Averaging in Banach spaces

Theorem 5.4.5 The Dunford-Pettis Theorem)

6.2 Averaging in Banach spaces

F(t) =a²_k

I_k

∂²ϕ

∂²x(u_t, v_t) +∂²ϕ

∂²y(u_t, v_t) + 2_k ∂²ϕ

∂x∂y(u_t, v_t)

ds.

By Lemma 6.1.5 (ii),F(t)≥0. HenceF(1)≥F(0)≥0 and thus(6.4) holds.

To complete the proof whenp > 2 we plug x=f_n and y =g_n in (6.2).

Integrating both sides of this inequality and using (6.4) we obtain 1

0

(p−1)^p|fn(s)|^p− |gn(s)|^pds≥0.

The case when 1 < p < 2 now follows by duality: with f_n, g_n as before choose g_n∈L_q(Bn) so thatg_nq = 1 and

1 0

g_n(s)g_n(s)ds=g_np. Theng_n =n

j=1b_jh_j for some (b_j)ⁿ_j=1 and g_np=

n j=1

|I_j|_ja_jb_j ≤ f_np n j=1

_jb_jh_j

q ≤(q−1)f_np.

The constant p^∗−1 in Burkholder’s theorem is sharp, although we will not prove this here.

132 6 TheLp-Spaces for 1≤p <∞

Definition 6.2.1.TheRademacher functions(r_k)^∞_k=1 are defined on [0,1] by r_k(t) = sgn (sin 2^kπt).

Alternatively, the sequence (r_k)^∞_k=1 can be described as r1(t) =

1 if t∈[0,¹₂)

−1 if t∈[¹₂,1) r₂(t) =

1 if t∈[0,¹₄)∪[¹₂,³₄)

−1 if t∈[¹₄,¹₂)∪[³₄,1) ...

r_k+1(t) =

1 if t∈"2^k

s=1[^2s_2k+1⁻²,^2s_2k+1⁻¹)

−1 if t∈"2^k

s=1[^2s_2k+1⁻¹,_2k+1^2s ).

That is,

r_k+1=

2^k

s=1

h₂k+s, k= 0,1,2, . . .

Thus (r_k)^∞_k=1is a block-basic sequence with respect to the Haar basis in every L_p for 1≤p <∞. The key properties we need are the following:

• r_k(t) =±1 a.e. for allk,

•

r_k1r_k2(t). . . r_km(t)dt= 0, wheneverk₁< k₂<· · ·< k_m.

The Rademacher functions were first introduced by Rademacher in 1922 [191] with the idea of studying the problem of finding conditions under which a series of real numbers

±an, where the signs were assigned randomly, would converge almost surely. Rademacher showed that if

|a_n|²<∞then indeed

±a_n converges almost surely. The converse was proved in 1925 by Khintchine and Kolmogoroff [111].

For our purposes it will be convenient to replace the concrete Rademacher functions by an abstract model. To that end we will use the language and methods of probability theory.

Let us recall that arandom variable is a real-valued measurable function on some probability space (Ω,Σ,P). The expectation(or mean) of a random variablef is defined by

Ef =

Ω

f(ω)dP(ω).

A finite set of random variables {f_j}ⁿj=1 on the same probability space is independent if

P

$n j=1

f_j ∈B_j

= n j=1

P(f_j∈B_j)

for all Borel setsB_j. Therefore if (f_j)ⁿ_j=1 are independent, E

f₁f₂· · ·f_n

=E(f₁)E(f₂)· · ·E(f_n).

An arbitrary set of random variables is said to be independent if any finite subcollection of the set is independent.

Definition 6.2.2.A Rademacher sequence is a sequence of mutually inde- pendent random variables (ε_n)^∞_n=1 defined on some probability space (Ω,P) such thatP(ε_n= 1) =P(ε_n =−1) = ¹₂ for everyn.

The terminology is justified by the fact that the Rademacher functions (rn)^∞_n=1 are a Rademacher sequence on [0,1]. Thus,

1 0

n

i=1

r_i(t)x_idt=E n

i=1

ε_ix_i=

Ω

n i=1

ε_i(ω)x_idP.

