × · · · × (0, tn). Apply (2.38), with τ = π(t). This gives
Using (2.24) n times, we obtain that for all f ∈ S0(Rn) Z Rn+ [Rσf (t)]pdt = Z Rn |f (x)|pdx, (2.41)
for 0 < p < ∞. By (2.25), we also have
kRσf k∞= kf k∞.
For 0 < p, s < ∞ and σ ∈ Pn, we define the space Lp,sσ (Rn) as the class
of all functions f ∈ S0(Rn) such that
kf kLp,s σ ≡ Z Rn+ h π(t)1/pRσf (t)is dt π(t) 1/s < ∞ (see [9]). We also set
Lp,s(Rn) = \
σ∈Pn
Lp,sσ (Rn). It was proved in [48] that
kf kp,s≤ 21/s−1/pkf kLp,sσ ,
for 0 < s ≤ p < ∞ and σ ∈ Pn. Theorem 2.10 below improves the constant
in this inequality. The proof of this theorem is very similar to the proof of the preceding inequality given in [30, pp. 54–55].
Let M(Rn+) denote the class of all measurable non-negative functions on
Rn+, which are non-increasing in each variable.
Lemma 2.9. Let σ ∈ Pn, 0 < s, p < ∞, and f ∈ M(Rn+) ∩ L p,s
σ (Rn+). There
exists a sequence {fk} in M(Rn+) ∩ L p,s
σ (Rn+), such that:
(i) for each k, fk is positive and strictly decreasing in each variable;
(ii) kfkkp,s→ kf kp,s;
(iii) kfkkLp,sσ → kf kLp,sσ .
Proof. Fix α > 1/p and set g(t) =h n Y k=1 (1 + tk) i−α , t ∈ Rn+. We have kgkLp,s σ = Z ∞ 0 us/p−1(1 + u)−sαdun/s.
By our choice of α, this integral converges so that g ∈ Lp,sσ (Rn+). Set
fk= f +
1
Clearly fk∈ M(Rn+) ∩ L p,s
σ (Rn+) and statement (i) holds. Note that
|f − fk| ≤ 1
k, (2.42)
on Rn
+. Further, the sequence {fk} is decreasing and the function
t 7→ Yn k=1 ts/p−1k f1(t)s, t ∈ Rn+,
belongs to L1(Rn+). Hence, by the dominated convergence theorem,
state-ment (iii) holds. According to (2.42), fk → f in measure on Rn+. This
implies that fk∗ → f∗ a.e. on R+ by Proposition 2.4. The sequence {fk∗} is
decreasing and the function
u 7→ us/p−1f1∗(u)s, u > 0,
belongs to L1(R+). To obtain statement (ii), we again use the dominated
convergence theorem.
Theorem 2.10. Let f ∈ S0(Rn) and σ ∈ Pn. For all 0 < s ≤ p < ∞,
kf kp,s≤ kf kLp,sσ . (2.43)
Proof. Set F (t) = Rσf (t). By Lemma 2.9 we may suppose that
mesn{t ∈ Rn+: F (t) = y} = 0, (2.44)
for all y ≥ 0. Fix a > 1 and set
Aν = {t ∈ Rn+ : f∗(a−ν+1) ≤ F (t) < f∗(a−ν)},
for ν ∈ Z. Let t ∈ Aν. Then f∗(a−ν+1) ≤ F (t) ≤ F (s) for all s ∈ Rn+ such
that 0 < sk≤ tk, k = 1, . . . , n. Thus,
π(t) ≤ mesn{s ∈ Rn+ : F (s) ≥ f∗(a−ν+1)}.
By our assumption (2.44), this gives
for all t ∈ Aν. This inequality implies kf ksLp,s σ = Z Rn+ π(t)s/p−1F (t)sdt =X ν∈Z Z Aν π(t)s/p−1F (t)sdt ≥ ≥ X ν∈Z a(s/p−1)(−ν+1) Z Aν F (t)sdt = = as/p−1X ν∈Z a−ν(s/p−1) Z a−ν+1 a−ν f∗(u)sdu ≥ ≥ as/p−1X ν∈Z Z a−ν+1 a−ν us/p−1f∗(u)sdu = as/p−1kf ksp,s.
