A review of functional analysis tools for PDEs

A review of functional analysis tools for PDEs

Master 2, 2021-2022 University Paris-Dauphine

David Gontier

(version: September 14, 2021).

A review of PDEdeDavid Gontierest mis à disposition selon les termes de lalicence Creative Commons Attribution- Pas d’Utilisation Commerciale - Partage dans les Mêmes Conditions 4.0 International.

INTRODUCTION

The present manuscript contains the notes for a one-week course (15h) given at University Paris- Dauphine, entitled «A review in PDE» .

The presentation of the notes, the results, and most of the remarks are taken from the following books:

• Analysis by E. Lieb and M. Loss (in English) [LL01];

• Analyse Fonctionelle by H. Brezis (in French) [Bre99];

• Éléments d’analyse fonctionnelle by F. Hirsh and G. Lacombe (in French, with many exerci- ces) [HL09].

Some other references are

• Théorie des Distributions by L. Schwartz (in French, for the chapter on distributions) [Sch66];

• Partial Differential Equations by L.C. Evans (in English, very complete, hence quite long) [Eva10].

• Elliptic partial differential equations of second order by D. Gilbarg and N.S. Trudinger (in En- glish, when you cannot find the results in the other books) [GT15].

Most of the proofs of the theorems are simplified, and there will be links to the corresponding

theorems in these books.

CONTENTS

1 The Lebesgue L

^p

spaces 6

1.1 Notation and first facts . . . . 6

1.1.1 Basics in measure theory . . . . 6

1.1.2 Integrable functions . . . . 7

1.1.3 The «powerful» theorems . . . . 8

1.2 The L

^p

spaces . . . . 9

1.2.1 Definitions . . . . 9

1.2.2 The useful inequalities . . . . 10

1.2.3 Convolution in Lebesgue spaces . . . . 12

1.2.4 Convolution as a smoothing operator . . . . 13

1.3 L

^p

spaces as Banach spaces . . . . 15

1.3.1 Basics in Banach spaces . . . . 15

1.3.2 Completion of L

^p

spaces . . . . 16

1.3.3 Separability of L

^p

spaces . . . . 17

1.3.4 Duality in L

^p

spaces . . . . 17

1.4 Topologies of L

^p

spaces . . . . 18

1.4.1 Basics in topologies in Banach spaces . . . . 18

1.4.2 Weak convergence which are not strong in L

^p

. . . . 18

1.4.3 Banach-Alaoglu theorem in L

^p

. . . . 19

1.5 Lower-semi continuity and convexity . . . . 20

1.6 Additional exercices . . . . 21

2 Distributions 22 2.1 Distributions . . . . 22

2.1.1 Definition and examples . . . . 22

2.1.2 Operations on distributions . . . . 23

2.2 Example: the Poisson’s equation . . . . 26

2.2.1 Integration by parts on domains . . . . 26

2.2.2 Green’s functions and the Poisson’s equation . . . . 26

2.3 Sobolev spaces W

^m,p

(Ω) and H

^m

(Ω) . . . . 28

2.3.1 Definition . . . . 28

2.3.2 Completion of Sobolev spaces . . . . 28

2.4 Sobolev spaces W

₀^m,p

(Ω) and H

₀^m

(Ω) . . . . 29

2.4.1 Definition . . . . 29

2.4.2 Poincaré’s inequalities . . . . 29

CONTENTS 5

3 Hilbert spaces, and Lax-Milgram theorem 32

3.1 Hilbert spaces . . . . 32

3.1.1 The dual of a Hilbert space . . . . 33

3.1.2 Basis of an Hilbert space . . . . 34

3.2 Lax-Milgram theorem . . . . 34

3.3 Some examples of applications . . . . 36

3.3.1 The Laplace equation . . . . 36

3.3.2 The Dirichlet equation in a bounded domain . . . . 37

3.3.3 Neumann problem on bounded domain . . . . 38

4 Complements on Sobolev spaces 39 4.1 Basics in operator theory . . . . 39

4.2 Extension operators . . . . 39

4.2.1 Extension operator on half-space . . . . 40

4.2.2 Extension operators on domains with smooth boundary . . . . 41

4.3 Sobolev embeddings . . . . 42

4.3.1 Sobolev embeddings on the whole space . . . . 42

4.3.2 Morrey’s embedding in the whole space . . . . 43

4.3.3 Compact embedding in bounded domains . . . . 44

4.4 Trace operators . . . . 45

4.4.1 Trace operators on the half-space . . . . 46

4.4.2 Trace operators on bounded domains . . . . 46

5 Optimisation 48 5.1 Euler-Lagrange equations . . . . 48

5.1.1 Application, defocusing NLS in a bounded domain . . . . 49

5.1.2 Application, focusing NLS in a bounded domain . . . . 51

5.2 Spectral decomposition of compact symmetric operators . . . . 52

5.2.1 Basic notions in operator theory . . . . 52

5.2.2 Decomposition of compact symmetric operators . . . . 53

5.2.3 Application: the spectrum of Dirichlet Laplacien in bounded domains . . . . . 54

6 Fourier transform 56 6.1 Fourier transform in L

¹

. . . . . 56

6.2 Fourier transform in L

²

. . . . 59

6.3 Fourier transform for distributions . . . . 60

6.4 Application: the heat kernel . . . . 61

CHAPTER 1

THE LEBESGUE L

^P

SPACES

In this Chapter, we define the Lebesgue L

^p

spaces. We focus on the special case where the measure is the Lebesgue one on R

^d

.

1.1 Notation and first facts

First, we recall basic facts about measure theory and integration. The reader who is not familiar with the theory can refer to [LL01, Chapter 1].

