Exercises for Chapter 4

4.5 Exercises for Chapter 4 119 1

r = α

p+1−α

q withα∈ [0,1]

and prove that

f

r ≤f^α

pf^1−α

q . 4.5 Let 1≤p <∞and 1≤q≤ ∞.

1. Prove thatL¹()∩L^∞()is a dense subset ofL^p().

2. Prove that the set

$f ∈L^p()∩L^q(); fq ≤1% is closed inL^p().

3. Let(f_n)be a sequence inL^p()∩L^q()and letf ∈L^p(). Assume that f_n→f inL^p()andf_nq ≤C.

Prove thatf ∈ L^r()and thatf_n → f inL^r()for everyrbetweenpand q, r =q.

4.6 Assume||<∞.

1. Letf ∈L^∞(). Prove that limp→∞fp= f∞.

2. Letf ∈ ∩1≤p<∞L^p()and assume that there is a constantCsuch that fp≤C ∀1≤p <∞.

Prove thatf ∈L^∞().

3. Construct an example of a functionf ∈ ∩1≤p<∞L^p()such thatf /∈L^∞() with=(0,1).

4.7 Let 1≤q ≤p ≤ ∞. Leta(x)be a measurable function on. Assume that au∈L^q()for every functionu∈L^p().

Prove thata∈L^r()with

r=

⎧⎨

⎩ pq

p−q ifp <∞, q ifp= ∞. [Hint:Use the closed graph theorem.]

4.8 LetX⊂L¹()be a closed vector space inL¹(). Assume that

X⊂

1<q≤∞

L^q().

1. Prove that there exists somep >1 such thatX⊂L^p().

[Hint:For every integern≥1 consider the set X_n=

f ∈X∩L^1+(1/n)();f

1+(1/n)≤n

.] 2. Prove that there is a constantCsuch that

fp ≤Cf1 ∀f ∈X.

4.9 Jensen’s inequality.

Assume||<∞. Letj :R→(−∞,+∞]be a convex l.s.c. function,j ≡ +∞.

Letf ∈L¹()be such thatf (x)∈D(j )a.e. andj (f )∈L¹(). Prove that j

1

||

f

≤ 1

||

j (f ).

4.10 Convex integrands.

Assume||<∞. Let 1≤p <∞and letj :R→Rbe a convex and continuous function. Consider the functionJ :L^p()→(−∞,+∞]defined by

J (u)=

⎧⎨

⎩

j (u(x))dx ifj (u)∈L¹(), +∞ ifj (u) /∈L¹().

1. Prove thatJ is convex.

2. Prove thatJ is l.s.c.

[Hint:Start with the casej ≥0 and use Fatou’s lemma.]

3. Prove that the conjugate functionJ:L^p()→(−∞,+∞]is given by J(f )=

j(f (x))dx ifj(f )∈L¹(), +∞ ifj(f ) /∈L¹().

[Hint:When 1< p <∞considerJ_n(u)=J (u)+¹_n

|u|^pand determineJ_n.]

4. Let∂j(resp.∂J )denote the subdifferential ofj (resp.J) (see Problem 2). Let u∈L^p()and letf ∈L^p(); prove that

f ∈∂J (u)⇐⇒f (x)∈∂j (u(x)) a.e. on. 4.11 The spacesL^α()with0< α <1.

Let 0< α <1. Set L^α()=

u:→R; uis measurable and|u|^α ∈L¹()

4.5 Exercises for Chapter 4 121 and

[u]α =

|u|^α 1/α

.

1. Check thatL^α is a vector space but that[ ]α is not a norm. More precisely, prove that ifu, v∈L^α(),u≥0 a.e. andv ≥0 a.e., then

[u+v]α ≥[u]α+ [v]α. 2. Prove that

[u+v]^αα≤ [u]^αα+ [v]^αα ∀u, v∈L^α().