3. Some geometric results
In this section, we will examine some of the properties of the spaces Vk= L1ˆxk(R
n−1)[L∞
xk(R)], k = 1, . . . , n,
where, as above, ˆxk denotes the point in Rn−1 which is obtained from a
given point x ∈ Rnby removal of its kth coordinate. Recall from Section 1, that the corresponding norms where used by Gagliardo, and later also by Fournier, to prove embeddings of Sobolev spaces. More complicated mixed norms related to embeddings of Sobolev type spaces will be studied in latter sections of this thesis.
In Section 7 below we will study embeddings of the space ∩nk=1Vk, and
in particular we prove a refinement of Fournier’s inequality (1.6) for the Vk-norms (see Theorem 7.10 and Remark 7.11 below).
For E ⊂ Rn and 1 ≤ k ≤ n, we let ΠkE ⊂ Rn−1 be the orthogonal
projection of E onto the coordinate hyperplane xk = 0. Throughout this
work we study geometric properties of sets, in particular the measures of the projections. An important role is played by the following inequality proved by Loomis and Whitney [34].
Theorem 3.1. For any Fσ-set E ⊂ Rn there holds the inequality
(mesnE)n−1≤ n
Y
k=1
mesn−1ΠkE. (3.1)
We shall also use the following elementary lemma.
Lemma 3.2. Let n ≥ 2 and 1 ≤ k ≤ n. Assume that E ⊂ Rn and D ⊂ Rn−1 are measurable in Rn and Rn−1 respectively. Then the set
E0= {x ∈ E : ˆxk∈ D}
is measurable in Rn.
Proof. It is sufficient to consider the case k = n. In this case E0 = E ∩ (D × R).
Since the Cartesian product of two measurable sets is measurable, the
mea-surability of E0 follows.
For a point ˆxk∈ Rn−1, denote by E(ˆxk) the ˆxk-section of the set E:
E(ˆxk) = {xk ∈ R : (xk, ˆxk) ∈ E}.
Assume that E ⊂ Rnis measurable. By Fubini’s theorem, for any 1 ≤ k ≤ n and almost all ˆxk ∈ Rn−1, the sections E(ˆxk) are measurable in R, and the
functions
defined a.e. on Rn−1 are measurable.
The essential projection of a measurable set E onto the hyperplane xk= 0
is defined to be the set Π∗kE of all points ˆxk ∈ Rn−1 such that E(ˆxk) is
measurable and mk(ˆxk) > 0. Since the function mk is measurable, the
essential projection Π∗kE is measurable.
Observe that inequality (3.1) holds also if ΠkE is replaced by the essential
projection Π∗kE. That is, for all measurable sets E ⊂ Rn, there holds the inequality (mesnE)n−1 ≤ n Y k=1 mesn−1Π∗kE. (3.2)
Indeed, for any ˆxk∈ Rn−1 the section E(ˆxk) is an Fσ-set in R and therefore
it is measurable. Thus, Π∗kE consists exactly of all points ˆxk for which
mk(ˆxk) > 0. Put E0 = n \ k=1 {x ∈ E : ˆxk∈ Π∗kE}.
Then E0is measurable by Lemma 3.2. Note also that ΠkE0⊂ Π∗kE. Further,
we have mesnE0 = mesnE. Namely,
E \ E0= n [ k=1 {x ∈ E : ˆxk 6∈ Π∗kE} = n [ k=1 {x ∈ E : mk(ˆxk) = 0},
and each of the sets in the last union have measure 0. Let E00 be an Fσ
-subset of E0 with mesnE00 = mesnE. Then ΠkE00 ⊂ Π∗kE. Using (3.1) we
then get (mesnE)n−1 = (mesnE00)n−1≤ n Y k=1 mesn−1ΠkE00 ≤ n Y k=1 mesn−1Π∗kE, so (3.2) is proved.
