Multiplication and Composition Operators on Weak Lp

Multiplication and Composition Operators on Weak L _p spaces

Ren´ e Erlin Castillo

, Fabio Andr´ es Vallejo Narvaez

and Julio C. Ramos Fern´ andez

Abstract

In a self-contained presentation, we discuss the WeakLp spaces.

Invertible and compact multiplication operators on WeakL_p are char- acterized. Boundedness of the composition operator on WeakLp is also characterized.

Keywords: Compact operator, multiplication and composition oper- ator, distribution function, WeakLp spaces.

Contents

1 Introduction 2

2 Weak L_p spaces 2

3 Convergence in measure 7

4 An interpolation result 13

5 Normability of WeakLp for p >1 25

6 Multiplication Operators 39

7 Composition Operator 49

A Appendix 52

02010 Mathematics subjects classification primary 47B33, 47B38; secondary 46E30

1 Introduction

One of the real attraction of WeakLp space is that the subject is sufficiently concrete and yet the spaces have fine structure of importance for applications.

WeakL_p spaces are function spaces which are closely related toL_pspaces. We do not know the exact origin of WeakLp spaces, which is a apparently part of the folklore. The Book by Colin Benett and Robert Sharpley[4] contains a good presentation of WeakL_p but from the point of view of rearrangement function. In the present paper we study the WeakLp space from the point of view of distribution function. This circumstance motivated us to undertake a preparation of the present paper containing a detailed exposition of these function spaces. In section 6 of the present paper we first prove a charac- terization of the boundedness of M_u in terms of u, and show that the set of multiplication operators on WeakL_p is a maximal abelian subalgebra of B WeakLp

, the Banach algebra of all bounded linear operators on WeakLp. For the systemic study of the multiplication operator on different spaces we refereed to ([1], [2], [5] [3], [10], [18], [21]).

We use it to characterize the invertibility of Mu on WeakLp. The compact multiplication operators are also characterized in this section.

In section 7 a necessary and sufficient condition for the boundedness of com- position operator CT is given. For the study of composition operator on different function spaces we refereed to ([6], [11], [12], [16], [18], [19]).

2 Weak L

spaces

Definition 2.1. Forf a measurable function onX, the distribution function of f is the function Df defined on [0,∞) as follows:

D_f(λ) :=µ

{x∈X :|f(x)|> λ}

. (1)

The distribution function D_f provides information about the size of f but not about the behavior off itself near any given point. For instance, a func- tion on Rⁿ and each of its translates have the same distribution function. It follows from definition 2.1 that D_f is a decreasing function of λ (not neces- sarily strictly).

Let (X, µ) be a measurable space andf andg be a measurable functions on (X, µ) then D_f enjoy the following properties: For allλ₁, λ₂ >0:

1. |g| ≤ |f| µ-a.e. implies thatD_g ≤D_f; 2. D_cf(λ₁) = D_f

λ₁

|c|

for all c∈C/{0};

3. D_f+g(λ₁+λ₂)≤D_f(λ₁) +D_g(λ₂);

4. D_{f g}(λ₁λ₂)≤D_f(λ₁) +D_g(λ₂).

For more details on distribution function see ([7] and [15]).

Next, Let (X, µ) be a measurable space, for 0< p <∞, we consider WeakL_p :=

f :µ({x∈X :|f(x)|> λ})≤ C

λ p

,

for some C >0. Observe that WeakL∞=L∞. WeakL_p as a space of functions is denoted by L_(p,∞). Proposition 2.1. Let f ∈WeakL_p with 0< p <∞. Then

kfk_L(p,∞) = inf

C >0 :D_f(λ)≤ C

λ p

=

sup

λ>0

λ^pD_f(λ) 1/p

= sup

λ>0

λ{D_f(λ)}^1/p. Proof. Let us define

λ= inf

C >0 :D_f(α)≤ C

α p

,

and

B =

sup

α>0

α^pD_f(α) 1/p

. Since f ∈WeakL_p, then

D_f(α)≤ C

α p

, for some C >0, then

C >0 :D_f(α)≤ C

α p

∀α >0

6=∅.

On the other hand

α^pD_f(α)≤B^p,

thus {α^pD_f(α) :α >0} is bounded above by B^p and so B ∈R. Therefore

λ = inf

C >0 :D_f(α)≤ C

α p

α >0

≤B. (2)

Now, let >0, then there exists C such that λ≤C < λ+, and thus

D_f(λ)≤ C^p

λ^p < (λ+)^p λ^p , then

sup

λ>0

λ^pD_f(λ)<(λ+)^p

sup

λ>0

λ^pD_f(λ) 1/p

≤λ

B < λ, (3)

by (2) and (3) B =λ.

Definition 2.2. For 0< p <∞ the space L_(p,∞) is defined as the set of all µ−measurable functions f such that

kfkL_(p,∞) = inf

C >0 :Df(λ)≤ C

λ p

∀λ >0

=

sup

λ>0

λ^pD_f(λ) 1/p

= sup

λ>0

λ{D_f(λ)}^1/p,

is finite. Two functions in L_(p,∞) will be considered equal if they are equal µ−a.e.

