Comments on Chapter 4

F_|⊂

i

B(g_i, ε) withg_i ∈L^p().

Set

¯ g_i(x)=

g_i(x) in,

0 onR^N\.

It is clear thatFis covered by the ballsB(g¯_i,2ε)inL^p(R^N).

Remark13.The converse of Corollary 4.27 is also true (see Exercise 4.34). Therefore we have a complete characterization of compact sets inL^p(R^N).

We conclude with a useful application of Theorem 4.26:

Corollary 4.28.LetGbe a fixed function inL¹(R^N)and let F =G B,

whereB is a bounded set inL^p(R^N)with1 ≤ p < ∞. ThenF_| has compact closure inL^p()for any measurable setwith finite measure.

Proof. ClearlyFis bounded inL^p(R^N). On the other hand, if we writef =G u withu∈Bwe have

τ_hf −fp= (τ_hG−G) up≤Cτ_hG−G1, and we conclude with the help of the following lemma:

Lemma 4.3.LetG∈L^q(R^N)with1≤q <∞. Then

hlim→0τ_hG−Gq=0.

Proof. Givenε >0, there exists (by Theorem 4.12) a functionG₁∈C_c(R^N)such thatG−G₁q< ε.

We write

τ_hG−Gq≤ τ_hG−τ_hG₁q+ τ_hG₁−G₁q+ G₁−Gq

≤2ε+ τhG1−G1q. Since limh→0τhG1−G1q =0 we see that

lim sup

h→0 τ_hG−Gq≤2ε ∀ε >0.

4.5 Comments on Chapter 4 115 Theorem 4.29 (Egorov).Assume thatis a measure space with finite measure.

Let(f_n)be a sequence of measurable functions onsuch that f_n(x)→f (x)a.e. on(with|f (x)|<∞a.e.).

Then∀ε > 0 ∃A⊂measurable such that|\A| < εandf_n → f uniformly onA.

For a proof, see Exercise 4.14, P. Halmos [1], G. B. Folland [2], E. Hewitt–

K. Stromberg [1], R. Wheeden–A. Zygmund [1], K. Yosida [1], A. Friedman [3], etc.

2. Weakly compact sets inL¹.

SinceL¹is not reflexive, bounded sets ofL¹do not play an important role with respect to the weak topologyσ (L¹, L^∞). The following result provides a useful characterization of weakly compact sets ofL¹.

Theorem 4.30 (Dunford–Pettis).LetFbe a bounded set inL¹(). ThenF has compact closure in the weak topologyσ (L¹, L^∞)if and only ifFis equi-integrable, that is,

(a)

⎧⎨

⎩

∀ε >0 ∃δ >0 such that

A

|f|< ε ∀A⊂,measurable with|A|< δ, ∀f ∈F and

(b)

⎧⎨

⎩

∀ε >0 ∃ω⊂, measurable with|ω|<∞such that

\ω

|f|< ε ∀f ∈F.

For a proof and discussion of Theorem 4.30 see Problem 23 or N. Dunford–

J. T. Schwartz [1], B. Beauzamy [1], J. Diestel [2], I. Fonseca–G. Leoni [1], and also J. Neveu [1], C. Dellacherie–P. A. Meyer [1] for the probabilistic aspects; see also Exercise 4.36.

3. Radon measures.

As we have just pointed out, bounded sets ofL¹enjoy no compactness properties.

To overcome this lack of compactness it is sometimes very usefulto embedL¹into a large space: the space of Radon measures.

Assume, for example, thatis a bounded open set ofR^N with the Lebesgue measure. Consider the spaceE =C()with its normu = sup_x∈ |u(x)|. Its dual space, denoted byM(), is called the space ofRadon measureson.The weaktopology onM()is sometimes called the “vague” topology.

We shall identifyL¹()with a subspace ofM(). For this purpose we introduce the mappingL¹()→M()defined as follows. Givenf ∈L¹(), the mapping u∈C()→

f u dxis a continuous linear functional onC(), which we denote Tf, so that

Tf, uE,E=

f u dx ∀u∈E.

ClearlyT is linear, and, moreover,T is anisometry, since Tf_M()= sup

u∈E u≤1

f u= f1 (see Exercise 4.26).