Let f be a measurable function on Rn and let 1 ≤ k ≤ n. By Fubini’s theorem, for almost all ˆxk ∈ Rn−1 the sections fxˆk (see Section 2.4) are
measurable functions on R. Moreover, as we observed above, the function Ψk(ˆxk) = kfˆxkkL∞(R)= ess supxk∈R|f (xk, ˆxk)|
(defined on Rn−1) is measurable. It is sufficient to show this in the case
when f is a bounded function with compact support. In this case we have Ψk(ˆxk) = lim
m→∞kfxˆkkLm(R),
and the functions
ˆ
are measurable by Fubini’s theorem. Thus, the definition of the spaces Vk
(see Section 1) is correct. Now we shall show that the norms in Vk have a
simple geometric interpretation.
Theorem 3.3. Let f be a measurable function on Rn, and put E = {(x, y) ∈ Rn+1: x ∈ Rn, 0 ≤ y ≤ |f (x)|}.
For k ∈ {1, . . . , n}, we let Ek denote the essential projection of E onto
the hyperplane xk = 0. Then f ∈ Vk if and only if Ek is measurable and
mesnEk< ∞. Moreover, in this case
mesnEk= kf kVk. (3.3)
Proof. It is enough to give the proof for k = n. Denote
Mn(ˆxn) = ess supxn∈R|f (ˆxn, xn)| (3.4)
for all ˆxn where this essential supremum is defined, and put Mn(ˆxn) = 0
otherwise. We have
En(ˆxn) ⊂ [0, Mn(ˆxn)], (3.5)
for all ˆxn ∈ Rn−1 (with the obvious interpretation if Mn(ˆxn) equals 0 or
∞). Indeed, suppose (ˆxn, y) ∈ En. We have
E(ˆxn, y) = {xn∈ R : (ˆxn, xn, y) ∈ E}.
By definition of En, the set E(ˆxn, y) ⊂ R is measurable and mes1E(ˆxn, y) >
0. But for all xn ∈ E(ˆxn, y), we have 0 ≤ y ≤ |f (ˆxn, xn)|, and thus y ≤
Mn(ˆxn). This proves (3.5).
Further, if Mn(ˆxn) > 0, then we have
[0, Mn(ˆxn)) ⊂ En(ˆxn). (3.6)
Indeed, suppose Mn(ˆxn) > 0 and let y ∈ [0, Mn(ˆxn)). Note that
E(ˆxn, y) = {xn∈ R : 0 ≤ y ≤ |f(ˆxn, xn)|}.
Since y < Mn(ˆxn), this set has positive linear measure, so by definition
(ˆxn, y) ∈ En. That is, (3.6) holds.
By (3.5) and (3.6), En(ˆxn) is an interval with the length
mes1En(ˆxn) = Mn(ˆxn) (3.7)
for all ˆxnwhere Mn(ˆxn) > 0. Suppose that Enis measurable and mesnEn<
∞. By Fubini’s theorem and (3.7), we have mesnEn= Z Rn−1 mes1En(ˆxn)dˆxn= Z Rn−1 Mn(ˆxn)dˆxn= kf kVn.
So we have proved (3.3), and thus f ∈ Vn. To prove the converse, suppose
that f ∈ Vn. Then Mn ∈ L1(Rn−1), and kMnkL1(Rn−1) = kf kVn. We are
done if we prove (3.3). To this end, we consider the sets En0 = {(ˆxn, y) ∈ Rn: 0 ≤ y < Mn(ˆxn)}
and
En00= {(ˆxn, y) ∈ Rn: 0 ≤ y ≤ Mn(ˆxn)}.
These sets are measurable and
mesnEn0 = mesnEn00= kMnkL1(Rn−1)= kf kVn.
Note also that
En0 ⊂ En⊂ En00.
These observations imply that En is measurable, and that (3.3) holds.
Next, we let n = 2 and show that the space V1∩ V2 is not invariant under
rotation. To make clear what we mean, suppose that f ∈ V1∩ V2, and set
g(x, y) = fx − y√ 2 , x + y √ 2 . (3.8)
The vectors (x, y) and ((x − y)/√2, (x + y)/√2) have the same length, and the angle between them is π/4. Thus, the graph of g is obtained by rotating the graph of f the angle π/4 around the origin. We prove the following proposition by finding f ∈ V1∩ V2 such that the function g defined by (3.8)
does not belong to V1∩ V2.