The WeakL_p =L(p,∞) are larger than the L_p spaces, we have the following.

Proposition 2.2. For any 0< p <∞ and any f ∈L_p we have L_p ⊂L(p,∞),

and hence

f _L

(p,∞) ≤

f _L

p.

(This is just a restatement of the Chebyshev inequality).

Proof. If f ∈L_p, then λ^pµ

{x∈X :|f(x)|> λ}

≤ Z

{|f|>λ}

|f|^pdu≤ Z

X

|f|^pdu= f

p Lp, therefore

µ

{x∈X :|f(x)|> λ}

≤

kfk_Lp λ

p

. (4)

Hence f ∈ Weak L_p =L_(p,∞), which means that

L_p ⊂L(p,∞). (5)

Next, from (4) we have

sup

λ>0

{λ^pD_f(λ)}

1/p

≤ f

Lp

f

L_(p,∞) ≤

f Lp.

Remark 2.1. The inclusion (5) is strict, indeed, let f(x) = x^−1/p on (0,∞) (with the Lebesgue measure). Note

m

x∈(0,∞) : 1

|x|^1/p > λ

=m

x∈(0,∞) :|x|< 1 λ^p

= 2λ^−p. Thus f ∈WeakL_p(0,∞), but

Z ∞ 0

1 x^1/p

p

dx= Z ∞

0

dx

x → ∞, then f /∈L_p(0,∞).

Proposition 2.3. Let f, g ∈L(p,∞). Then 1.

cf

L(p,∞) =|c|

f

L(p,∞) for any constant c,

2.

f+g

L(p,∞) ≤2

f

p

L(p,∞) + g

p L(p,∞)

1/p

. Proof. (1) For c >0 we have

µ

{x∈X :|cf(x)|> λ}

=µ

x∈X :|f(x)|> λ c

, thus

Dcf(λ) =Df

λ c

. And thus

kcfkL_(p,∞) =

sup

λ>0

λ^pDcf(λ) 1/p

=

sup

λ>0

λ^pD_f λ

c 1/p

=

sup

cw>0

c^pw^pD_f(w) 1/p

=c

sup

cw>0

w^pD_f(w) 1/p

, then

kcfkL_(p,∞) =ckfkL_(p,∞). (2) Note that

n

x∈X :|f(x)+g(x)|> λ o

⊆

x∈X :|f(x)|> λ 2

[

x∈X :|g(x)|> λ 2

. Hence

µ

{x∈X :|f(x) +g(x)|> λ}

≤µ

x∈X :|f(x)|> λ 2

+µ

x∈X :|g(x)|> λ 2

, then

λ^pDf+g(λ)≤λ^pDf

λ 2

+λ^pDg

λ 2

λ^pDf+g(λ)≤2^p

sup

λ>0

λ^pDf(λ) + sup

λ>0

λ^pDg(λ)

,

therefore

sup

λ>0

λ^pD_f+g(λ) 1/p

≤2 f

p

L(p,∞)+

g

p L(p,∞)

1/p

f +g

L(p,∞) ≤2

f

p

L(p,∞)+

g

p L(p,∞)

1/p

.

Remark 2.2. Proposition 2.3 (2) tell us that k.k_L(p,∞) define a quasi-norm onL_(p,∞).

Definition 2.3. A quasi-norm is a functional that is like a norm except that it does only satisfy the triangle inequality with a constant C ≥1, that is

kf +gk ≤C

kfk+kgk .

3 Convergence in measure

Next, we discus some convergence notions. The following notion is of impor- tance in probability theory.

Definition 3.1. Let f, f_n (n = 1,2,3, ...) be a measurable functions on the measurable space(X, µ). The sequence{f_n}_n∈Nis said to converge in measure to f (f_n→^µ f) if for all >0 there exists an n₀ ∈N such that

µ

{x∈X :|fn(x)−f(x)|> }

< for all n≥n0. (6) Remark 3.1. The preceding definition is equivalent to the following state- ment.

For all >0,

n→∞lim µ

{x∈X :|f_n(x)−f(x)|> }

= 0. (7)

Cleary (7) implies (6). To see the convergence given > 0, pick 0 < δ <

and apply (6) for this δ.

There exists an n0 ∈N such that µ

{x∈X :|f_n(x)−f(x)|> δ}

< δ,

holds for n≥n₀. Since µ

{x∈X :|f_n(x)−f(x)|> }

≤µ

{x∈X :|f_n(x)−f(x)|> δ}

. We concluded that

µ

{x∈X :|f_n(x)−f(x)|> }

< δ, for all n ≥n₀. Let n → ∞ to deduce that

lim sup

n→∞

µ

{x∈X :|f_n(x)−f(x)|> }

≤δ. (8)

Since (8) holds for all 0< δ < (7) follows by letting δ→0.