Historically, the subject of finding estimates for averages over all choices of signs was initiated in 1923 by the classical Khintchine inequality [110], but the usefulness of a probabilistic viewpoint in studying theL_p-spaces seems to have been fully appreciated quite late (around 1970).

Theorem 6.2.3 (Khintchine’s Inequality). There exist constantsA_p,B_p (1 ≤ p < ∞) such that for any finite sequence of scalars (a_i)ⁿ_i=1 and any n∈N we have

A_pⁿ

i=1

|a_i|²1/2

≤ n i=1

a_ir_i

p≤ⁿ

i=1

|a_i|²1/2

if 1≤p <2,

and ⁿ

i=1

|a_i|²1/2

≤ n i=1

a_ir_i

p≤B_pⁿ

i=1

|a_i|²1/2

if p >2.

We will not prove this here but it will be derived as a consequence of a more general result below. Theorem 6.2.3 was first given in the stated form by Littlewood in 1930 [141] but Khintchine’s earlier work (of which Littlewood was unaware) implied these inequalities as a consequence.

Remark 6.2.4.(a) Khintchine’s inequality says that (r_i)^∞_i=1 is a basic se- quence equivalent to the₂-basis in everyL_p for 1≤p <∞. InL_∞, though, one readily checks that (ri)^∞_i=1 is isometrically equivalent to the canonical 1-basis.

(b) (ri)^∞_i=1 is an orthonormal sequence inL2, which yields the identity

n i=1

airi

2

= ⁿ

i=1

|ai|²1/2

,

134 6 TheLp-Spaces for 1≤p <∞

for any choice of scalars (a_i). But (r_i)^∞_i=1is not a complete system inL₂, that is, [r_i]=L₂ (for instance, notice that the functionr₁r₂ is orthogonal to the subspace [r_i]). However, one can obtain a complete orthonormal system for L₂ using the Rademacher functions by adding to (r_n) the constant function r₀= 1 and the functions of the formr_k1r_k2. . . r_knfor anyk₁< k₂<· · ·< k_n. This collection of functions are theWalsh functions.

Thus we can also interpret Khintchine’s inequality as stating that all the norms{·p: 1≤p <∞}are equivalent on the linear span of the Rademacher functions in Lp. It turns out that in this form the statement can be general- ized to an arbitrary Banach space. This generalization was first obtained by Kahane in 1964 [101].

Theorem 6.2.5 (Kahane-Khintchine Inequality). For each 1≤p <∞ there exists a constant C_p such that, for every Banach spaceX and for any finite sequence(x_i)ⁿ_i=1 inX, the following inequality holds:

E n i=1

ε_ix_i≤ E

n i=1

ε_ix_i^p1/p

≤C_pE n i=1

ε_ix_i.

We will prove the Kahane-Khintchine inequality (and this will imply the Khintchine inequality by takingX =RorX=C) but first we shall establish three lemmas on our way to the proof. To avoid repetitions, in all three lemmas (Ω,Σ,P) will be a probability space and X will be a Banach space. Let us recall that an X-valued random variable on Ω is a functionf : Ω→X such that f⁻¹(B)∈ Σ for every Borel setB ⊂X. f issymmetric if P(f ∈B) = P(−f ∈B) for all Borel subsetsB ofX.

Lemma 6.2.6.Letf : Ω→X be a symmetric random variable. Then for all x∈X we have

P

f+x ≥ x

≥1/2.

Proof. Let us take any x ∈ X. For every ω ∈ Ω, using the convexity of the norm of X, clearly f(ω) +x+x−f(ω) ≥ 2x. Then, either f(ω) +x ≥ x orx−f(ω) ≥ x. Hence

1≤P

f+x ≥ x +P

x−f ≥ x .

Sincef is symmetric,x+f andx−f have the same distribution and so the lemma follows.

Let (εi)^∞_i=1 be a Rademacher sequence on Ω. Given n ∈ N and vectors x₁, . . . , xn in X, we shall consider Λm: Ω−→X (1≤m≤n) defined by

Λ_m(ω) = m i=1

ε_i(ω)x_i.