Proposition 3.4. The space V1∩ V2 is not invariant under rotation.
Proof. Put E = (−1, 1) × ∞ [ k=1 (k, k + 2−k) and set f (x, y) = χE(x, y), on R2. We have kf (·, y)k∞= ∞ X k=1 χ(k,k+2−k)(y), and kf (x, ·)k∞= χ(−1,1)(x). Thus, kf kV1 = ∞ X k=1 Z R χ(k,k+2−k)(y)dy = 1, and kf kV2 = Z R χ(−1,1)(x)dx = 2.
So f ∈ V1∩ V2. For this function f , the function g from (3.8) is given by g(x, y) = χ(−1,1)x − y√ 2 X∞ k=1 χ(k,k+2−k) x + y √ 2 ,
and as we noted above, it is obtained by rotating f . The proof is complete if we check that g does not belong to V1∩ V2.
Fix x0≥ 1/
√
2 and let k ≥ 1 be the integer contained in (x0 √ 2 − 1, x0 √ 2]. Then g(x0, y) = 1, for all y ∈ (k √ 2 − x0, k √ 2 − x0+ 2−k+1/2). (3.9)
Indeed, for y in this interval it holds that x0√+ y 2 ∈ (k, k + 2 −k) and x0− y √ 2 ∈ (x0 √ 2 − k − 2−k, x0 √ 2 − k) ⊂ (−2−k, 1), where the last inclusion holds since k ∈ (x0
√ 2 − 1, x0 √ 2]. This proves (3.9). It follows that kg(x, ·)k∞= 1, x ≥ 1/ √ 2,
which implies that kgkV2 = ∞. (In fact, a similar argument shows that also
kgkV1 = ∞.) The proof is complete.
Remark 3.5. With the above proposition in mind, we point out that ∩n
k=1Vk contains W11(Rn) (according to (1.4)), and this Sobolev space is
clearly invariant under rotation. Indeed, there holds the obvious inequality kDuf k1 ≤ n X k=1 k ∂f ∂xk k1, Duf (x) = u · ∇f (x),
for f ∈ W11(Rn), where Duf is the weak directional derivative of f in the
direction of a given unit vector u. This shows that if f ∈ W11(Rn), and if g is obtained by rotating f a given angle around the origin, then g belongs to the same Sobolev space.
Let X be a linear space of measurable real-valued functions on Rn, which is generated by a norm k · kX. We say that such a space is rearrangement
invariant (r.i. for short) if, whenever f ∈ X and g is equimeasurable with f , then g ∈ X and
kf kX = kgkX
Remark 3.6. It follows immediately from Proposition 3.4 that V1∩ V2 is
not a r.i. space.
Given any set W ⊂ S0(Rn), we define the r.i. hull of W as the smallest
r.i. space that contains W .
Remark 3.7. It is well known that Ln0,1(Rn) is the r.i. hull of W11(Rn). Since W11(Rn) ⊂ n \ k=1 Vk⊂ Ln 0,1 (Rn),
it follows that Ln0,1(Rn) is the r.i. hull also of ∩nk=1Vk. The direct proof of
4. The spaces Λσ
In this section we consider a one parameter family of spaces denoted Λσ.
These spaces were introduced in [29]. Let σ ∈ R. Recall from Section 1 that a function f ∈ S0(R) belongs to Λσ if
kf kΛσ ≡ sup
t>0
tσ∆f(t) < ∞.
In Section 4.1 we see how Λσ relates to known spaces. We also give an equivalent definition of k · kΛσ for σ < 0, and show that in this case all
functions in this space belong to L∞(R). The main results in Section 4.2 are Theorems 4.8 and 4.10, which show how functions in Λσ can be approxi-mated by simple functions (defined below) and by continuous functions with compact support.
4.1. Some general properties of the spaces Λσ. Propositions 4.1, 4.2, and 4.3 below, state embeddings of Λσfor different values of σ. These results were obtained in [29]; for the sake of completeness, we give the proofs.
First we determine to what extent k·kΛσ satisfies the properties of a norm.