Remark 3.2. Convergence in measure is a more general property than con- vergence in either L_p or L(p,∞), 0 < p < ∞, as the following proposition indicates:

Proposition 3.1. Let 0< p≤ ∞ and f_n, f be in L(p,∞).

1. If f_n, f are in L_p and f_n→f in L_p, then f_n→f in L(p,∞). 2. If f_n→f in L(p,∞) then f_n→^µ f.

Proof. (1) Fix 0< p <∞. Proposition 2.2 gives that for all >0 we have:

µ

{x∈X :|f_n(x)−f(x)|> }

≤ 1 ^p

Z

X

|f_n−f|^pdµ

^pµ

{x∈X :|f_n(x)−f(x)|> }

≤ kf_n−fk^p_L

p

sup

λ>0

λ^pD_fn−f(λ)≤ kf_n−fk^p_Lp, and thus

kfn−fkL_(p,∞) ≤ kfn−fkLp.

This shows that convergence inL_p implies convergence in WeakL_p. The case p=∞ is tautological.

(2) Give >0 find an n₀ ∈N such that forn > n₀, we have

kf_n−fk_L(p,∞) =

sup

λ>0

λ^pD_fn−f(λ) 1/p

< ^1p+1, then taking λ=, we conclude that

^pµ

{x∈X :|f_n(x)−f(x)|> }

< ^p+1, for n > n₀.

Hence

µ

{x∈X :|f_n(x)−f(x)|> }

< for n > n0.

Example 3.1. Fix 0< p <∞. On [0,1] define the functions f_k,j =k^1/pχ(^j−1_k ^,j_k) k≥1, 1≤j ≤k.

Consider the sequence {f_1,1, f_2,1, f_2,2, f_3,1, f_3,2, f_3,3, ...}.

Observe that

m

{x∈[0,1] :fk,j(x)>0}

= 1 k, thus

k→∞lim m

{x∈[0,1] :fk,j(x)>0}

= 0, that is f_k,j −→^m 0.

Likewise, Observe that kfk,jkL_(p,∞) =

sup

λ>0

λ^pm

{x∈[0,1] :fk,j(x)> λ}1/p

≥

sup

k≥1

k−1 k

1/p

= 1.

Which implies that fk,j does not converge to 0 in L(p,∞).

It turns out that every sequence convergent inL(p,∞) has a subsequence that converges µ-a.e. to the same limit.

Theorem 3.1. Let fn and f be a complex-valued measurable functions on a measure space (X,A, µ) and suppose f_n −→^µ f. Then some subsequence of f_n converges to f µ−a.e.

Proof. For all k = 1,2, ...choose inductively n_k such that µ

{x∈X :|f_n(x)−f(x)|>2^−k}

<2^−k, (9) and such that n₁ < n₂ < ... < n_k < ... Define the sets

A_k=n

x∈X :|f_nk(x)−f(x)|>2^−ko , (9) implies that

µ

∞

[

k=m

A_k

!

≤

∞

X

k=m

µ(A_k)≤

∞

X

k=m

2^−k= 2^1−m, (10) for all m = 1,2,3, ... It follows from (10) that

µ

∞

[

k=1

A_k

!

≤1<∞. (11)

Using (10) and (11), we conclude that the sequence of the measure of the sets

_∞ S

k=m

A_k

m∈N

converges as m → ∞to µ

∞

\

m=1

∞

[

k=m

A_k

!

= 0. (12)

To finish the proof, observe that the null set in (12) contains the set of all x∈X for which fn_k(x) does not converge tof(x).

Remark 3.3. In many situations we are given a sequence of functions and we would like to extract a convergent subsequence. One way to achieve this is via the next theorem which is a useful variant of theorem 3.1. We first give a relevant definition.

Definition 3.2. We say that a sequence of measurable functions{f_n}n∈N on the measure space (X,A, µ) is Cauchy in measure if for every > 0 there exists an n₀ ∈N such that for n, m > n₀ we have

µ

{x∈X :|f_n(x)−f_m(x)|> }

< .

Theorem 3.2. Let(X,A, µ)be a measure space and let{f_n}n∈Nbe a complex valued sequence on X, that is Cauchy in measure. Then some subsequence of fn converges µ−a.e.

Proof. The proof is very similar to that of theorem 3.1 for all k = 1,2,3, ...

choose n_k inductively such that µ

{x∈X :|f_nk(x)−f_nk+1(x)|>2^−k}

<2^−k, (13) and such that n₁ < n₂ < n₃ < ... < n_k < n_k+1 < ... Define

A_k =n

x∈X :|f_nk(x)−f_nk+1(x)|>2^−ko . As shown in the proof of theorem 3.1 (13) implies that

µ

∞

\

m=1

∞

[

k=m

A_k

!