1.1.1 Basics in measure theory

The open ball of R

^d

of center x ∈ R

^d

and radius r > 0 is denoted by B(x, r) = n

y ∈ R

^d

, kx − yk

_Rd

< r o .

The Borel sigma-algebra of R

^d

is the one generated by the family of all open balls of R

^d

. There is a natural measure on this sigma-algebra, called the Lebesgue measure, denoted Leb

d

(A), L

^d

(A),

|A| or dx(A) , which is the one for which

Leb

_d

(B(x, r)) := |B(x, r)| := | S

^d−1

|

d r

^d

, with | S

^d−1

| := 2π

^d/2

Γ(d/2) ,

where Γ is the usual Euler’s Gamma function. Recall that Γ(x + 1) = xΓ(x) for all x > 0 , and that Γ(1/2) = √

π while Γ(1) = 1 . This gives the usual well-known formulae Leb

₁

(B(x, r)) = 2r, Leb

₂

(B(x, r)) = πr

²

, Leb

₃

(B(x, r)) = 4

3 πr

³

, etc.

By construction, the Lebesgue measure is translation invariant: Leb

_d

(A) = Leb

_d

(A + y) for all y ∈ R

^d

. We say that a property P : R

^d

→ { True , False } holds almost everywhere (and write a.e. ) if P

⁻¹

{ False } is (contained in a Borel set) of measure 0 .

Example 1.1 (Countable sets have 0 measure) . For all x ∈ R

^d

, we have Leb

_d

({x}) = 0. By countable additivity, we deduce that if C ∈ R

^d

is countable, then C is Borel-measurable, and Leb

d

(C) = 0.

For instance, since Q is countable, the assertion « x is irrational» holds almost everywhere.

1.1. Notation and first facts 7 In the sequel, Ω always denotes an non empty open set of R

^d

.

We say that a function f : Ω → R is (Borel or Lebesgue)- measurable if, for all λ ∈ R, the set {f > λ} := {x ∈ Ω, f (x) > λ}

is Borel-measurable.

Exercice 1.2

Prove that if f is measurable, then for allλ∈R, the set {f < λ}is measurable.

Hint: prove that{f ≤λ+ 1/n}is measurable, and that {f < λ}=S

n≥1{f ≤λ+ 1/n}

1.1.2 Integrable functions We now focus on Integrable functions.

Definition 1.3 (Lebesgue integration) . A positive measurable function f : Ω → R

+

is (Lebesgue)- integrable if the function F

f

(λ) := Leb

d

({f > λ}) is Riemann integrable. In this case, its integral

is ˆ

Ω

f(x)dx :=

ˆ

∞ 0

F

_f

(λ)dλ. (1.1)

Remark 1.4. The function F

_f

: R

+

→ R

+

is positive and decreasing, so the Riemann sums always converge (why?). Being integrable only means that the limit is not +∞, that is ´

Ω

f = ´

_∞

0

F

_f

< ∞.

This formula is formally easy to understand from the Fubini’s theorem (see Theorem 1.11 below).

Indeed, if 1 (x > 0) denotes the Heaviside function, we have ˆ

_∞

0

F

f

(λ)dλ = ˆ

_∞

0

ˆ

Ω

1 (f (x) > λ)dx

dλ = ˆ

Ω

ˆ

_∞

0

1 (f (x) > λ)dλ

dx = ˆ

Ω

f (x)dx.

If f : Ω → R is not positive valued, we introduce

f

₊

:= max{f, 0} and f

−

:= max{−f, 0}.

These two functions are positive valued, and we have f = f

₊

− f

−

and |f| = f

₊

+ f

−

. In this case, we say that f is integrable if f

₊

and f

−

are integrable.

Exercice 1.5

Prove that a measurable functionf is integrable iff|f|is integrable.

We admit the following result.

Lemma 1.6. If f is Riemann integrable, then it is Lebesgue integrable, and the two integrals coincide.

Remark 1.7. There is a slight change of notation between the Riemann and Lebesgue integral in the case the one-dimensional case d = 1. If f is positive, we can write

ˆ

[a,b]

f (x)dx (Lebesgue notation) or

ˆ

_b

a

f (x)dx (Riemann notation)

The first integral is always positive, while the second one is positive if a < b, and negative if b < a.

It is unclear from Definition 1.3 that the integral is linear: ´

(f + g) = ´ f + ´

g . It is however the

case (this is a consequence of Theorem ?? below). So we can use Lebesgue integration as usual .

1.1. Notation and first facts 8

1.1.3 The «powerful» theorems

There are three powerful theorems that describe how limits of functions behave with the integral.

These theorems are enunciated in [Bre99, Thm IV.1 and IV.2] and proved in [LL01, Thm 1.6, 1.7 and 1.8]

Theorem 1.8: Monotone Convergence Theorem

If (f

j

) is a sequence of measurable functions, increasing in the sense f

j

(x) ≤ f

j+1

(x) a.e., then f (x) := lim f

j

(x) is measurable, and

ˆ

Ω

f(x)dx = lim

j→∞

ˆ

Ω

f

j

(x)dx.

In other words, increasing sequence implies ´

lim = lim ´

. The last value can be infinite, in which case f is not integrable.

Proof. Replacing f

_j

by f

_j

− f

₁

, we may assume that f

_j

≥ 0 is positive valued.

We set F

_j

:= F

_fj

= Leb

_d

({f

_j

> λ}) . Since f

_j+1

(x) ≥ f

_j

(x) , we have F

_j+1

(λ) > F

_j

(λ) . So (F

_j

) is a monotone family of decreasing functions, which converges point-wise to F

f

(λ) . We leave the rest of the proof to the reader. It is a classical exercise in the theory of Riemann integration.