We have kf kΛσ ≥ 0 for all f ∈ Λσ since ∆f is non-negative. Moreover,
∆f = 0 on R+ if and only if f∗= 0 on R+. Therefore kf kΛσ = 0 if and only
if f = 0 a.e. on Rn. Furthermore, by (2.12), we have kλf kΛσ = |λ|kf kΛσ,
for all λ ∈ R. However, we will show that if σ ≤ 0 then there is no constant c such that the “triangle inequality”,
kf + gkΛσ ≤ c(kf kΛσ + kgkΛσ), (4.1)
holds for all f, g ∈ Λσ. Set fn= nχ(0,1], h2 = χ(1,2], and hn+1 = hn+ χ(1,2n],
n ≥ 2. Using induction we prove that
∆fn = nχ(1/2,1], ∆fn+hn = χ(1/2,2n−1], and ∆hn = n−1 X k=1 χ((2k−1)/2,2k−1].
So, if α ≥ 0, then kfnkΛ−α = n2α, kfn+ hnkΛ−α = 2α, and khnkΛ−α = 2α.
Clearly there is no constant c for which
kfnkΛ−α ≤ c(kfn+ hnkΛ−α+ khnkΛ−α),
for all n ≥ 2, so (4.1) is not satisfied when σ ≤ 0. For σ > 0, (4.1) holds with c = 4σ/(σ ln 2). To prove this, we will use the fact that for σ > 0, the
space Λσ coincides with the Marcinkiewicz space, as stated in the following proposition.
Proposition 4.1. Let σ > 0 and set r = 1/σ. Then Λσ = Lr,∞(R) and kf kΛσ ≤ kf kr,∞≤ 2
σ
σ ln 2kf kΛσ. (4.2)
Proof. The first inequality in (4.2) is immediate for all f ∈ Lr,∞(R). Let
f ∈ Λσ(R). By (2.34), kf kr,∞≤ 2σ ln 2supt>0 tσ Z ∞ t ∆f(u) du u .
The second inequality in (4.2) now follows by inequality (2.3). Let σ > 0 and set r = 1/σ. Suppose f, g ∈ Lr,∞(R). By (2.13) we have
kf + gkr,∞≤ sup
t>0
t1/r(f∗(t/2) + g∗(t/2)) ≤ 21/r(kf kr,∞+ kgkr,∞).
This inequality and Proposition 4.1 now give kf + gkΛσ ≤
4σ
σ ln 2(kf kΛσ + kgkΛσ), (4.3) for all f, g ∈ Λσ, i.e. (4.1) holds when σ > 0.
Define the space W , called weak-L∞, as the class of all f ∈ S0(R) such
that
kf kW = sup
t>0
[f∗∗(t) − f∗(t)] < ∞. (4.4) This space was introduced in [4] by Bennett, DeVore, and Sharpely. Proposition 4.2. The spaces Λ0 and W coincide and
1
2kf kΛ0 ≤ kf kW ≤ 2kf kΛ0.
Proof. Let f ∈ W . The first inequality follows immediately from the first inequality in (2.36). Therefore W ⊂ Λ0. Suppose f ∈ Λ0. Fix t > 0. By the second inequality in (2.36) we have
f∗∗(t) − f∗(t) ≤ 2 t
Z t
0
∆f(u)du ≤ 2kf kΛ0.
The second inequality now follows. This gives Λ0 ⊂ W .
As above, C(R) denotes the class of all bounded continuous functions on R. For 0 < α ≤ 1 we define Lip α to be the space of all functions f ∈ C(R) for which (recall (2.18))
kf kLip α≡ sup
δ>0
ω(f ; δ)
δα < ∞. (4.5)
Proposition 4.3. Let 0 < α ≤ 1. If f ∈ S0(R) ∩ Lip α then f ∈ Λ−α and
Proof. Fix t > 0. By inequality (2.23) in Remark 2.8 we have ∆f(t) ≤ ω(f∗; t) ≤ ω(f ; t)
and then
t−α∆f(t) ≤ kf kLip α.
Taking supremum over all t > 0 we obtain the inequality stated in the
proposition.
The next proposition gives an equivalent definition of the space Λσ, when σ < 0.
