Lebesgue spaces

UPMC Basic functional analysis

Master 1, MM05E 2011-2012

Lebesgue spaces

In the sequel,(X,M, µ)stands for a measure space.

1) Interpolation inequality. Let1≤p≤q≤+∞.

a) Show that if f ∈L^p(X, µ)∩L^q(X, µ), thenf ∈L^r(X, µ)for allr∈[p, q], and that

kfk_r≤ kfk^α_pkfk^1−α_q ,

whereα∈[0,1]is defined by ¹_r =^α_p +^1−α_q .

b) Show that ifµ(X)<+∞andf ∈L^p(X, µ), thenf ∈L^r(X, µ)for allr∈[1, p], and that there exists a constantC >0 (independent off) such that

kfkr≤Ckfkp.

2) Generalized Hölder inequality. Letf1∈L^p1(X, µ), . . . , fk∈L^pk(X, µ)be such that 1

p₁ +· · ·+ 1 p_k =: 1

r ≤1.

Show that the productQk

i=1f_i belongs toL^r(X, µ)and

k

Y

i=1

f_{i r}

≤

k

Y

i=1

kfikpi.

3) Let1≤p0<+∞.

a) Show that if f ∈L^p0(X, µ)∩L^∞(X, µ), then

p→∞lim kfkp=kfk_∞.

b) Let f ∈L^p(X, µ)for allp∈[p0,+∞)such thatkfkp→ ∞asp→ ∞. Show thatf /∈L^∞(X, µ).

c) Letf ∈L^p(X, µ)for allp∈[p0,+∞)such thatf /∈L^∞(X, µ). Show thatkfkp→ ∞asp→ ∞.

4) Continuity of the translation inL^p(R^N). LetX =R^N,M=L(R^N)be theσ-algebra of all Lebesgue measurable subsets ofR^N, andµ=L^N be the Lebesgue measure. Let1≤p <∞andf ∈L^p(R^N). For each h∈R^N, we define the translation off by

τ_hf(x) :=f(x−h) ∀x∈R^N.

Show that

lim

|h|→0kτhf−fkp= 0.

5) Assume thatµis a probability measure, i.e.,µ(X) = 1. Letf :X →[0,+∞)be a function inL¹(X, µ).

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a) Using Hölder’s inequality, show that ifµ({f >0})<1, thenkfkp→0 asp→0.

b) Show that

p→0lim Z

X

f^pdµ=µ({f >0}).

c) Show that for allp∈(0,1), and ally∈(0,+∞), then

|y^p−1|

p ≤y+|logy|.

d) From now on, we assume thatf >0 onX, and thatlogf ∈L¹(X, µ). Show that

p→0lim Z

X

f^p−1 p dµ=

Z

X

logf dµ.

e) Show that

p→0limkfkp= exp Z

X

logf dµ

.

6) Jensen’s inequality. Assume thatµ is a probability measure,i.e., µ(X) = 1. Letϕ: (a, b)→Rbe a convex function (with−∞ ≤a < b≤+∞).

a) Show that

ϕ(t)−ϕ(s)

t−s ≤ ϕ(u)−ϕ(t) u−t whenevera < s < t < u < b.

b) Deduce thatϕis continuous, and that for eachs∈(a, b), there existsβs∈Rsuch that

ϕ(t)≥ϕ(s) +β_s(t−s)

for everyt∈(a, b).

c) Letf :X →(a, b)such thatf ∈L¹(X, µ). Show thatϕ◦f is measurable and that

ϕ Z

X

f dµ

≤ Z

X

ϕ◦f dµ.

7) Let1≤p <∞andp⁰=p/(p−1). Show that for everyu∈L^p(X, µ), then

kukp= sup Z

X

|uv|dµ:v∈L^p0(X, µ),kvkp⁰ ≤1

.

Ifp=∞andX isσ-finite, show that for everyu∈L^∞(X, µ), then

kuk_∞= sup Z

X

|uv|dµ:v∈L¹(X, µ),kvk1≤1

.

We recall thatX isσ-finite is there exists an increasing sequence of measurable sets (Xn)n∈N⊂Msuch that X =S

n∈N andµ(X_n)<∞for each n∈N.

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