Theorem 1.9: Fatou’s theorem

Let (f

j

) be a sequence of positive measurable functions. Then f := lim inf

j→∞

f

j

is positive, measurable, and

0 ≤ ˆ

Ω

f(x)dx ≤ lim inf

j→∞

ˆ

Ω

f

_j

(x)dx.

In other words, positivity implies ´

lim inf ≤ lim inf ´

. A mnemotechnic trick is that the sum of minima is always lower than the minimum of the sum .

Proof. Define g

_k

(x) := inf

j≥k

f

_j

(x) . The sequence g

_k

is measurable, increasing, with lim

k→∞

g

_k

= lim inf

j→∞

f

j

= f . By the Monotone Convergence Theorem 1.8, we have

ˆ

Ω

f(x)dx = lim

k→∞

ˆ

Ω

g

k

(x)dx.

Now, we see that g

k

≤ f

j

for all k ≤ j, so ´

g

k

≤ inf

j≥k

´ f

j

, and the result follows.

Finally, we have the master Theorem.

Theorem 1.10: Dominated Convergence Theorem

Let (f

_j

) be a sequence of measurable functions which converges point-wise to f a.e. Assume there is an integrable function G so that |f

_j

|(x) ≤ G(x) ( domination ). Then |f | ≤ G(x) and

ˆ

Ω

f(x)dx = lim

j→∞

ˆ

Ω

f

_j

(x)dx.

In other words, domination implies lim ´

= ´ lim.

Proof. We only do the proof for positive functions f

_j

. By Fatou’s theorem 1.9, we have lim inf

ˆ f

_j

≥

ˆ

f.

1.2. The L

^p

spaces 9 On the other hand, since G− f

_j

is also a family of positive functions, we have, by Fatou’s lemma again

lim inf ˆ

G − f

j

≥ ˆ

G − f, so lim sup ˆ

f

j

≤ ˆ

f.

This proves lim sup ´ f

_j

≤ ´

f ≤ lim inf ´

f

_j

, and the result follows.

To see why the domination is important, the reader should keep in mind the following three counterexamples .

• The mass goes to infinity . Let ψ ∈ C

₀^∞

(R

^d

, R

⁺

) and e ∈ R

^d

\ {0} . Then f

j

(x) := ψ(x − je) converges point-wise to f = 0 . However, ´

f

_j

= ´

ψ > 0 , while ´

f = 0 .

• The mass spreads over . Consider now f

_j

(x) = j

^−d

ψ(x/j) . Again, f

_j

converges point-wise to f = 0 (the convergence is even uniform). However, ´

f

j

= ´

ψ > 0, while ´

f = 0.

• The mass concentrates . Take f

j

(x) = j

^d

ψ(jx) . Then f

j

converge point-wise to f for all x 6= 0 , so a.e. However, ´

f

_j

= ´

ψ > 0 , while ´

f = 0 .

The following theorem is of different nature, but we include it here, as it is also powerful . We skip the proof, which can be found in [LL01, Thm 1.12]

Theorem 1.11: Fubini’s theorem If f : R

^d

× R

^s

→ R

⁺

is measurable, then

ˆ

R^d+s

f (x, y)d

^d+s

((x, y)) = ˆ

R^s

ˆ

R^d

f(x, y)d

^d

x

d

^s

y = ˆ

R^d

ˆ

R^s

f (x, y)d

^s

y

d

^d

x.

1.2 The L

^p

spaces

We now introduce the Lebesgue L

^p

(Ω) space. We focus on the case 1 ≤ p ≤ ∞ . The case p = ∞ is always a bit special .

1.2.1 Definitions For 1 ≤ p < ∞ , we define

L

^p

(Ω) := {f measurable from Ω to C , kf k

_L^p

< ∞} , where kf k

_L^p

:=

ˆ

Ω

|f|

^p

(x)dx

1/p

,

and for p = ∞ , we set

L

^∞

(Ω) := {f measurable from Ω to C, bounded a.e. }, kf k

_L^∞

:= inf{λ ≥ 0, Leb

_d

({|f | > λ}) = 0}.

We have

• kf k

_Lp

= 0 iff f = 0 almost-everywhere;

• kλfk

_L^p

= |λ| · kf k

_L^p

;

• kf + gk

_L^p

≤ kf k

_L^p

+ kgk

_L^p

(Minkowski inequality, see Theorem 1.16 below).

This proves that the map k · k is a semi -norm, in the sense that kf k

_L^p

= 0 does not imply f = 0 ,

but only that f = 0 almost everywhere. To cure this problem, we introduce the equivalent relation

f ∼ g iff f = g almost everywhere, and define L

^p

(E) := L

^p

(E)/ ∼ . In practice, this means that

elements of L

^p

(E) are not functions, but classes of functions. However, we usually say a function f in

L

^p

(E), with the convention that f is only defined almost everywhere. For instance, we say f ∈ L

^p

(E )

is continuous to state that there is a continuous representation of f .

1.2. The L

^p

spaces 10

1.2.2 The useful inequalities

After the powerful theorems come the powerful inequalities.

Theorem 1.12: Jensen’s inequality

Let J : R

⁺

→ R be a convex function, f : Ω → R be measurable, and µ : Ω → R

⁺

be measurable with ´

Ω

µ = 1 , then ˆ

Ω

J (f )µ ≥ J ˆ

Ω

f µ

.