Proposition 4.4. Let σ < 0. Then f ∈ Λσ if and only if there exists a constant A such that for all t > 0
kf k∞≤ f∗(t) + At−σ. (4.6)
Moreover, if A0 ≥ 0 is the smallest constant such that inequality (4.6) holds
for all t > 0, then
(2−σ− 1)A0≤ kf kΛσ ≤ 2−σA0. (4.7)
Proof. Suppose (4.6) holds. Then
∆f(t) ≤ kf k∞− f∗(2t) ≤ (2t)−σA,
and thus
kf kΛσ ≤ 2−σA.
So, f ∈ Λσ and the right-hand side inequality in (4.7) follows. Let now
f ∈ Λσ. For any N > 0, f∗(2−Nt) − f∗(t) = N X k=1 ∆f(2−kt) ≤ t−σkf kΛσ N X k=1 2kσ. Let N → ∞. By (2.15) we obtain kf k∞≤ f∗(t) + t−σ kf kΛσ 1 − 2σ. Thus, (4.6) holds.
If A0 = 0, then (4.7) follows immediately. Suppose A0 > 0 and fix
ε ∈ (0, A0). By definition of A0 there exists t0> 0 such that
kf k∞> f∗(t0) + (A0− ε)t−σ0 .
Take N > 0 such that f∗(2−Nt0) > kf k∞− ε. We then have
A0− ε < tσ0 f ∗ (2−Nt0) − f∗(t0) + ε = εtσ0 + kf kΛσ N X k=1 2kσ.
Since ε ∈ (0, A0) was arbitrary, it follows that A0 ≤ kf kΛσ N X k=1 2kσ
which implies the left-hand side inequality in (4.7).
Corollary 4.5. Let σ < 0. Then Λσ ⊂ L∞(R).
Proof. Let f ∈ Λσ. By Proposition 4.4, there exists a constant A > 0 for which inequality (4.6) holds. This implies that f ∈ L∞(R). 4.2. Approximation in Λσ. The main results in this section are Theorem 4.8 and Theorem 4.10 below. They show in particular how functions in Λσ can be approximated by simple functions, and by continuous functions with compact support. We first give a negative result on the separability of Λσ. Proposition 4.6. Let σ > 0. Then the space Λσ is not separable.
Proof. We need only find an uncountable subfamily S ⊂ Λσ with the prop-erty that
kf − gkΛσ ≥ 1, (4.8)
for all functions f, g ∈ S such that f is not equivalent to g. Indeed, suppose S = {fξ}ξ∈I is such a family and let {gn}∞n=1 be any countable sequence in
Λσ. Set
r = σ ln 2 22σ+1.
Then the balls Bξ = B(fξ, r) are pairwise disjoint. Indeed, suppose g ∈
Bξ∩ Bη for some ξ, η ∈ I, ξ 6= η. By inequality (4.3) we would then have
kfξ− fηkΛσ ≤ 4
σ
σ ln 2(kfξ− gkΛσ + kg − fηkΛσ) < 1,
which is a contradiction. Since S is uncountable, there must then exist balls Bξ which does not contain any of the functions gn. Therefore the sequence
{gn}∞n=1 can not be dense in Λσ.
To construct such a family S ⊂ Λσ we set
fξ(t) = (t − ξ)−σχ(ξ,1](t),
for 0 < ξ < 1 and t 6= ξ. Then
fξ∗(t) = t−σχ(0,1−ξ] so that sup t>0 tσ∆fξ(t) ≤ sup t>0 tσfξ∗(t) = 1,
and therefore fξ∈ Λσ. Let 0 < ξ < η < 1. By (4.2) 2σ+1 σ kfξ− fηkΛσ ≥ supt>0 tσ(fξ− fη)∗(t) ≥ sup t>0 tσ (fξ− fη)χ(0,η] ∗ (t) = 1, so we can let S consist of the functions 2σ+1σ−1fξ, ξ ∈ (0, 1).