4.12 L^pis uniformly convex for1< p≤2(by the method of C. Morawetz).

1. Let 1 < p <∞. Prove that there is a constantC(depending only onp) such that

|a−b|^p≤C(|a|^p+ |b|^p)^1−s

|a|^p+ |b|^p−2 a+b

2 ^ps

∀a, b∈R, wheres=p/2.

2. Deduce thatL^p()is uniformly convex for 1< p≤2.

[Hint:Use question 1 and Hölder’s inequality.]

4.13

1. Check that

|a+b| − |a| − |b|≤2|b| ∀a, b∈R. 2. Let(f_n)be a sequence inL¹()such that

(i) f_n(x)→f (x)a.e.,

(ii) (f_n)is bounded inL¹()i.e.,f_n1≤M ∀n.

Prove thatf ∈L¹()and that

nlim→∞

{|fn| − |fn−f|} =

|f|.

[Hint:Use question 1 witha =f_n−f andb=f, and consider the sequence ϕ_n=|f_n| − |f_n−f| − |f|.]

3. Let(fn)be a sequence inL¹()and letf be a function inL¹()such that (i) fn(x)→f (x)a.e.,

(ii) f_n1→ f. Prove thatf_n−f1=0.

4.14 The theorems of Egorov and Vitali.

Assume || < ∞. Let(f_n)be a sequence of measurable functions such that f_n→f a.e. (with|f|<∞a.e.).

1. Letα >0 be fixed. Prove that

meas[|f_n−f|> α|] −→

n→∞ 0.

2. More precisely, let

Sn(α)=

k≥n

[|fk−f|> α]. Prove that|S_n(α)| −→

n→∞ 0.

3. (Egorov). Prove that

∀δ >0 ∃A⊂ measurable such that

|A|< δandf_n→f uniformly on\A.

[Hint:Given an integerm ≥ 1, prove with the help of question 2 that there exists_m⊂, measurable, such that|_m|< δ/2^mand there exists an integer N_msuch that

|f_k(x)−f (x)|< 1

m ∀k≥N_m, ∀x∈\_m.] 4. (Vitali). Let(f_n)be a sequence inL^p()with 1≤p <∞. Assume that

(i) ∀ε >0 ∃δ >0 such that

A|f_n|^p < ε ∀nand∀A⊂measurable with

|A|< δ.

(ii) fn→f a.e.

Prove thatf ∈L^p()and thatfn→f inL^p().

4.15 Let=(0,1).

1. Consider the sequence(f_n)of functions defined byf_n(x)=ne^−nx. Prove that (i) f_n→0 a.e.

(ii) f_nis bounded inL¹().

(iii) f_n0 inL¹()strongly.

(iv) f_n 0 weaklyσ (L¹, L^∞).

More precisely, there is no subsequence that converges weaklyσ (L¹, L^∞).

2. Let 1< p <∞and consider the sequence(g_n)of functions defined byg_n(x)= n^1/pe^−nx. Prove that

(i) gn→0 a.e.

(ii) (gn)is bounded inL^p().

(iii) g_n0 inL^p()strongly.

(iv) g_n0 weaklyσ (L^p, L^p).

4.5 Exercises for Chapter 4 123 4.16 Let 1< p <∞. Let(f_n)be a sequence inL^p()such that

(i) f_nis bounded inL^p().

(ii) f_n→f a.e. on.

1. Prove thatf_n f weaklyσ (L^p, L^p).

[Hint:First show that if f_n fweaklyσ (L^p, L^p)andf_n → f a.e., then f =fa.e. (use Exercise 3.4).]

2. Same conclusion if assumption (ii) is replaced by (ii) f_n−f1→0.

3. Assume now (i), (ii), and||<∞. Prove thatf_n−fq →0 for everyqwith 1≤q < p.

[Hint:Introduce the truncated functionsT_kf_nor alternatively use Egorov’s the- orem.]