The theorem is also valid if µ is a measure. Taking f (x) = x and µ = P

n

i=1

λ

i

δ

xi

(sum of Dirac masses) with P

λ

_i

= 1 , we obtain

n

X

i=1

λ

i

J (x

i

) ≥ J

n

X

i=1

λ

i

x

i

! ,

which is a well-known property of convex functions. Jensen’s inequality is somehow a n = ∞ version of this inequality.

Proof. Assume J differentiable for simplicity. Since J is convex, we have for all a, b ∈ R, J(a) ≥ J (b) + J

⁰

(b)(a − b).

Taking b = ´

Ω

f µ , and a = f(x) gives J (f (x)) ≥ J

ˆ

Ω

f µ

+ J

⁰

ˆ

Ω

f µ

×

f(x) − ˆ

Ω

f µ

.

We multiply this inequality by µ(x) (which is positive) and integrate. The term in bracket vanishes since ´

Ω

µ = 1 , and the result follows.

Theorem 1.13: Hölder’s inequality Let 1 ≤ p, q ≤ ∞ be such that

1 p + 1

q = 1, or, equivalently, q = p p − 1 .

Let f ∈ L

^p

(Ω) and q ∈ L

^q

(Ω) . Then f g ∈ L

¹

(Ω) , and

kf gk

_L1(Ω)

≤ kfk

_Lp(Ω)

kgk

_Lq(Ω)

.

In the case p = q = 2 , we recover the Cauchy-Schwarz inequality ´

f g ≤ kf k

_L2

kgk

_L2

. In the sequel, we denote by

p

⁰

:= p

p − 1 (the dual exponent of p). (1.2)

Proof. Without loss of generality, we may assume that f and g are positive. We introduce G :=

g/kgk

_L^q

, which satisfies kGk

_L^q

= 1. We then set µ(x) = G

^q

(x), F(x) := f (x)/G

^q/p

(x) and J(t) := t

^p

, which is convex. We apply Jensen’s inequality to (J, F, µ) , which gives

ˆ

Ω

f G

^q/p

p

G

^q

≥ ˆ

Ω

f G

^q/p

G

^q

p

.

1.2. The L

^p

spaces 11 with (we use that q(1 −

¹_p

) = 1 in the last equality)

ˆ

Ω

f G

^q/p

p

G

^q

= ˆ

Ω

f

^p

, and ˆ

Ω

f G

^q/p

G

^q

p

= ˆ

Ω

f G

^q(1−1p)

p

= ˆ

Ω

f G

p

.

So ˆ

Ω

f g kgk

_L^q

p

= ˆ

Ω

f G

p

≤ ˆ

Ω

f

^p

, hence kf gk

^p_L1

= ˆ

Ω

f g

p

≤ kf k

^p_Lp

kgk

^p_Lq

.

The following form of the Hölder’s inequality is often used.

Theorem 1.14: Holder’s inequality in the general case Let 1 ≤ p, q, r ≤ ∞ be such that

1 p + 1

q = 1 r . Let f ∈ L

^p

(Ω) and g ∈ L

^q

(Ω) . Then f g ∈ L

^r

(Ω) , and

kf gk

_L^r

(Ω) ≤ kf k

_Lp(Ω)

kgk

_Lq(Ω)

.

Proof. We use Hölder’s inequality with F = |f|

^r

∈ L

^p/r

(E) and G = |g|

^r

∈ L

^q/r

(E) . We have 1/(p/r) + 1/(q/r) = 1 , so F G ∈ L

¹

, and

kf gk

^r_Lr

= ˆ

Ω

|f |

^r

|g|

^r

= kF Gk

_L1

≤ kF k

_Lp/r

kGk

_Lq/r

= ˆ

Ω

|f |

^p

r/p

ˆ

Ω

|g|

^q

r/q

= (kf k

_Lp

· kgk

_Lq

)

^r

.

Another corollary which is very useful is the following.

Theorem 1.15: Interpolation

If 1 ≤ p

₁

≤ p

₂

≤ ∞ , and f ∈ L

^p1

(Ω) ∩ L

^p2

(Ω) , then for all p ∈ [p

₁

, p

₂

] , we have f ∈ L

^p

(Ω) with kf k

_L^p

≤ kf k

^α_Lp1

kf k

^1−α_Lp2

, where 0 ≤ α ≤ 1 is chosen so that 1

p = α

p

₁

+ 1 − α p

₂

.

Proof. Write f = f

^α

f

^(1−α)

, with f

^α

∈ L

^p1/α

and f

^(1−α)

∈ L

^p2/(1−α)

, and apply the previous result.

Finally, we prove Minkowski’s inequality.

Theorem 1.16: Minkowski’s inequality For all f, g ∈ L

^p

(Ω) with 1 ≤ p ≤ ∞ , we have

kf + gk

_Lp

≤ kf k

_Lp

+ kgk

_Lp

. In particular, the map f → kf k is convex.

Proof. First, we have the easy bound |f + g|

^p

≤ (2 max{f, g})

^p

≤ |2f |

^p

+ |2g|

^p

, so f + g is indeed in L

^p

(Ω) . Then, we have

ˆ

Ω

|f + g|

^p

= ˆ

Ω

|f + g| · |f + g|

^p−1

≤ ˆ

Ω

|f| · |f + g|

^p−1

+ ˆ

Ω

|g| · |f + g|

^p−1

.

1.2. The L

^p

spaces 12 We then use Hölder’s inequality with f ∈ L

^p

and |f + g|

^p−1

∈ L

^q

with q =

_p−1^p

, and get

ˆ

Ω

|f | · |f + g|

^p−1

≤ ˆ

Ω

|f |

^p

1/p

ˆ

Ω

|f + g|

^p

^p−1

p

.