Observe that if σ ≤ 0, it makes no sense to speak about approximation “in the norm” k · kΛσ. Indeed, if σ < 0 set α ≡ −σ and fn= χ(0,n]. Then
kfnkΛ−α = sup t>0 t−αχ(n/2,n](t) =2 n α . So, fn∈ Λ−α and kfnkΛ−α → 0, as n → ∞. If σ = 0 we set
gn(x) = 1 −log2(1 + x) n χ(0,2n−1](x). For 0 < t ≤ (2n− 1)/2 we have ∆gn(t) = 1 nlog2 1 + 2t 1 + t < 1 n and for ((2n− 1)/2) < t ≤ 2n− 1 we have
∆gn(t) = g ∗ n(t) ≤ gn((2n− 1)/2) < 1 − 1 nlog2(2 n−1) = 1 n.
So, kgnkΛ0 < 1/n. Thus gn ∈ Λ0 and kgnkΛ0 → 0, as n → ∞. These
examples shows that even if kf kΛσ is “small”, it can still happen that f is
“big”.
Let w be a positive continuous function on R+. We say that a function
f ∈ S0(R) belongs to the space Λ(w) if
kf kΛ(w)≡ sup
t>0
w(t)∆f(t) < ∞.
If w(t) = tσ, then k · kΛ(w) = k · kΛσ. We will give two theorems on how
a function f ∈ Λ(w) can be approximated by a function g. The approx-imation will not be in the sense that kf − gkΛ(w) is small. As the above
example shows, this does not imply that g is “close” to f . Instead we will ensure that g approximates f in measure and at the same time that kgkΛ(w)
approximates kf kΛ(w). Observe that these results are similar to those
ob-tained for functions of bounded variation (see [49, p. 225]). There is no additional complication of the proofs resulting from the replacement of Λσ by Λ(w).
By a simple function we mean a real-valued, measurable and everywhere finite function f on R which takes only finitely many values and which has the property that for every c 6= 0, the level set {x ∈ R : f (x) = c} has finite measure. It is well known that bounded measurable functions can be
uniformly approximated by simple functions. We will use this property in the following form.
Lemma 4.7. Let f ∈ S0(R). Suppose that |f (x)| ≤ M for all x ∈ R, and
|{x : f (x) 6= 0}| < ∞. (4.9)
Then for every ε > 0 there exists a simple function g such that: (i) |g(x)| ≤ M , for all x ∈ R;
(ii) {x : |g(x)| = M } = {x : |f (x)| = M }; (iii) {x : g(x) 6= 0} = {x : f (x) 6= 0}; (iv) |f (x) − g(x)| < ε, for all x ∈ R.
Proof. Fix ε > 0. We can assume that M/ε ∈ N. Set g(x) = f (x) if f (x) = 0 or |f (x)| = M . Let E = {x : 0 < |f (x)| < M } and set
g(x) = hf (x) ε i +1 2 ε,
for all x ∈ E (here [a] denotes the integral part of a number a). Then for all x ∈ E
f (x) − ε
2 < g(x) ≤ f (x) + ε 2.
This implies statement (iv). Furthermore, −M < f (x) < M on E and therefore −M ε ≤ hf (x) ε i ≤ M ε − 1, for all x ∈ E. It follows that
−M +ε
2 ≤ g(x) ≤ M − ε 2
on E. Thus |g(x)| < M on E, and statements (i) and (ii) hold. We also have that g(x) 6= 0 on E, which implies statement (iii). Finally, g satisfies our definition of a simple function. Indeed, clearly g is measurable and everywhere finite. Moreover, by (iii) and (4.9),
|{x : g(x) = c}| ≤ |{x : f (x) 6= 0}| < ∞,
for all c 6= 0.
Our first main result in this section reads:
Theorem 4.8. Let f ∈ Λ(w). For every ε > 0 there exists a simple function g on R which satisfies:
(i) |{x ∈ R : |f (x) − g(x)| > ε}| < ε; (ii) kf kΛ(w)− kgkΛ(w)
Proof. We can assume that kf k∞> 0. Then we have kf kΛ(w)> 0. Fix 0 <
ε < min(kf kΛ(w), kf k∞). We will construct a function f1 that approximates
f and which has certain good properties that allow us to approximate it with a simple function g. To construct f1we first define the function f0as follows.