4.17 Brezis–Lieb’s lemma.

Let 1< p <∞.

1. Prove that there is a constantC(depending onp) such that |a+b|^p− |a|^p− |b|^p≤C

/|a|^p−1|b| + |a| |b|^p−10

∀a, b∈R. 2. Let(f_n)be a bounded sequence inL^p()such thatf_n → f a.e. on. Prove

thatf ∈L^p()and that

nlim→∞

$|f_n|^p− |f_n−f|^p%

=

|f|^p.

[Hint:Use question 1 witha =f_n−f andb=f. Note that by Exercise 4.16,

|f_n−f|0 weakly inL^pand|f_n−f|^p−10 weakly inL^p.]

3. Deduce that if(f_n)is a sequence inL^p()satisfying (i) f_n(x)→f (x) a.e.,

(ii) fnp→ fp, thenfn−fp→0.

4. Find an alternative method for question 3.

4.18 Rademacher’s functions.

Let 1≤p≤ ∞and letf ∈L^p_loc(R).Assume thatfisT-periodic, i.e.,f (x+T )= f (x) a.e.x ∈R.

Set

f = 1 T

T

0

f (t )dt.

Consider the sequence(u_n)inL^p(0,1)defined by

u_n(x)=f (nx), x ∈(0,1).

1. Prove thatun f inL^p(0,1)with respect to the topologyσ (L^p, L^p).

2. Determine limn→∞u_n−fp. 3. Examine the following examples:

(i) u_n(x)=sinnx,

(ii) u_n(x)=f (nx)wheref is 1-periodic and f (x)=

α forx ∈(0,1/2), β forx ∈(1/2,1).

The functions of example (ii) are calledRademacher’s functions.

4.19

1. Let(f_n)be a sequence inL^p()with 1< p <∞and letf ∈L^p(). Assume that

(i) fn f weaklyσ (L^p, L^p), (ii) fnp→ fp.

Prove thatfn→f strongly inL^p().

2. Construct a sequence(f_n)inL¹(0,1), fn ≥0, such that:

(i) f_n f weaklyσ (L¹, L^∞), (ii) f_n1→ f1,

(iii) f_n−f10.

Compare with the results of Exercise 4.13 and with Proposition 3.32.

4.20 Assume||<∞. Let 1≤p <∞and 1≤q <∞.

Leta:R→Rbe a continuous function such that

|a(t )| ≤C{|t|^p/q+1} ∀t∈R. Consider the (nonlinear) mapA:L^p()→L^q()defined by

(Au)(x)=a(u(x)), x∈.

1. Prove thatAis continuous fromL^p()strong intoL^q()strong.

2. Take =(0,1)and assume that for every sequence(u_n)such thatu_n u weaklyσ (L^p, L^p)thenAu_n Auweaklyσ (L^q, L^q).

Prove thatais an affine function.

[Hint:Use Rademacher’s functions; see Exercise 4.18.]

4.21 Given a functionu₀:R→R, setu_n(x)=u₀(x+n).

1. Assumeu₀ ∈ L^p(R)with 1 < p < ∞. Prove that u_n 0 inL^p(R)with respect to the weak topologyσ (L^p, L^p).

4.5 Exercises for Chapter 4 125 2. Assumeu₀∈ L^∞(R)and thatu₀(x)→ 0 as|x| → ∞in the following weak

sense:

for everyδ >0 the set [|u₀|> δ] has finite measure.

Prove thatu_n 0 inL^∞(R)weakσ (L^∞, L¹).

3. Takeu₀=χ_(0,1).

Prove that there exists no subsequence(u_nk)that converges inL¹(R)with respect toσ (L¹, L^∞).

4.22

1. Let(f_n)be a sequence inL^p()with 1< p≤ ∞and letf ∈L^p().

Show that the following properties are equivalent:

(A) fn f inσ (L^p, L^p).