This gives kf + gk

^p_Lp

≤ (kf k

_L^p

+ kgk

_L^p

) kf + gk

^p−1_Lp

, which is also kf + gk

_L^p

≤ kf k

_L^p

+ kgk

_L^p

. 1.2.3 Convolution in Lebesgue spaces

In this section, we take Ω = R

^d

(convolution is only defined on the full space). Let f, g be two complex-valued functions. We define the convolution f ∗ g by

(f ∗ g)(x) :=

ˆ

R^d

f (y)g(x − y)dy.

The reader can check that f ∗ g = g ∗ f, and that (f ∗ g) ∗ h = f ∗ (h ∗ g): the convolution is commu- tative and associative .

Convolution from L

^p

× L

^q

→ L

^r

Thanks to Hölder’s inequality 1.13, we see that for fixed x ∈ R

^d

, the integrand defining the convolution is integrable whenever f ∈ L

^p

( R

^d

) and g ∈ L

^q

( R

^d

) with 1/p + 1/q = 1 , and we have

∀x ∈ R

^d

, |(f ∗ g)(x)| ≤ ˆ

R^d

|f (y)g(x − y)| dy ≤ kfk

_L^p

kg(x − ·)k

_L^q

= kf k

_L^p

kgk

_L^q

. So the function f ∗ g is bounded, that is f ∗ g ∈ L

^∞

( R

^d

) , with

kf ∗ gk

_L^∞

≤ kfk

_L^p

kgk

_L^q

.

On the other hand, if f ∈ L

¹

(R

^d

) and g ∈ L

¹

(R

^d

), we have by Fubini theorem that f ∗ g ∈ L

¹

(R

^d

) with

kf ∗ gk

_L1

= kf k

_L1

kgk

_L1

.

The generalisation of these two (in)-equalities is called Young’s inequality (see [LL01, Theorem 4.2]).

Theorem 1.17: Young’s inequality, first form Let 1 ≤ p, q, r ≤ ∞ be such that

1 p + 1

q = 1 + 1 r . If f ∈ L

^p

( R

^d

) and g ∈ L

^q

( R

^d

) , then f ∗ g ∈ L

^r

( R

^d

) , and

kf ∗ gk

_L^r

≤ kf k

_L^p

kgk

_L^q

.

One other way to state this theorem is as follows.

Theorem 1.18: Young’s inequality, second form Let 1 ≤ p, q, r ≤ ∞ with

1 p + 1

q + 1 r = 2.

Let f ∈ L

^p

( R

^d

) , g ∈ L

^q

( R

^d

) and h ∈ L

^r

( R

^d

) . Then the function (f ∗ g)h is in L

¹

( R

^d

) , and

¨

(R^d)²

f(x)g(y − x)h(y)dxdy

≤ k(f ∗ g)hk

_L1

≤ kf k

_L^p

kgk

_L^q

khk

_L^r

.

1.2. The L

^p

spaces 13 The fact that these two inequalities are equivalent comes from the duality result presented in Theorem 1.24 below. In this second version, the variable r plays the role of r/(r − 1) of the first version). We prove the second version following [LL01, Theorem 4.2].

Proof. Without loss of generality, we may assume that f, g, h are positive. Let p

⁰

, q

⁰

, r

⁰

be the dual powers of p, q, r , see Eqn. (1.2), and set



 

 

α(x, y) := f (x)

^p/r0

g(y − x)

^q/r0

β(x, y) := g(y − x)

^q/p0

h(y)

^r/p0

γ(x, y) := h(y)

^r/q0

f (x)

^p/q0

. We have

1 r

⁰

+ 1

p

⁰

+ 1 q

⁰

=

1 − 1

r

+

1 − 1 p

+

1 − 1

q

= 3 − 2 = 1,

so we can use Hölder’s inequality (on (R

^d

)

²

) and get ˆ

(R^d)²

α(x, y)β(x, y)γ(x, y)dxdy ≤ kαk

_Lr0

((R^d)²)

kβk

_Lp0

((R^d)²)

kγk

_Lq0

((R^d)²)

.

The integrand in the left is also (we focus on the f terms for the computation) α(x, y)β(x, y)γ(x, y) = f (x)

p

r0+_q^p0

· · · = f (x)

^p(1−

1 p0)

· · · = f (x)g(y − x)h(y).

On the other hand, we have, by Fubini’s theorem, that kαk

^r0

L^r0((R^d)²)

=

¨

(R^d)²

f (x)

^p

g(y − x)

^q

dxdy = kf k

^p_Lp

kgk

^q_Lq

,

so indeed α ∈ L

^r0

((R

^d

)

²

). Writing similar inequalities for β and γ gives the result.

1.2.4 Convolution as a smoothing operator We now prove that, in general, f ∗ g is more regular than f . Smoothing sequences

Let j ∈ C

₀^∞

( R

^d

) be such that j is radial decreasing, with j(x) = 0 for all |x| > 1 , and ´

R^d

j = 1 . The fact that such functions exist is classical. For ε > 0 , we set

j

ε

(x) := 1 ε

^d

j

x ε

.

Since j is compactly supported in B(0, 1) , the function j

ε

is compactly supported in B(0, ε) . The scaling is chosen so that ´

R^d

j

_ε

= ´

R^d

j = 1 . The family (j

_ε

) is called a smoothing sequence , a mollifier, or an approximation of the Dirac (see Example 2.6 below).

Approximation by smooth functions

In the sequel, we say that a function f is smooth if f ∈ C

^∞

( R

^d

) . For a multi-index α = (α

1

, · · · , α

d

) ∈ N

^d

, we set

D

^α

f :=

∂

∂x

₁

α1

· · · ∂

∂x

_d

αd

f.