= 0, (14)

for x /∈

∞

S

k=m

A_k and i≥j ≥j₀ ≥m (and j₀ large enough) we have

|f_ni(x)−f_nj(x)| ≤

i−1

X

l=j

|f_nl(x)−f_nl+1(x)| ≤

i

X

l=j

2^−l ≤2^1−j ≤2^1−j0. This implies that the sequence {f_ni(x)}i∈N is Cauchy for every x in the set _∞

S

k=m

A_k c

and therefore converges for all such x. We define a function

f(x) =















j→∞lim f_nj(x) when x /∈

∞

T

m=1

∞

S

k=m

A_k

0 when x∈

∞

T

m=1

∞

S

k=m

A_k Then f_nj →f almost everywhere.

Proposition 3.2. If f ∈ WeakL_p and µ

{x ∈ X : f(x) 6= 0}

< ∞, then f ∈ L_q for all q < p. On the other hand, if f ∈ WeakL_p∩L∞ then f ∈L_q for all q > p.

Proof. If p <∞, we write Z

X

|f(x)|^qdµ=q

∞

Z

0

λ^q−1D_f(λ)dλ

=q

1

Z

0

λ^q−1D_f(λ)dλ+q

∞

Z

1

λ^q−1D_f(λ)dλ.

Note that µ

{x∈X :|f(x)|> λ}

≤µ

{x∈X :f(x)6= 0}

.

Therefore µ

{x∈X :|f(x)|> λ}

≤C, then

Z

X

|f(x)|^qdµ≤qC

1

Z

0

λ^q−1dλ+qC

∞

Z

1

λ^q−p−1dλ=C+qCλ^q−p q−p

i∞ 1

<∞

Therefore f ∈Lq.

If f ∈WeakL_p ∩L∞. Then Z

X

|f(x)|^qdµ=q

∞

Z

0

λ^q−1D_f(λ)dλ

=q

M

Z

0

λ^q−1D_f(λ)dλ+q

∞

Z

M

λ^q−1D_f(λ)dλ, where M =esssup|f(x)|. Note that

µ

{x∈X :|f(x)|> λ}

= 0 for λ > M, since f ∈WeakL_p∩L_∞, therefore

q

∞

Z

M

λ^q−1Df(λ)dλ= 0 and Df(λ)≤ f

p L(p,∞)

λ^p .

Then Z

X

|f(x)|^qdµ=q

M

Z

0

λ^q−1D_f(λ)dλ≤q f

p L_(p,∞)

M

Z

0

λ^q−p−1dλ

= q

f

p

L(p,∞)M^q−p q−p <∞,

then Z

X

|f(x)|^qdµ≤ ∞.

Thus f ∈L_q.

Proposition 3.3. Let f ∈ WeakL_p0 ∩WeakL_p1 with p₀ < p < p₁. Then f ∈L_p.

Proof. Let us write

f =f χ{|f|≤1}+f χ{|f|>1} =f₁+f₂.

Observe that f₁ ≤ f and f₂ ≤ f. In particular f₁ ∈ Weak L_p0 and f₂ ∈ WeakL_p1. Also, write that f₁ is bounded and

µ

{x∈X :f₂(x)6= 0}

=µ

{x∈X :|f(x)|>1}

< C <∞.

Therefore by proposition 3.2, we have f₁ ∈ L_p and f₂ ∈ L_p. Since L_p is a linear vector space, we conclude that f ∈L_p.

4 An interpolation result

It is a useful fact that if a function is in L_p(X, µ)∩L_q(X, µ), then it also lies inL_r(X, µ) for allp < r < q. The usefulness of the spacesL(p,∞) can be seen from the following sharpening of this statement:

Proposition 4.1. Let 0< p < q ≤ ∞ and let f in L(p,∞)∩L(q,∞). Then f is in L_r for all p < r < q and

f Lr ≤

r

r−p+ r q−r

1/r

f

1r−1 q 1 p−1

q

L(p,∞)

f

p1−1 r 1 p−1

q

L(q,∞) (15)

whit the suitable interpolation when q =∞.

Proof. Let us take first q <∞. We know that D_f(λ)≤min

f

p Lp,∞

λ^p , f

q Lq,∞

λ^q

!

, (16)

set

B =

f

q Lq,∞

f

p Lp,∞

!q−p¹

. (17)

We now estimate theL_r norm of f. By (16), (17), we have f

r Lr =r

∞

Z

0

λ^r−1D_f(λ)dλ

≤r

∞

Z

0

λ^r−1min f

p Lp,∞

λ^p , f

q Lq,∞

λ^q

! dλ

=r

B

Z

0

λ^r−1−p f

p

Lp,∞ dλ+r

∞

Z

B

λ^r−1−q f

q

Lq,∞ dλ (18)

= r

r−q f

p

Lp,∞B^r−p+ r q−r

f

q

Lq,∞B^r−q

= r

r−p + r q−r

f

p L(p,∞)

^q−r_q−p f

q L(q,∞)

^r−p_q−p .

Observe that the integrals converge, since r−p > 0 and r−q <0.