(B)

⎧⎪

⎨

⎪⎩

f_np≤C and

Ef_n→

Ef ∀E⊂, Emeasurable and|E|<∞. 2. Ifp=1 and||<∞prove that (A)⇔(B).

3. Assumep=1 and|| = ∞. Prove that (A)⇒(B).

Construct an example showing that in general, (B)(A).

[Hint:Use Exercise 4.21, question 3.]

4. Let(f_n)be a sequence inL¹()and letf ∈L¹()with|| = ∞. Assume that (a) f_n≥0 ∀nandf ≥0 a.e. on,

(b)

fn→

f, (c)

Efn→

Ef ∀E⊂, Emeasurable and|E|<∞.

Prove thatf_n f inL¹()weaklyσ (L¹, L^∞).

[Hint:Show that

Ff_n→

Ff ∀F ⊂, F measurable and|F| ≤ ∞.]

4.23 Letf :→ Rbe a measurable function and let 1≤ p≤ ∞. The purpose of this exercise is to show that the set

C=$

u∈L^p(); u≥f a.e.% is closed inL^p()with respect to the topologyσ (L^p, L^p).

1. Assume first that 1≤p <∞. Prove thatC is convex and closed in the strong L^ptopology. Deduce thatCis closed inσ (L^p, L^p).

2. Takingp= ∞, prove that

C =

⎧⎨

⎩u∈L^∞()

uϕ≥

f ϕ ∀ϕ∈L¹() with f ϕ∈L¹() and ϕ ≥0 a.e.

⎫⎬

⎭.

[Hint:Assume first thatf ∈ L^∞(); in the general case introduce the sets ω_n = [|f|< n].]

3. Deduce that whenp= ∞,Cis closed inσ (L^∞, L¹).

4. Letf₁, f₂∈L^∞()withf₁≤f₂a.e. Prove that the set C=$

u∈L^∞(); f₁≤u≤f₂ a.e.% is compact inL^∞()with respect to the topologyσ (L^∞, L¹).

4.24 Letu∈L^∞(R^N). Let(ρ_n)be a sequence of mollifiers. Let(ζ_n)be a sequence inL^∞(R^N)such that

ζn_∞≤1 ∀n and ζn→ζ a.e. onR^N. Set

vn=ρn (ζnu) and v=ζ u.

1. Prove thatv_n v inL^∞(R^N)weakσ (L^∞, L¹).

2. Prove that

B|v_n−v| →0 for every ballB.

4.25 Regularization of functions inL^∞().

Let⊂R^Nbe open.

1. Letu∈L^∞(). Prove that there exists a sequence(u_n)inC_c^∞()such that (a) u_n∞≤ u_∞ ∀n,

(b) u_n→ua.e. on, (c) un

u inL^∞()weakσ (L^∞, L¹).

2. Ifu≥0 a.e. on, show that one can also take (d) u_n≥0 on ∀n.

3. Deduce thatC^∞_c ()is dense inL^∞()with respect to the topologyσ (L^∞, L¹).

4.26 Let⊂R^N be open and letf ∈L¹_loc().

1. Prove thatf ∈L¹()iff A=sup

f ϕ;ϕ∈C_c(), ϕ_∞≤1

<∞. Iff ∈L¹()show thatA= f1.

2. Prove thatf⁺∈L¹()iff B =sup

f ϕ;ϕ∈C_c(), ϕ_∞≤1 andϕ≥0

<∞. Iff⁺∈L¹()show thatB= f⁺1.

3. Same questions whenC_c()is replaced byC_c^∞().

4.5 Exercises for Chapter 4 127 4. Deduce that

-

f ϕ=0 ∀ϕ ∈C_c^∞() .

⇒[f =0 a.e.]

and -

f ϕ≥0 ∀ϕ ∈C_c^∞(), ϕ≥0 .

⇒[f ≥0 a.e.].