Recall that if f is smooth, the order of the derivatives is irrelevant, thanks to Schwartz’ Lemma. The

following Theorem shows that convolution smooths functions (see [LL01, Theorem 2.16]).

1.2. The L

^p

spaces 14

Theorem 1.19: Convolution smooths functions

Let 1 ≤ p ≤ ∞ , and let (j

_ε

) be a smoothing sequence. For all f ∈ L

^p

( R

^d

) , we set f

_ε

:= f ∗ j

_ε

. Then

• f

_ε

is smooth, and D

^α

(f ∗ j

_ε

) = f ∗ (D

^α

j

_ε

) ;

• f

_ε

∈ L

^p

( R

^d

) with kf

_ε

k

_L^p

≤ kfk

_L^p

;

• if in addition p < ∞ , then kf − f

ε

k

_L^p

→ 0 as ε → 0

⁺

.

Before we give the proof, we emphasise that the last result is false if p = ∞ . Indeed, take f a bounded discontinuous function, and assume that kf − f

_ε

k

∞

→ 0 . Then, f would the limit of the continuous functions f

ε

for the uniform convergence. This would imply that f is continuous (the uniform limit of continuous functions is continuous), a contradiction.

Proof. The inequality kf

_ε

k

_L^p

≤ kf k

_L^p

is Young’s inequality, together with the fact that kjk

_L1

= 1 . Let us prove the first point. We focus on the case p = 1 . Let e

_i

be the i -th canonical vector of R

^d

. We have

1

t (f

ε

(x + te

i

) − f

ε

(x)) = ˆ

R^d

f (y) 1

t (j

ε

(x − y + te

i

) − j

ε

(x − y))

dy.

Since j

_ε

is smooth, the term in bracket converges pointwise to (∂

_i

j

_ε

) (x − y) as t → 0 . In addition, using the mean value theorem, there is c in the segment [x − y, x − y + te

i

] so that

f (y) 1

t (j

_ε

(x + te

₁

− y) − j

_ε

(x − y))

= |f (y)(∂

_i

j

_ε

)(c)| ≤ k(∂

_i

j

_ε

)k

_L^∞

|f (y)|,

which is integrable in y , and independent of t . So, by the Dominated Convergence Theorem 1.10, we can take the limit t → 0 , and we obtain

∂

_i

(f ∗ j

_ε

) = f ∗ (∂

_i

j

_ε

) . The result follows by induction.

For the last point, we focus again on the case p = 1 . Consider first the case where f (x) = 1 (x ∈ A) , where A is a half-open rectangular set, of the form

A = (a

1

, b

1

] × · · · × (a

d

, b

d

].

Since j

_ε

is compactly supported in B(0, ε) with ´

j = 1 , the function f

_ε

:= f ∗ j

_ε

satisfies 1 (x ∈ A

⁻_ε

) ≤ f

_ε

(x) ≤ 1 (x ∈ A

⁺_ε

), with A

^±_ε

:= (a

₁

∓ ε, b

₁

± ε] × · · · × (a

_d

∓ ε, b

_d

± ε].

In particular, we have

kf

_ε

− f k

_L1

≤ max{k 1 (x ∈ A

⁺_ε

) − 1 (x ∈ A)k, k 1 (x ∈ A) − 1 (x ∈ A

⁻_ε

)k} ≈ Cε − −− →

ε→0

0.

So the result holds for f (x) = 1 (x ∈ A) . By linearity, it also holds for any finite linear combination of such functions (such combination are called really simple functions ). It is a result in measure theory that such combinations are dense in L

¹

(see [LL01, Theorem 1.18]). So, by density, the result holds for all f ∈ L

¹

.

A direct corollary is the following density result, valid for p < ∞ (see [LL01, Lemma 2.19]).

Theorem 1.20: Smooth functions are dense in L

^p

For all 1 ≤ p < ∞ , and all Ω ⊂ R

^d

, the set C

₀^∞

(Ω) is dense in L

^p

(Ω).

Again, the result is false in L

^∞

, see the paragraph after Theorem 1.19.

1.3. L

^p

spaces as Banach spaces 15

Proof. We notice that if f ∈ L

^p

(Ω) , then the function

f e (x) :=

( f (x) if x ∈ Ω, 0 else ,

is in L

^p

(R

^d

). This function is called the extension of f . Let η > 0. By the previous result, ( f e

ε

) is a family of smooth functions which converge to f e in L

^p

( R

^d

) . Restricting to Ω shows that there is ε > 0 so that f

_η

:= f e

_ε

satisfies kf

_η

− fk

_Lp(Ω)

< η .

We now prove that we can choose compactly supported functions. The Urysohn’s Lemma (see [LL01, Lemma 2.19]) states that there is a sequence of positive compactly supported functions (χ

j

) ∈ C

₀^∞

(Ω) so that, for all x ∈ Ω , we have χ

_j+1

(x) ≥ χ

_j

(x) and lim

j→∞

χ

_j

(x) = 1 . We set f

_j

(x) := χ

_j

(x)f

_η

(x) , which is smooth and compactly supported. The sequence |f

_η

− f

_j

|

^p

converges point-wise to 0 and is dominated by |f

_η

|

^p

. The Dominated Convergence Theorem 1.10 shows that kf

_η

− f

j

k

_L^p

→ 0 . So, for j large enough, we have kf

_η

− f

_j

k

_L^p

< η . This proves that f

_j

∈ C

₀^∞

(Ω) satisfies

kf − f

j

k

_L^p

≤ kf − f

η

k

_L^p

+ kf

_η

− f

j

k

_L^p

≤ 2η.