The case q = ∞ is easier. Since D_f(λ) = 0 for λ > kfk_L∞ we need to use only the inequality

D_f(λ)≤λ^−p f

p L(p,∞),

for λ≤ kfk_L∞ in estimating the first integral in (18). We obtain f

r

Lr ≤ r r−p

f

p L(p,∞)

f

r−p L∞,

Which is nothing other than (15) whenq =∞. This complete the proof.

Note that (15) holds with constant 1 if L(p,∞) and L(q,∞) are replaced by Lp

and L_q, respectively. It is often convenient to work with functions that are only locally in some L_p space. This leads to the following definition.

Definition 4.1. For 0 < p < ∞, the space L^p_loc(Rⁿ,|.|) or simply L^p_loc(Rⁿ) (where |.| denote the Lebesgue measure) is the set of all Lebesgue-measurable functions f on Rⁿ that satisfy

Z

K

|f(x)|^pdx <∞, (19)

for any compact subset K of Rⁿ. Functions that satisfy (19) with p= 1 are called locally integrable functions on Rⁿ.

The union of all L_p(Rⁿ) spaces for 1≤p≤ ∞ is contained in L¹_loc(Rⁿ).

More generally, for 0< p < q <∞ we have the following:

L_q(Rⁿ)⊆L^q_loc(Rⁿ)⊆L^p_loc(Rⁿ).

Functions in L_p(Rⁿ) for 0 < p <1 may not be locally integrable. For exam- ple, takef(x) = |x|^−α−nχ{x:|x|≤1}which is inLp(Rⁿ) whenp < n/(n+α), and observe thatf is not integrable over any open set inRⁿcontaining the origen.

In what follows we will need the following useful result.

Proposition 4.2. Let {a_j}j∈N be a sequence of positives reals.

a)

∞

P

j=1

aj

!θ

≤

∞

P

j=1

a^θ_j for any 0≤θ ≤1. If

∞

P

j=1

a^θ_j <∞.

b)

∞

P

j=1

a^θ_j ≤

∞

P

j=1

a_j

!θ

for any 1≤θ <∞. If

∞

P

j=1

a_j <∞.

c)

N

P

j=1

a_j

!θ

≤N^θ−1

N

P

j=1

a^θ_j when 1≤θ < ∞.

d)

N

P

j=1

a^θ_j

!

≤N^1−θ

N

P

j=1

a_j

!θ

when 0≤θ ≤1.

Proof. (a) We proceed by induction. Note that if 0≤θ ≤1, then θ−1≤0, also a₁+a₂ ≥a₁ and a₁+a₂ ≥a₂ from this we have (a₁+a₂)^θ−1 ≤a^θ−1₁ and (a1+a2)^θ−1 ≤a^θ−1₂ and thus

a₁(a₁+a₂)^θ−1 ≤a^θ₁ and a₂(a₁+a₂)^θ−1 ≤a^θ₂. Hence

a₁(a₁+a₂)^θ−1+a₂(a₁+a₂)^θ−1 ≤a^θ₁+a^θ₂,

next, pulling out the common factor on the left hand side of the above in- equality, we have

(a₁+a₂)^θ−1(a₁+a₂)≤a^θ₁+a^θ₂, (a₁+a₂)^θ ≤a^θ₁+a^θ₂. Now, suppose that

n

X

j=1

a_j

!θ

≤

n

X

j=1

a^θ_j,

holds. Since

n

X

j=1

a_j +a_n+1 ≥a_n+1,

and n

X

j=1

a_j+a_n+1 ≥

n

X

j=1

a_j,

we have

n

X

j=1

a_j +a_n+1

!θ−1

≤a^θ−1_n+1,

and

n

X

j=1

a_j+a_n+1

!θ−1

≤

n

X

j=1

a_j

!θ−1

.

Hence

n

X

j=1

aj +an+1

!^θ−1 _n X

j=1

aj +an+1

!

≤a^θ_n+1+

n

X

j=1

aj

!θ

n

X

j=1

a_j +a_n+1

!θ

≤a^θ_n+1+

n

X

j=1

a_j

!θ

≤a^θ_n+1+

n

X

j=1

a^θ_j =

n+1

X

j=1

a^θ_j.

Since

∞

P

j=1

a^θ_j <∞, we have

∞

X

j=1

a_j

!θ

≤

∞

X

j=1

a^θ_j.

(b) Since

∞

P

j=1

a_j <∞, then lim

j→∞a_j = 0,

which implies that there exists n₀ ∈N such that

0< a_j <1 if j ≥n₀, since 1≤θ <∞, we obtain

a^θ_j < a_j for all j ≥n₀. From this we have

∞

X

j=1

a^θ_j <∞.

Consider the sequence {a^θ_j}j∈N, since 1≤θ, then 0< 1

θ ≤1 by part (a)

∞

X

j=1

a^θ_j

!¹_θ

≤

∞

X

j=1

a^θ_j¹_θ

=

∞

X

j=1

a_j,

and thus

∞

X

j=1

a^θ_j ≤

∞

X

j=1

a_j

!θ

.