Lemma 6.2.7.For allλ >0, P

max

m≤nΛ_m> λ

≤2P

Λ_n> λ . Proof. Givenλ >0, form= 1, . . . , nput

Ω^(λ)_m =

ω∈Ω :Λ_m(ω)> λ and Λ_j(ω) ≤λ for all j= 1, . . . , m−1 . Since{ω∈Ω : max_m≤nΛ_m(ω)> λ}=∪ⁿm=1Ω^(λ)_m , by the disjointedness of the sets Ω^(λ)m it follows that

P max

m≤nΛ_m> λ

= n m=1

P(Ω^(λ)_m ). (6.5) Therefore,

P

Λ_n> λ

= n m=1

P

Ω^(λ)_m ∩(Λ_n> λ)

. (6.6)

Notice that every Ω^(λ)m can be written as the union of sets of the type {ω∈Ω :ε_j(ω) =δ_j for 1≤j≤m}

for some choices of signsδ_j =±1.For each of these choices of signsδ₁, . . . , δ_m we observe that by Lemma 6.2.6,

P m j=1

δ_jx_j+ n j=m+1

ε_jx_j>

m j=1

δ_jx_j

≥1 2. Summing over the appropriate signs (δ₁, . . . , δ_m) it follows that

P

Ω^(λ)_m ∩(Λ_n ≥ Λ_m)

≥ 1

2P(Ω^(λ)_m ).

Thus,

P

Ω^(λ)_m ∩(Λ_n> λ)

≥ 1

2P(Ω^(λ)_m ).

Summing inmand combining (6.5) and (6.6) we finish the proof.

Lemma 6.2.8.For allλ >0,

P

Λn>2λ

≤4 P

Λn> λ2

.

Proof. We will keep the notation that we introduced in the previous lemma.

Notice that for each 1 ≤ m ≤ n, the random variable n

i=mε_ix_i is in- dependent of each of ε₁, . . . , ε_m and hence for all λ > 0 the events {ω : n

i=mε_i(ω)x_i> λ}and Ω^(λ)m are independent. Observe as well that if some

136 6 TheLp-Spaces for 1≤p <∞

ω ∈ Ω^(λ)m further satisfies Λ_n(w) >2λ, thenΛ_n(ω)−Λ_m−1(ω) > λ (for m= 1, take Λ0 = 0). Therefore, sinceP(n

i=mεixi> λ)≤2P(Λn> λ) for eachm= 1, . . . , nby Lemma 6.2.7, we have

P

Ω^(λ)_m ∩(Λ_n>2λ)

≤P(Ω^(λ)_m )Pⁿ

i=m

ε_ix_i> λ

≤2P(Ω^(λ)_m )P

Λ_n> λ . Summing inmand using again Lemma 6.2.7 we obtain

P

Λ_n>2λ

≤P max

m≤nΛ_m> λ P

Λ_n> λ

≤4 P

Λ_n> λ2

. Proof of Theorem 6.2.5. Fix 1≤p < ∞and let {xi}ⁿi=1 be any finite set of vectors inX. Without loss of generality we will suppose thatEn

i=1εixi= 1. Then, by Chebyshev’s inequality,

P

Λ_n>8

≤1

8. (6.7)

Using Lemma 6.2.8 repeatedly we obtain P

Λ_n>2·8

≤4(1/8)², P

Λ_n>2²·8

≤4³(1/8)⁴, P

Λn>2³·8

≤4⁷(1/8)⁸, and so on. Hence, by induction, we deduce that

P

Λn>2ⁿ·8

≤4²ⁿ⁻¹(1/8)²ⁿ≤4²ⁿ(1/8)²ⁿ = (1/2)²ⁿ. Therefore,

E n i=1

ε_ix_i^p = _∞

0

P

Λ_n^p> t dt

= _∞

0

p t^p−1P

Λ_n> t dt

= 8

0

p t^p−1P

Λ_n> t dt+

∞ n=1

2ⁿ·8 2ⁿ⁻¹·8

p t^p−1P

Λ_n> t dt

≤ 8

0

p t^p−1dt+ ∞ n=1

(1/2)²ⁿ⁻¹ 2ⁿ·8

2ⁿ⁻¹·8

p t^p−1dt

≤8^p

1 + ∞ n=1

(1/2)²ⁿ⁻¹2^np

=C_p^p.