4.27 Let⊂R^Nbe open. Letu, v∈L¹_loc()withu =0 a.e. on a set of positive measure. Assume that

-

ϕ∈C_c^∞()and

uϕ >0 .

⇒ -

vϕ≥0 .

. Prove that there exists a constantλ≥0 such thatv=λu.

4.28 Letρ∈L¹(R^N)with

ρ =1. Setρ_n(x)=n^Nρ(nx). Letf ∈L^p(R^N)with 1≤p <∞. Prove thatρ_n f →f inL^p(R^N).

4.29 LetK ⊂ R^N be a compact subset. Prove that there exists a sequence of functions(u_n)inC_c^∞(R^N)such that

(a) 0≤u_n≤1 onR^N, (b) u_n=1 onK,

(c) suppun⊂K+B(0,1/n),

(d) |D^αun(x)| ≤Cαn^|α|∀x ∈R^N,∀multi-indexα(whereCαdepends only onα and not onn).

[Hint:Letχ_nbe the characteristic function ofK+B(0,1/2n); takeu_n=ρ_2n χ_n.]

4.30 Young’s inequality.

Let 1≤p≤ ∞, 1≤q≤ ∞be such that ¹_p+¹_q ≥1.

Set ¹_r = _p¹+¹_q −1, so that 1≤r≤ ∞.

Letf ∈L^p(R^N)andg∈L^q(R^N).

1. Prove that for a.e.x ∈R^N, the functiony →f (x−y) g(y)is integrable onR^N. [Hint:Setα=p/q, β=q/pand write

|f (x−y)g(y)| = |f (x−y)|^α|g(y)|^β/

|f (x−y)|^1−α|g(y)|^1−β0 .] 2. Set

(f g)(x)=

R^Nf (x−y)g(y)dy.

Prove thatf g∈L^r(R^N)and thatf gr ≤ fpgq. 3. Assume here that_p¹ +¹_q =1. Prove that

f g∈C(R^N)∩L^∞(R^N)

and, moreover, if 1< p <∞then(f g)(x)→0 as|x| → ∞.

4.31 Letf ∈L^p(R^N)with 1≤p <∞. For everyr >0 set fr(x)= 1

|B(x, r)|

B(x,r)

f (y)dy, x ∈R^N.

1. Prove thatf_r ∈ L^p(R^N)∩C(R^N)and thatf_r(x)→ 0 as|x| → ∞(rbeing fixed).

2. Prove thatf_r →f inL^p(R^N)asr→0.

[Hint:Writef_r =ϕ_r f for some appropriateϕ_r.]

4.32

1. Letf, g∈L¹(R^N)and leth∈L^p(R^N)with 1≤p ≤ ∞. Show thatf g = g f and(f g) h=f (g h).

2. Letf ∈L¹(R^N). Assume thatf ϕ =0 ∀ϕ ∈ C_c^∞(R^N). Prove thatf =0 a.e. onR^N. Same question forf ∈L¹_loc(R^N).

3. Leta ∈ L¹(R^N)be a fixed function. Consider the operatorT_a : L²(R^N) → L²(R^N)defined by

T_a(u)=a u.

Check thatT_a is bounded and thatT_aL(L²) ≤ aL¹(R^N). ComputeT_a◦T_b and prove thatT_a◦T_b=T_b◦T_a ∀a, b∈L¹(R^N). Determine(T_a),T_a◦(T_a) and(T_a)◦T_a. Under what condition onais(T_a)=T_a?

4.33 Fix a functionϕ∈C_c(R),ϕ ≡0, and consider the family of functions F=

∞ n=1

{ϕn},

whereϕn(x)=ϕ(x+n), x∈R.

1. Assume 1≤p <∞. Prove that∀ε >0∃δ >0 such that

τ_hf −fp< ε ∀f ∈Fand∀h∈Rwith|h|< δ.