(c) By H¨older’s inequality we have

N

X

j=1

aj ≤

N

X

j=1

1

!¹⁻¹_{θ N} X

j=1

a^θ_j

!¹_θ

=N^θ−1θ

N

X

j=1

a^θ_j

!¹_θ ,

then

N

X

j=1

aj

!^θ

≤N^θ−1

N

X

j=1

a^θ_j.

(d) On more time, by H¨older’s inequality

N

X

j=1

a^θ_j ≤

N

X

j=1

1

!^1−θ _N X

j=1

a^θ_j¹_θ

!θ

=N^1−θ

N

X

j=1

a_j

!θ

.

Proposition 4.3. Let f₁, . . . , f_N be in L(p,∞) then a)

N

P

j=1

f_j L(p,∞)

≤N

N

P

j=1

kf_jk_L(p,∞) for 1≤p < ∞.

b)

N

P

j=1

f_{j L}

(p,∞)

≤N^1p

N

P

j=1

kf_jk_L(p,∞) for 0< p <1.

Proof. First of all, note that for α >0 and N ≥1

|f₁|+...+|f_N| ≥ |f₁+f₂+...+f_N|> α≥ α N. Thus

n

x∈X :|f₁+f₂+. . .+f_N|> αo

⊂n

x∈X :|f1|> α N

o

∪n

x∈X :|f1|> α N

o

∪. . .∪n

x∈X :|fN|> α N

o .

Then µ

{x∈X :|f1+f2 +...+fN|> α}

≤

N

X

j=1

µ n

x∈X :|fj|> α N

o ,

that is

D^P_fj(α)≤

N

X

j=1

D_fjα N

.

Hence

N

X

j=1

f_j

p L(p,∞)

= sup

α>0

α^pD^P_fj(α)≤

N

X

j=1

sup

α>0

α^pD_fjα N

=

N

X

j=1

sup

α>0

α^pDN fj(α)

=

N

X

j=1

kN f_jk^p_L

(p,∞) =N^p

N

X

j=1

kf_jk^p_L

(p,∞), thus

N

X

j=1

f_j L(p,∞)

≤N

N

X

j=1

kf_jk^p_L

(p,∞)

!¹_p .

By proposition (4.2) (a) since 0< ¹_p <1 we have

N

X

j=1

f_j L(p,∞)

≤N

N

X

j=1

kf_jk_L(p,∞)

! .

(b) As in part (a) we have

N

X

j=1

f_j

p

L_(p,∞) ≤N^p

N

X

j=1

f_j

p L(p,∞)

! .

Since 0< p <1, then 1< ¹_p, next by proposition 4.2 (c) we have

N

X

j=1

f_j

L_(p,∞) ≤N

N

X

j=1

f_j

p L(p,∞)

!¹_p

≤N(N^1p−1)

N

X

j=1

f_j

p L(p,∞)

¹_p

=N^1p

N

X

j=1

f_j L(p,∞).

Proposition 4.4. Give a measurable function f on(X, µ)andλ >0, define f_λ =f χ_{|f|>λ} and f^λ =f−f_λ =f_λ =f χ_{|f|≤λ}.

a) Then

D_fλ(α) =

D_f(α) when α > λ D_f(λ) when α≤λ.

D_f^λ(α) =

0 when α≥λ D_f(α)−D_f(λ) when α < λ b) If f ∈L_p(X, µ). Then

f_λ

p Lp =p

∞

Z

λ

α^p−1D_f(α)dα+λ^pD_f(λ),

f^λ

p Lp =p

λ

Z

0

α^p−1D_f(α)dα−λ^pD_f(λ),

Z

λ<|f|≤δ

|f|^pdµ=p

δ

Z

λ

α^p−1D_f(α)dα−δ^pD_f(α) +λ^pD_f(λ).

c) Iff is in L(p,∞) thenf^λ is inL_q(X, µ)for anyq > pand f_λ is inL_q(X, µ) for any q < p. ThusL_(p,∞) ⊆L_p0+L_p1 when 0< p₀ < p < p₁ ≤ ∞.

Proof. (a) Note D_fλ(α) =µ

{x:|f(x)|χ{|f|>λ}(x)> α}

=µ

{x:|f(x)|> α}∩{x:|f|> λ}

, if α > λ, then {x:|f(x)|> α} ⊆ {x:|f|> λ}, thus

D_fλ(α) =µ

{x:|f(x)|> α}∩{x:|f|> λ}

=µ

{x:|f(x)|> α}

=D_f(α).

If α≤λ, then {x:|f(x)|> λ} ⊆ {x:|f|> α}, thus D_fλ(α) =µ

{x:|f(x)|> α}∩{x:|f|> λ}

=µ

{x:|f(x)|> λ}

=D_f(λ).