Take t∗ > 0 such that
|w(t∗)∆f(t∗) − kf kΛ(w)| <
ε
4. (4.10)
Take t0 ∈ (0, min(t∗, ε/2)) and define f0 as
f0(x) = f∗(t0), f (x) > f∗(t0) f (x), −f∗(t0) ≤ f (x) ≤ f∗(t0) −f∗(t 0), f (x) < −f∗(t0).
By (2.14), there exists t1 > 2t∗ such that λ ≡ f0∗(t1) < min(ε/2, f∗(t0)).
Define f1 as f1(x) = f0(x) − λ, f0(x) > λ 0, −λ ≤ f0(x) ≤ λ f0(x) + λ, f0(x) < −λ.
We will show that f1 approximates f . If f (x) = f0(x) then |f (x) − f1(x)| =
|f0(x) − f1(x)| ≤ λ < ε/2, so |{x : |f (x) − f1(x)| > ε 2}| ≤ |{x : f (x) 6= f0(x)}| = = |{x : |f (x)| > f∗(t0)}| ≤ t0≤ ε 2, (4.11)
where the second inequality holds by (2.10). By considering the three cases t ∈ (0, t0/2], t ∈ (t0/2, t0], and t ∈ (t0, ∞) one can verify that ∆f0(t) ≤
∆f(t), for all t > 0. Moreover, by considering the three cases t ∈ (0, t1/2],
t ∈ (t1/2, t1], and t ∈ (t1, ∞) one can also verify that ∆f1(t) ≤ ∆f0(t) for
all t > 0. Thus
kf1kΛ(w) ≤ kf kΛ(w). (4.12)
Observe that f0∗(t) = min(f∗(t), f∗(t0)). Since t0 ≤ t∗ we then have
f0∗(t∗) = f∗(t∗) and f0∗(2t∗) = f∗(2t∗). (4.13)
We also note that f1∗(t) = max(0, f0∗(t) − λ). Since t1 ≥ 2t∗ we have
f0∗(2t∗) ≥ f0∗(t1) = λ and then
f1∗(t∗) = f0(t∗) − λ and f1∗(2t∗) = f0(2t∗) − λ.
By these two equalities and (4.13) we see that
By (4.14) and (4.10) we obtain kf kΛ(w)≤ ε
4 + kf1kΛ(w). (4.15)
It remains only to approximate f1 by a simple function. First we observe
that kf1k∞< ∞. Moreover,
m ≡ |{x : f1(x) = kf1k∞}| > 0. (4.16)
Indeed, since λ < f∗(t0), we see that
{x : |f1(x)| = kf1k∞} = {x : |f0(x)| = f∗(t0)} = {x : |f (x)| ≥ f∗(t0)}.
So (4.16) holds by (2.11). We also note that by (2.10),
M ≡ |{x : f1(x) 6= 0}| = |{x : |f0(x)| > λ}| ≤ t1 < ∞. (4.17)
Fix ε1∈ (0, ε/2) such that for all t ∈ [m/2, M ],
8ε1w(t) < ε. (4.18)
By Lemma 4.7 there exists a simple function g on R such that
|f1(x) − g(x)| ≤ ε1 (4.19)
for all x ∈ R,
{x : |g(x)| = kgk∞} = {x : |f1(x)| = kf1k∞} (4.20)
and
{x : g(x) 6= 0} = {x : f1(x) 6= 0}. (4.21) By the triangle inequality
|{x : |f (x) − g(x)| > ε}| ≤ ≤ |{x : |f (x) − f1(x)| > ε 2}| + |{x : |f1(x) − g(x)| > ε 2}| ≤ ε 2,
where the last inequality holds by (4.11) and (4.19). Thus, statement (i) is true. According to (4.19) it holds that
g(x) − ε1≤ f1(x) ≤ g(x) + ε1,
for all x ∈ R. It follows that
g∗(t) − ε1≤ f1∗(t) ≤ g ∗
(t) + ε1,
which in turn implies that
|∆f1(t) − ∆g(t)| ≤ 2ε1, (4.22)
for all t > 0.
By (4.12), (4.10), and (4.14) we have kf1kΛ(w) ≤ kf kΛ(w)≤ ε