1.3 L

^p

spaces as Banach spaces

We now focus on the completion properties of L

^p

(Ω) spaces. We first recall some basic notions of Banach spaces. We then focus on the special case of L

^p

(Ω) spaces.

1.3.1 Basics in Banach spaces

A normed vectorial space (E, k · k

_E

) is a Banach space if it is complete , that is if all Cauchy sequences have limits. This is equivalent to

X

n∈N

kx

_n

k

_E

< ∞ = ⇒ X

n∈N

x

_n

converges in E.

A linear form L : E → C is continuous (or bounded ) if there is C ∈ R

⁺

so that

∀x ∈ E, |L(x)| ≤ Ckxk

_E

.

The set of all continuous forms is called the dual of E , and is denoted by E

^∗

. It is a Banach space when equipped with the norm

kLk

_op

:= kLk

_E^∗

:= sup {|L(x)|, x ∈ E, kxk

_E

≤ 1} = sup

|L(x)|

kxk

_E

, x ∈ E \ {0}

.

We sometime write

hL, xi

_E⁰_,E

:= L(x).

We recall the following (classical) Theorem, which we use all over these notes.

Theorem 1.21

Let F ⊂ E be a dense vectorial space in E , and let L : F → C be a linear functional on F such that

kLk

_op,F

:= sup{|L(x)|, x ∈ F, kxk

_E

= 1} < ∞.

Then there is a unique extension L e ∈ E

^∗

so that L e = L on F . In addition, k Lk e

_op

= kLk

_op,F

.

1.3. L

^p

spaces as Banach spaces 16 The bidual of E is the dual of the dual, that is E

^∗∗

:= (E

^∗

)

^∗

. We always have E ⊂ E

^∗∗

with the identification E 3 x 7→ M

x

∈ E

^∗∗

, where

M

_x

: E

^∗

→ C , M

_x

: L 7→ L(x).

If E = E

^∗∗

, we say that E is reflexive .

We say that E is separable if there is a countable dense set in E. This means that there is a countable family (x

n

)

n∈N

in E such that, for all x ∈ E and all ε > 0 , there is n ∈ N so that kx − x

_n

k

_E

< ε .

1.3.2 Completion of L

^p

spaces

We now focus on L

^p

(Ω) space. We start with the following (see [LL01, Theorem 2.7]).

Theorem 1.22: L

^p

is complete

Let 1 ≤ p ≤ ∞ , and let (f

j

) be a Cauchy sequence in L

^p

(Ω). Then there is f ∈ L

^p

(Ω) and a subsequence φ : N → N so that

• kf

_φ(j)

− fk

_L^p

→ 0;

• f

_φ(j)

(x) → f (x) almost everywhere.

In particular, the space L

^p

(Ω) is a Banach space (it is complete).

Proof. We prove the result for p < ∞ only. Let (f

j

) be a Cauchy sequence in L

^p

(Ω) . Up to a subsequence, we may assume that kf

_j

− f

_j+1

k

_L^p

≤ 1/2

^j

(why?). We introduce

F

_`

(x) := |f

₁

(x)| +

`−1

X

j=1

|f

_j+1

(x) − f

_j

(x)|.

By Minkowski’s inequality, F

_`

is in L

^p

(Ω) , and kF

_`

k

_L^p

≤ kf

₁

k

_L^p

+

`−1

X

j=1

kf

_j+1

− f

j

k

_L^p

≤ kf

₁

k

_L^p

+ 1.

The sequence (F

_`

) is positive and increasing. By the Monotone Convergence Theorem 1.8, the function F (x) := lim

`→∞

F

`

(x) is in L

^p

(Ω), and kF k

_L^p

≤ kf

₁

k

_L^p

+ 1.

Next, we notice that (the sum telescopes) f

_`

(x) = f

₁

(x) +

`−1

X

k=1

(f

_j+1

(x) − f

_j

(x)) .

For all fixed x ∈ Ω , the sum on the right converges absolutely in C, which is complete, hence has a limit as ` → ∞ . We set f (x) := lim

`→∞

f

`

(x). By definition, the sequence (f

`

) converges pointwise to f . In addition, we have the domination |f

_`

(x)| ≤ F

_`

(x) ≤ F (x) , which is in L

^p

(Ω) . So, by the Dominated Convergence Theorem 1.10, we have kf k

_L^p

= lim

`→∞

kf

_`

k

_L^p

< ∞ . In particular, f ∈ L

^p

(Ω) . Finally, the sequence |f − f

`

|

^p

converges pointwise to 0, and we have the domination

|f − f

_`

|

^p

≤ (|f| + |f

_`

|)

^p

≤ 2

^p

(|f |

^p

+ |f

_`

|

^p

) ≤ 2

^p

(|f |

^p

+ |F |

^p

) ,

which is integrable. Using again the Dominated Convergence Theorem shows that kf − f

_`

k

_L^p

→ 0 , as

wanted.

1.3. L

^p

spaces as Banach spaces 17

1.3.3 Separability of L

^p

spaces Theorem 1.23: Separability of L

^p

For all 1 ≤ p < ∞ , the space L

^p

(Ω) is separable. The space L

^∞

(Ω) is not separable.

Proof. Consider first 1 ≤ p < ∞ , and consider the set A of really simple functions of the form f = P

N

j=1

f

_j

1 (x ∈ A

_j

) , with f

_j

∈ ( Q + i Q ) and A

_j

the rectangles with rational boundaries. The set A is countable, and dense in L

^p

(Ω) for all 1 ≤ p < ∞ (see [LL01, Theorem 1.18]). So L

^p

(Ω) is separable.