And thus

D_fλ(α) =

D_f(α) when α > λ

Df(λ) when α≤λ. (20)

Next, consider

D_fλ(α) =µ

{x:|f(x)|χ_{|f|≤λ}(x)> α}

=µ

{x:|f(x)|> α} ∩ {x:|f| ≤λ}

,

if α≥λ then {x:|f|> α} ∩ {x:|f(x)| ≤λ}=∅, thus D_f^λ(α) = 0.

If α < λ, then D_fλ(α) = µ

{x:|f(x)|> α} ∩ {x:|f(x)| ≤λ}

=µ

{x:|f(x)|> α} ∩ {x:|f(x)|> λ}^c

=µ

{x:|f(x)|> α}\{x:|f(x)|> λ}

=µ

{x:|f(x)|> α}

−µ

{x:|f(x)|> λ}

=D_f(α)−D_f(λ).

And hence

D_f^λ(α) =

0 when α≥λ

D_f(α)−D_f(λ) when α < λ. (21)

(b) If f ∈L_p(X, µ), then f_λ

p Lp =p

∞

Z

0

α^p−1D_fλ(α)dα=p

λ

Z

0

α^p−1D_fλ(α)dα+p

∞

Z

λ

α^p−1D_fλ(α)dα,

By part (a)(20) we have f_λ

p Lp =p

λ

Z

0

α^p−1D_f(λ)dα+p

∞

Z

λ

α^p−1D_f(α)dα

=λ^pD_f(λ) +p

∞

Z

λ

α^p−1D_f(α)dα.

Also f^λ

p Lp =p

∞

Z

0

α^p−1D_fλ(α)dα =p

λ

Z

0

α^p−1D_fλ(α)dα+p

∞

Z

λ

α^p−1D_fλ(α)dα,

by part (a) (21) we obtain f^λ

p Lp =p

λ

Z

0

α^p−1

D_f(α)−D_f(λ)

dα=p

λ

Z

0

α^p−1D_f(α)dα−λ^pD_f(λ).

Next, Z

λ<|f|≤δ

|f|^pdµ

= Z

|f|>λ

|f|^pdµ− Z

|f|>δ

|f|^pdµ

= Z

X

|f|^pχ{|f|>λ}dµ− Z

X

|f|^pχ{|f|>δ}dµ

= Z

X

|fλ|^pdµ− Z

X

|fδ|^pdµ

=p

∞

Z

λ

α^p−1D_f(α)dα+λ^pD_f(λ)−p

∞

Z

δ

α^p−1D_f(α)dα−δ^pD_f(δ)

=p





∞

Z

λ

α^p−1D_f(α)dα−

∞

Z

δ

α^p−1D_f(α)dα



+λ^pD_f(λ)−δ^pD_f(δ)

=p

δ

Z

λ

α^p−1D_f(α)dα−δ^pD_f(α) +λ^pD_f(λ).

(c) We known that

D_f(α)≤ f

p L(p,∞)

α^p , then if q > p

f^λ

q Lq =q

λ

Z

0

α^q−1D_f(α)dα−λ^qD_f(λ)

≤q

λ

Z

0

α^q−1 f

p L(p,∞)

α^p dα−λ^qD_f(λ)

=q f

p L(p,∞)

λ^q−p

q−p −λ^qD_f(λ)≤q f

p L(p,∞)

λ^q−p

q−p <∞.

And thus f^λ ∈Lq if q > p.

Now, if q < p, then f_λ

q Lq =q

∞

Z

λ

α^q−1D_f(α)dα+λ^qD_f(λ)

≤q f

p L(p,∞)

∞

Z

λ

α^q−p−1dα+λ^qDf(λ)

=qλ^q−p p−q

f

p

L(p,∞) +λ^qD_f(λ)<∞.

Thus f_λ ∈L_q if q < p.

Finally, since f ∈L(p,∞) and

f =f^λ+f_λ,

where f^λ ∈L_p1 if p < p₁ and f_λ ∈L_p0 if p₀ < p. Then

L(p,∞) ⊆L_p0 +L_p1 when 0< p₀ < p < p₁ ≤ ∞.

Proposition 4.5. Let (X, µ) be a measure space and letE be a subset of X with µ(E)<∞. Then

a) for 0< q < p we have Z

E

|f(x)|^qdµ≤ p p−q

µ(E)1−^q_p f

q

L(p,∞) for f ∈L(p,∞).

b) Conclude that if µ(X)<∞ and 0< q < p, then Lp(X, µ)⊆L(p,∞) ⊆Lq(X, µ).