UsingT we may identify L¹()with a subspace ofM(). Since M()is the dual space of the separable spaceC(), it has some compactness properties in the weaktopology. In particular, if(f_n)is a bounded sequence inL¹(),there exist a subsequence(f_nk)and a Radon measureμsuch thatf_nk μ in the weaktopology σ (E, E), that is,

f_nku→ μ, u ∀u∈C().

For example, a sequence inL¹ can converge to a Dirac measure with respect to the weak topology. Some futher properties of Radon measures are discussed in Problem 24.

The terminology “measure” is justified by the following result, which connects the above definition with the standard notion of measures in the set-theoretic sense:

Theorem 4.31 (Riesz representation theorem).Letμbe a Radon measure on. Then there is a unique signed Borel measureν on(that is, a measure defined on Borel sets of)such that

μ, u =

udν ∀u∈C().

It is often convenient to replace the spaceE=C()by the subspace E₀= {f ∈C();f =0 on the boundary of}.

The dual ofE₀ is denoted byM()(as opposed toM()). The Riesz repre- sentation theorem remains valid with the additional condition that|ν|(boundary of )=0.

On this vast and classical subject, see, e.g., H. L. Royden [1], W. Rudin [2], G. B. Folland [2], A. Knapp [1], P. Malliavin [1], P. Halmos [1], I. Fonseca–

G. Leoni [1].

4. The Bochner integral of vector-valued functions.

Letbe a measure space and letEbe a Banach space. The spaceL^p(;E)consists of all functionsf defined on with values intoE that are measurable in some appropriate senseand such that

f (x)^pdμ <∞(with the usual modification whenp = ∞). Most of the properties described in Sections 4.2 and 4.3 still hold under some additional assumptions onE. For example, ifEis reflexive and 1< p <

∞, thenL^p(;E)is reflexive and its dual space isL^p(;E). For more details,

4.5 Comments on Chapter 4 117 see K. Yosida [1], D. L. Cohn [1], E. Hille [1], B. Beauzamy [1], L. Schwartz [3].

The spaceL^p(;E)is very useful in the study of evolution equations whenis an interval inR(see Chapter 10).

5. Interpolation theory.

The most striking result, which began interpolation theory, is the following.

Theorem 4.32 (Schur, M. Riesz, Thorin).Assume thatis a measure space with

||<∞, and thatT :L¹()→L¹()is a bounded linear operator with norm M₁= T_L(L¹,L¹).

Assume, in addition, thatT : L^∞() → L^∞()is a bounded linear operator with norm

M_∞= T_L(L^∞,L^∞).

ThenT is a bounded operator fromL^p()intoL^p()for all1< p <∞, and its normM_psatisfies

Mp≤M₁^1/pM_∞^1/p.

Interpolation theory was originally discovered by I. Schur, M. Riesz, G. O. Thorin, J. Marcinkiewicz, and A. Zygmund. Decisive contributions have been made by a number of authors including J.-L. Lions, J. Peetre, A. P. Calderon, E. Stein, and E. Gagliardo. It has become auseful tool in harmonic analysis(see, e.g., E. Stein–

G. Weiss [1], E. Stein [1], C. Sadosky [1]) and in partial differential equations (see, e.g., J.-L. Lions–E. Magenes [1]). On these questions see also G. B. Folland [2], N. Dunford–J. T. Schwartz [1] (Volume 1 p. 520), J. Bergh–J. Löfström [1], M. Reed–B. Simon [1], (Volume 2, p. 27) and Problem 22.

6. Young’s inequality.

The following is an extension of Theorem 4.15.

Theorem 4.33 (Young).Assumef ∈L^p(R^N)andg∈L^q(R^N)with1≤p ≤ ∞, 1≤q ≤ ∞and ¹_r =_p¹ +_q¹−1≥0.

Thenf g∈L^r(R^N)andf gr ≤ fpgq. For a proof see, e.g., Exercise 4.30.

7. The notion of convolution—extended to distributions (see L. Schwartz [1] or A. Knapp [2])—plays a fundamental role in the theory of partial differential equa- tions. For example, the equationP (D)u=f inR^N, whereP (D)is any differential operator with constant coefficients, has a solution of the formu=E f, whereE is thefundamental solutionofP (D) (theorem of Malgrange–Ehrenpreis; see also Comment 2b in Chapter 1). In particular, the equationu=f inR³has a solution of the formu=E f, whereE(x)= −(4π|x|)⁻¹.