In the case p = ∞ , let us prove that L

^∞

( R ) is not countable. We consider the following set of functions. For a subset Q ∈ Z, we define

f

_Q

(x) =

( 1 if bxc ∈ Q

0 else .

Then (f

_Q

)

Q⊂Z

is an uncountable family, and Q 6= Q

⁰

implies kf

_Q

− f

_Q⁰

k

_∞

= 1 . So L

^∞

( R ) cannot be separable (why?). We can prove similarly that L

^∞

((−1, 1)) is not separable by considering the functions g

Q

(x) := f

Q

(x/(1 − x

²

)).

1.3.4 Duality in L

^p

spaces

We state the main result, which we will prove later in Section 3.1.1 in the case p = 2 . We refer to [LL01, Theorem 2.14] for the proof in the general case. Recall that the dual exponent of p is p

⁰

= p/(p − 1) (see Eqn. (1.2)).

Theorem 1.24: The dual of L

^p

(Ω)

For all 1 < p < ∞ , the dual space of L

^p

(Ω) is (L

^p

(Ω))

^∗

= L

^p0

(Ω) . (Case p = 1). The dual space of L

¹

(Ω) is L

¹

(Ω)

∗

= L

^∞

(Ω).

( Case p = ∞ ). We have the strict inclusion L

¹

(Ω) ( (L

^∞

(Ω))

^∗

.

For 1 < p < ∞ , the space L

^p

(Ω) is reflexive, while L

¹

(Ω) and L

^∞

(Ω) are not reflexive.

The dual for L

^∞

(Ω) is a set of measures. We do not elaborate on this point.

We postpone the proof until Section 3.1.1 (in the case p = 2 only), and just remark that the inclusion L

^p0

(Ω) ⊂ (L

^p

(Ω))

^∗

comes from Hölder’s inequality. Indeed, for all g ∈ L

^p0

(Ω) , one can consider the linear form L

_g

: L

^p

(Ω) → C defined by L

_g

(f ) := ´

Ω

gf . Thanks to Hölder’s inequality, we have

|L

_g

(f )| ≤ ˆ

Ω

|gf | ≤ kgk

_Lp0

kf k

_Lp

.

This proves that L

g

∈ (L

^p

(Ω)) ∗ , with kL

_g

k

_op

≤ kgk

_Lp0

. Let us prove that there is equality. Consider f

0

:= |g|

^p0−2

g. Since g ∈ L

^p0

(Ω) and since p = p

⁰

/(p

⁰

− 1), we have f

0

∈ L

^p

(Ω) with

kf

₀

k

^p_Lp

= ˆ

Ω

|f

₀

|

^p

= ˆ

Ω

|f

₀

|

p0 p0−1

=

ˆ

Ω

|g|

^p0

= kgk

^p0

L^p0

. On the other hand, we have

L

_g

(f

₀

) = ˆ

Ω

gf

₀

= ˆ

Ω

|g|

^p0

= kgk

^p0

L^p0

= kgk

_Lp0

kf

₀

k

_L^p

.

We deduce that kL

_g

k

_op

= kgk

_Lp0

. In practice, we identify L

_g

and g .

1.4. Topologies of L

^p

spaces 18

1.4 Topologies of L

^p

spaces

We now focus on the different topologies of L

^p

spaces.

1.4.1 Basics in topologies in Banach spaces

Let E be a Banach space. We can consider several topologies on E. The more natural one is the strong topology , defined by the following notion of convergence:

x

n

−−−→

n→∞

x, iff kx

_n

− xk

_E

→ 0. ( strong convergence ) .

We can also define the weak topology of E. This one is defined by the following notion:

x

n

, −−−→

^weak

n→∞

x, iff ∀L ∈ E

^∗

, hL, x

_n

− xi

_E^∗_,E

→ 0. ( weak convergence ) .

Finally, we sometime use the weak-∗ topology. This only applies if E = F

^∗

is already the dual space of another Banach space F . Then

x

_n

, −−−−→

^weak-*

n→∞

x, iff ∀f ∈ F, hx

_n

− x, fi

_F^∗_,F

→ 0. ( weak-* convergence if E = F

^∗

) . The weak-* topology will be used in the space E = L

^∞

(Ω) , which is the dual of F = L

¹

(Ω) .

Theorem 1.25

If x

n

→ x strongly in E, then x

n

→ x weakly-(*).

If E is reflexive, then weak-* convergence is equivalent to weak convergence.

If x

_n

→ x for any of these topologies, then (x

_n

) is bounded in E .

If x

_n

→ x strongly in E , and L

_n

→ L weakly-(*) in E

^∗

, then L

_n

(x

_n

) → L(x) in C.

Proof. For the first point, we write that

|L(x

_n

− x)| ≤ kLk

_E^∗

kx

_n

− xk

_E

−−−→

n→∞

0.

For the second point, we admit that since E = F

^∗

is reflexive, then so if F . This gives F = F

^∗∗

= E

^∗

, and the result follows. The third point is a non trivial result which we also admit (see [Bre99, Proposition III.12]). Finally for the last point, (L

_n

) converges to L , so it is bounded in E

^∗

. This gives

|L

_n

(x

_n

) − L(x)| ≤ |L

_n

(x

_n

− x)| + |(L

_n

− L)(x)| ≤

max

n

kL

_n

k

_E^∗

kx

_n

− xk

_E

+ |(L

_n

− L)(x)| .

Let n → ∞ , and the result follows.

1.4.2 Weak convergence which are not strong in L

^p

Let 1 ≤ p < ∞ , and consider the special case E = L

^p

(R). There are several ways for a subsequence (f

n

) to weakly converge to f , and to not strongly converge to f .

• The mass goes to infinity;

• The mass vanishes;

• Oscillations.