Proof. Letf ∈L(p,∞), then Z

E

|f|^qdµ

=q

∞

Z

0

λ^q−1µ

{x∈E :|f(x)|> λ}

dλ

≤q

µ(E)⁻¹_p

kfkL(p,∞)

Z

0

λ^q−1µ(E)dλ+q

∞

Z µ(E)⁻¹p

kfk_L

(p,∞)

λ^q−1D_f(λ)dλ

≤q

µ(E)⁻¹p

kfk_L

(p,∞)

Z

0

λ^q−1µ(E)dλ+q

∞

Z µ(E)⁻¹p

kfk_L

(p,∞)

λ^q−1kfk^p_L

(p,∞)

λ^p dλ

=

µ(E)−¹_p

kfk_L(p,∞)q

µ(E) + q p−q

µ(E)−¹_p

kfk_L(p,∞)^q−p kfk^p_L

(p,∞)

=

µ(E)1−^q

pkfk^q_L

(p,∞) + q

p−q

µ(E)1−^q

pkfk^q_L

(p,∞)

= p

p−q

µ(E)^1−q_p kfk^q_L

(p,∞). And thus

Z

E

|f|^qdµ≤ p p−q

µ(E)1−^q_p f

q L(p,∞). (b) If µ(X)<∞, then

Z

X

|f|^qdµ≤ p p−q

µ(X)^1−q_p f

q L(p,∞).

Hence

L_p ⊆L(p,∞) ⊆L_q.

Corolario 4.1. Let (X, µ) be a measurable space and let E be a subset of X with µ(E)<∞. Then

kfk_p/2 ≤

4µ(E)1/p f

L(p,∞). And thus L(p,∞) ⊆L_p/2.

Proof. Since 0< ^p₂ < p we can apply proposition 4.5 to obtain Z

E

|f|^p/2dµ≤ p

p− ^p₂[µ(E)]^1−p/2p f

p/2 L_(p,∞)

= 2[µ(E)]^1/2 f

p/2 L(p,∞)

kfk_p/2 ≤2^2/p[µ(E)]^1/p f

_L

(p,∞)

=

4µ(E)1/p f

L(p,∞). From this last result one can see that

L(p,∞) ⊆L_p/2.

5 Normability of Weak L

for p > 1

Let (X,A, µ) be a measure space and let 0 < p < ∞. Pick 0 < r < p and define

f

L(p,∞) = sup

0<µ(E)<∞

µ(E)−¹_r+¹_p



 Z

E

|f|^rdµ





1 r

,

where the supremum is taken over all measurable subsets E of X of finite measure.

Proposition 5.1. Let f be in L(p,∞). Then f

_L

(p,∞) ≤

f _L

(p,∞) ≤

p p−r

¹_r f

_L

(p,∞). Proof. By proposition 4.5 with q =r we have

f

L(p,∞) = sup

0<µ(E)<∞

µ(E)−¹_r+¹_p



 Z

E

|f|^rdµ





1 r

≤ sup

0<µ(E)<∞

µ(E)−¹

r+¹_p p p−r

µ(E)1−^r

p f

r L(p,∞)

¹_r

= sup

0<µ(E)<∞

µ(E)−¹

r+¹_p p p−r

¹_r

µ(E)¹_r−¹

p f

_L

(p,∞)

= p

p−r ¹_r

f L(p,∞). On the other hand by definition

µ(E)−¹_r+¹_p



 Z

E

|f|^rdµ





1 r

≤

f L(p,∞),

for allE ∈A such that µ(E)<∞now, let us considerA={x:|f(x)|> α}

for f ∈L(p,∞). Observe that µ(A)<∞. Then

f

p

L(p,∞) ≥







µ(A)−¹

r+¹_p



 Z

A

|f|^rdµ





1 r







p

≥

D_f(α)−^p_r+1



 Z

A

α^rdµ





p r

=

D_f(α)−^p_r+1

α^p

D_f(α)^p_r

=α^pD_f(α).

That is

α^pD_f(α)≤

f L(p,∞),

and thus

sup

α>0

α^pD_f(α)≤

f L(p,∞).

Lemma 5.1(Fatou forL(p,∞)). For all measurable functiongnonX we have

lim inf

n→∞ |g_n|

L(p,∞) ≤C_plim inf

n→∞

g_n L(p,∞). for some constant C_p that depends only on p∈(0,∞).

Proof.

lim inf

n→∞ |g_n|

L(p,∞) ≤

lim inf

n→∞ |g_n| L(p,∞)

= sup

0<µ(E)<∞

µ(E)−¹_r+¹_p



 Z

E

lim inf

n→∞ |gn|r

dµ





1 r

≤ sup

0<µ(E)<∞

µ(E)−¹_r+¹_p



 Z

E

lim inf

n→∞ |gn|^rdµ





1 r

.

By Fatou’s lemma

≤ sup

0<µ(E)<∞

µ(E)−¹

r+¹_p



lim inf

n→∞

Z

E

|g_n|^rdµ





1 r

≤lim inf

n→∞ sup

0<µ(E)<∞

µ(E)−¹

r+¹_p



 Z

E

|g_n|^rdµ





1 r

≤lim inf

n→∞

p p−r

¹_r g_n

L(p,∞)

= p

p−r ¹_r

lim inf

n→∞

g_n L(p,∞). Finally

lim inf

n→∞ |g_n|

L(p,∞) ≤ p

p−r ¹_r

lim inf

n→∞

g_n L(p,∞).