2. Prove thatFdoesnothave compact closure inL^p(R).

4.34 Let 1≤p <∞and letF⊂L^p(R^N)be a compact subset ofL^p(R^N).

1. Prove thatFis bounded inL^p(R^N).

2. Prove that∀ε >0 ∃δ >0 such that

τhf −fp < ε ∀f ∈Fand∀h∈R^N with|h|< δ.

3. Prove that∀ε >0 ∃⊂R^Nbounded, open, such that

4.5 Exercises for Chapter 4 129 f

L^p(R^N\)< ε ∀f ∈F. Compare with Corollary 4.27.

4.35 Fix a functionG∈L^p(R^N)with 1≤p <∞and letF =G B, whereB is a bounded set inL¹(R^N).

Prove thatF|has compact closure inL^p()for any measurable set⊂R^N with finite measure. Compare with Corollary 4.28.

4.36 Equi-integrable families.

A subset F ⊂ L¹()is said to beequi-integrable if it satisfies the following properties:⁶

Fis bounded inL¹(),

(a)

∀ε >0 ∃δ >0 such that

E|f|< ε

∀f ∈F, ∀E⊂, Emeasurable and|E|< δ, (b)

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

\ω|f|< ε ∀f ∈F. (c)

Let (_n)be a nondecreasing sequence of measurable sets in with|_n| <

∞ ∀nand such that=

n_n. 1. Prove thatFis equi-integrable iff

tlim→∞sup

f∈F

[|f|>t]|f| =0 (d)

and

nlim→∞sup

f∈F

\n

|f| =0.

(e)

2. Prove that ifF ⊂L¹()is compact, thenF is equi-integrable. Is the converse true?

4.37 Fix a functionf ∈L¹(R)such that _+∞

−∞ f (t )dt =0 and

_+∞

0

f (t )dt >0, and letun(x)=nf (nx)forx ∈I =(−1,+1).

1. Prove that

nlim→∞

I

u_n(x)ϕ(x)dx =0 ∀ϕ ∈C([−1,+1]).

6One can show that (a) follows from (b) and (c) if the measure spaceis diffuse (i.e.,has no atoms). Consider for example=R^Nwith the Lebesgue measure.

2. Check that the sequence(u_n)is bounded inL¹(I ). Show that no subsequence of(u_n)is equi-integrable.

3. Prove that there exists no functionu∈L¹(I )such that

klim→∞

I

u_nk(x)ϕ(x)dx=

I

u(x)ϕ(x)dx ∀ϕ∈L^∞(I ), along some subsequence(un_k).

4. Compare with the Dunford–Pettis theorem (see question A3 in Problem 23).

5. Prove that there exists a subsequence(u_nk)such thatu_nk(x)→ 0 a.e. onI as k→ ∞.

[Hint:Compute

[n^−1/2<|x|<1]|u_n(x)|dxand apply Theorem 4.9.]

4.38 SetI =(0,1)and consider the sequence(u_n)of functions inL¹(I )defined by

u_n(x)=

⎧⎪

⎨

⎪⎩

n ifx∈ⁿ⁻¹

j=0

/j n,^j_n +_n¹2

0 , 0 otherwise.

1. Check that|suppun| = ¹_nandun1=1.

2. Prove that

n→+∞lim

I

u_n(x)ϕ(x)dx=

I

ϕ(x)dx ∀ϕ∈C([0,1]).

[Hint:Start with the caseϕ ∈C¹([0,1]).]

3. Show that no subsequence of(u_n)is equi-integrable.

4. Prove that there exists no functionu∈L¹(I )such that

klim→∞

I

u_nk(x)ϕ(x)dx=

I

u(x)ϕ(x)dx ∀ϕ∈L^∞(I ), along some subsequence(un_k).

[Hint:Use a further subsequence(u_n

k)such that

k|suppu_n k|<1.]

5. Prove that there exists a subsequence(u_nk)such thatu_nk(x)→ 0 a.e. onI as k→ ∞.

Chapter 5