Suppose that H is a Hilbert space. The well-known Parallelogram Law states that for any two vectorsx, yin H we have

x+y²+x−y²

2 =x²+y².

This identity is a simple example of the power of averaging over signs and has an elementary generalization:

Proposition 6.2.9 (Generalized Parallelogram Law). Suppose that H is a Hilbert space. Then for every finite sequence(x_i)ⁿ_i inH,

E n i=1

ε_ix_i²= n i=1

x_i².

Proof. For any vectors (xi)ⁿ_i=1 inH we have E

n i=1

ε_ix_i²=E%ⁿ

i=1

ε_ix_i, n i=1

ε_ix_i&

= n i,j=1

xi, xjE(εiεj)

= n i=1

x_i².

Next we are going to study how the averages (En

i=1ε_ix_i^p)^1/p are situ- ated with respect to the sums (n

i=1x_i^p)^1/p using the concepts oftypeand cotype of a Banach space. These were introduced into Banach space theory by Hoffmann-Jørgensen [79] and their basic theory was developed in the early 1970s by Maurey and Pisier [147]; see [146] for historical comments. However, it should be said that the origin of these ideas was in two very early papers of Orlicz in 1933, [163] and [164]. Orlicz essentially introduced the notion of co- type for the spacesLp although he did not use the more modern terminology.

Definition 6.2.10.A Banach spaceX is said to haveRademacher typep(in short, typep) for some 1≤p≤2 if there is a constantCsuch that for every finite set of vectors{x_i}ⁿi=1 inX,

E n i=1

ε_ix_i^p1/p

≤C ⁿ

i=1

x_i^p1/p

. (6.8)

The smallest constant for which (6.8) holds is called thetype-pconstant of X and is denotedTp(X).

138 6 TheLp-Spaces for 1≤p <∞

Similarly, a Banach spaceXis said to haveRademacher cotypeq(in short, cotype q) for some 2 ≤ q ≤ ∞ if there is a constant C such that for every finite sequencex₁, x₂, . . . , x_n in X,

ⁿ

i=1

x_i^q1/q

≤C E

n i=1

ε_ix_i^q1/q

, (6.9)

with the usual modification of max_1≤i≤nx_i replacing ⁿ_i=1x_i^q1/q

whenq=∞. The smallest constant for which (6.9) holds is called thecotype-q constant ofX and is denotedC_q(X).

Remark 6.2.11.(a) The restrictions on p and q in the definitions of type and cotype respectively are natural since it is impossible to have typep >2 or cotypeq <2 even in a one-dimensional space. To see this, for eachntake vectors {x_i}ⁿi=1 all equal to some x ∈ X with x = 1. The combination of Khintchine’s inequality with (6.8) and (6.9) gives us the range of eligible values for pandq.

(b) Every Banach spaceX has type 1 withT₁(X) = 1 and cotype ∞with C_∞(X) = 1 by the triangle law. ThusX is said to have nontrivial type if it has typepfor some 1< p≤2; similarlyX is said to havenontrivial cotypeif it has cotypeq for some 2≤q <∞.

(c) The generalized Parallelogram Law (Proposition 6.2.9) says that a Hilbert spaceH has type 2 and cotype 2 withT₂(H) =C₂(H) = 1. In partic- ular a one-dimensional space has type 2 and cotype 2. But the Parallelogram Law is also a characterization of Banach spaces which are linearly isometric to Hilbert spaces, hence we deduce that a Banach space X is isometric to a Hilbert space if and only ifT2(X) =C2(X) = 1 (see Problem 7.6).

(d) By Theorem 6.2.5, theLp-average (En

i=1εixi^p)^1/pin the definition of type can be replaced by any otherLr-average (En

i=1εixi^r)^1/r(1≤r <

∞) and this has the effect only of changing the constant. The same comment applies to theL_q-average in the definition of cotype.

(e) If X has type p then X has type r for r < p and ifX has cotype q thenX has cotypesfors > q.

(f) The type and cotype of a Banach space are isomorphic invariants and are inherited by subspaces.

(g) Consider the unit vector basis (e_n)^∞_n=1 in _p (1≤p <∞) or c₀.Then for any signs (_k) we have

₁e₁+· · ·+_ne_np=n^p1 and

1e1+· · ·+nen_∞= 1.

Thus_p cannot have type greater thanpif 1≤p≤2 or cotype less thanpif 2≤p≤ ∞.

Proposition 6.2.12.If a Banach space X has typepthenX^∗ has cotypeq, where ¹_p+¹_q = 1 andC_q(X^∗)≤T_p(X).

Proof. Let us pick an arbitrary finite set{x^∗_i}ⁿi=1 in X^∗. Given >0 we can find x₁, . . . , x_n in X such that x_i = 1 and |x^∗_i(x_i)| ≥ (1−)x^∗_i for all i= 1, . . . , n. Thus

ⁿ

i=1

|x^∗_i(x_i)|^q1/q

≥(1−)ⁿ

i=1

x^∗_i^q1/q

. On the other hand,

ⁿ

i=1

|x^∗_i(x_i)|^q1_q

= sup n i=1

a_ix^∗_i(x_i) : n i=1

|a_i|^p ≤1

. For any scalars (a_i)ⁿ_i=1 withn

i=1|a_i|^p≤1 we have n

i=1

a_ix^∗_i(x_i) =

Ω

ⁿ

i=1

ε_ix^∗_iⁿ

i=1

ε_ia_ix_i dP

≤

Ω

n i=1

ε_ix^∗_i n i=1

ε_ia_ix_idP

≤

Ω

n i=1

ε_ix^∗_i^qdP¹_q

Ω

n i=1

ε_ia_ix_i^pdP¹_p

≤

Ω

n i=1

ε_ix^∗_i^qdP¹_q T_p(X)

ⁿ

i=1

|a_i|^p1_p . Therefore,

ⁿ

i=1

x^∗_i^q1_q

≤(1−)⁻¹Tp(X) E

n i=1

εix^∗_i^q1_q . Sincewas arbitrary, this shows C_q(X^∗)≤T_p(X).

Curiously, Proposition 6.2.12 does not have a converse statement. At the end of the section we shall give an example showing that if X has cotype q forq <∞thenX^∗ may not have typepwhere ¹_p+¹_q.

Next we want to investigate the type and cotype ofL_p for 1 ≤p < ∞. To do so we will estimate(n

i=1|f_i|²)^1/2pin relation with the Rademacher averages (En

j=1ε_jf_j^pp)^1/p on a genericL_p(µ)-space.

Theorem 6.2.13.For every finite set of functions{f_i}ⁿi=1 inL_p(µ)(1≤p <

∞),

140 6 TheLp-Spaces for 1≤p <∞

A_pⁿ

i=1

|f_i|²¹₂

p≤ E

n i=1

ε_if_i^p

p

1/p

≤B_pⁿ

i=1

|f_i|²¹₂

p

, whereAp,Bp are the constants in Khintchine’s inequality (in particularAp= 1 for2≤p <∞andBp= 1 for1≤p≤2).

Proof. For eachω∈Ω, from Khintchine’s inequality A_p

ⁿ

i=1

|f_i(ω)|²1/2

≤ E

n i=1

ε_if_i(ω)^p1/p

, whereA_p= 1 for 2≤p <∞. Now, using Fubini’s theorem

A^p_pⁿ

i=1

|f_i|²1/2^p

p≤

Ω

E n i=1

ε_if_i(ω)^pdµ

=E

Ω

n i=1

ε_if_i(ω)^pdµ

=E n i=1

ε_if_i^p

p

. The converse estimate is obtained similarly.

The next theorem is due to Orlicz for cotype [163, 164] and Nordlander for type [159]. Obviously, the language of type and cotype did not exist before the 1970s and their results were stated differently. Note the difference in behavior of theL_p-spaces whenp >2 orp <2.This is the first example where we meet some fundamental change around the indexp= 2 and, as the reader will see, it is really because when p/2 <1 the triangle law for positive functions in L_p/2reverses.

Theorem 6.2.14.

(a)If1≤p≤2,Lp(µ) has typepand cotype 2.

(b)If2< p <∞,L_p(µ)has type2 and cotypep.

Moreover,(a)and(b)are optimal.

Proof. (a) Let us prove first that if 1 ≤p ≤ 2, thenL_p(µ) has type p. We recall this elementary inequality:

Lemma 6.2.15.Let 0< r≤1. Then for any nonnegative scalars(α_i)ⁿ_i=1 we have

(α₁+· · ·+α_n)^r≤α^r₁+· · ·+α^r_n. (6.10)

This way, combining Theorem 6.2.13 with (6.10) we obtain E

n i=1

ε_if_i^p

p

_p¹

≤ⁿ

i=1

|f_i|²1/2

p

= n i=1

|fi|^21/2

p/2

≤ⁿ

i=1

|f_i|^p2/p^1/2

p/2

=

Ω

n i=1

|f_i|^pdµ 1/p

=ⁿ

i=1

f^pp

1/p

.

To show thatL_p(µ) has cotype 2 when 1≤p≤2 we need the reverse of Minkowski’s inequality:

Lemma 6.2.16.Let 0< r <1. Then

f+gr≥ fr+gr, wheneverf andg are nonnegative functions inL_r(µ).

Proof. Without loss of generality we can assume that f+gr = 1 and so dν= (f+g)^rdµis a probability measure. This implies

f_r=

Ω

f^rdµ 1/r

=

{f+g>0}

f^r

(f+g)^r(f +g)^rdµ 1/r

≤

{f+g>0}

f

f+g(f+g)^rdµ.

Analogously,

gr≤

{f+g>0}

g

f+g(f+g)^rdµ.

Thereforefr+gr≤1 =f+gr.

Now, combining Theorem 6.2.13 with Lemma 6.2.16,

A⁻_p¹ E

n i=1

ε_if_i^p

p

1/p

≥ⁿ

i=1

|f_i|²¹₂

p

= n i=1

|f_i|²

12

p/2

142 6 TheLp-Spaces for 1≤p <∞

≥ⁿ

i=1

f_i²

p/2

1/2

=ⁿ

i=1

fi²_p1/2

.

To obtain the cotype-2 estimate we just have to replace the Lp-average (En

j=1ε_jf_j^pp)^1/pby (En

j=1ε_jf_j²p)^1/2using Kahane’s inequality (at the small cost of a constant) .

(b) For each 2 < p < ∞, from Theorem 6.2.13 in combination with Ka- hane’s inequality there exists a constantC=C(p) so that

E n i=1

ε_if_i²

p

1/2

≤Cⁿ

i=1

|f_i|²¹₂

p

. Sincep/2>1, the triangle law now holds inL_p/2(µ) and hence

ⁿ

i=1

|f_i|²¹₂

p

= n i=1

|f_i|^21/2

p/2≤ⁿ

i=1

f_i²p/2

1/2

= ⁿ

i=1

f_i²p

1/2

. This shows that L_p(µ) has type 2. Therefore, from part (a) and Proposi- tion 6.2.12 it follows thatL_p(µ) has cotypep.

The last statement of the theorem follows from Remark 6.2.11 and the fact thatL_p(µ) contains_pas a subspace.

Example 6.2.17.To finish the section let us give an example showing that the concepts of type and cotype are not in duality, in the sense that the converse of Proposition 6.2.12 need not hold. The space C[0,1] fails to have nontrivial type because it contains a copy ofL₁, whereas its dual,M(K), has cotype 2 (we leave the verification of this fact to the reader).