L p Boundedness of the Commutator

Exercises

7.5 Commutators of Singular Integrals with BMO Functions

7.5.3 L p Boundedness of the Commutator

We note that if f has compact support and b is in BMO, then b f lies in L^q(Rⁿ) for all q<∞and therefore T(b f)is well defined whenever T is a singular integral operator. Likewise,[b,T]is a well defined operator onC₀^∞for all b in BMO.

Having obtained the crucial Lemma 7.5.5, we now pass to an important result concerning its L^pboundedness.

Theorem 7.5.6. Let T be as in Lemma 7.5.5. Then for any 1<p<∞there exists a constant C=Cp,nsuch that for all smooth functions with compact support f and all BMO functions b, the following estimate is valid:

[b,T](f)

L^p(Rⁿ)≤Cb

BMOf

L^p(Rⁿ). (7.5.9) Consequently, the linear operator

f →[b,T](f)

admits a bounded extension from L^p(Rⁿ)to L^p(Rⁿ)for all 1<p<∞with norm at most a multiple ofb

BMO.

Proof. Using the inequality of Theorem 7.4.4, we obtain for functions g, with|g|^δ locally integrable,

{Md(|g|^δ)¹δ >2^δ1λ} ∩ {M_δ^#(g)≤γλ} ≤2ⁿγ^δ {Md(|g|^δ)δ¹ >λ} (7.5.10) for allλ^,γ^,δ^>0. Then a repetition of the proof of Theorem 7.4.5 yields the second inequality:

M(|g|^δ)¹δ

L^p≤CnM_d(|g|^δ)δ¹

L^p ≤Cn(p)M_δ^#(g)

L^p (7.5.11)

for all p∈(p0,∞), provided M_d(|g|^δ)¹δ ∈L^p0(Rⁿ)for some p0>0.

For the following argument, it is convenient to replace b by the bounded function

bk(x) =

⎧⎪

⎨

⎪⎩

k if b(x)<k, b(x) if−k≤b(x)≤k,

−k if b(x)>−k, which satisfiesbk

BMO≤b

BMOfor any k>0; see Exercise 7.1.4.

For given 1<p<∞, select p0such that 1<p0<p. Given a smooth function with compact support f , we note that the function b_kf lies in L^p0; thus T(b_kf) also lies in L^p0. Likewise, b_kT(f)also lies in L^p0. Since M_δ is bounded on L^p0 for 0<δ ^<1, we conclude that

M_δ([bk,T](f))

L^p0≤C_δM_δ(bkT(f))

L^p0+M_δ(T(bkf))

L^p0

<∞. This allows us to obtain (7.5.11) with g= [bk,T](f). We now turn to Lemma 7.5.5, in which we pick 0<δ^<ε^<1. Taking L^pnorms on both sides of (7.5.6) and using (7.5.11) with g= [bk,T](f)and the boundedness of M_ε, T , and M²on L^p(Rⁿ), we deduce the a priori estimate (7.5.9) for smooth functions with compact support f and the truncated BMO functions bk.

The Lebesgue dominated convergence theorem gives that bk→b in L²of every compact set and, in particular, in L²(supp f). It follows that bkf →b f in L²and therefore T(bkf)→T(b f)in L²by the boundedness of T on L². We deduce that

for some subsequence of integers kj, T(bkjf)→T(b f)a.e. For this subsequence we have [bkj,T](f)→[b,T](f)a.e. Letting j→∞and using Fatou’s lemma, we deduce that (7.5.9) holds for all BMO functions b and smooth functions f with compact support.

Since smooth functions with compact support are dense in L^p, it follows that the commutator admits a bounded extension on L^pthat satisfies (7.5.9).

We refer to Exercise 7.5.4 for an analogue of Theorem 7.5.6 when p=1.

Exercises

7.5.1. Use Jensen’s inequality to show that M is pointwise controlled by M_{L log(1+L)}. 7.5.2. (a) (Young’s inequality for Orlicz spaces ) Letϕbe a continuous, real-valued, strictly increasing function defined on[0,∞)such thatϕ(0) =0 and limt→∞ϕ(t) =

∞. Letψ⁼ϕ⁻¹and for x∈[0,∞)define Φ(x) =

_x

0 ϕ(t)dt, Ψ(x) = _x

0 ψ(t)dt. Show that for s,t∈[0,∞)we have

st≤Φ^{(s) +}Ψ^(t).

(b) (cf. Exercise 4.2.3 ) Choose a suitable functionϕ in part (a) to deduce for s,t in [0,∞)the inequality

st≤(t+1)log(t+1)−t+e^s−s−1≤t log(t+1) +e^s−1. (c) (H¨older’s inequality for Orlicz spaces ) Deduce the inequality

f,g ≤2f

Φ(L)g

Ψ(L).

Hint: Give a geometric proof distinguishing the cases t>ϕ^{(s)and t}≤ϕ^{(s). Use} that for u≥0 we have₀^uϕ(t)dt+₀^ϕ(u)ψ(s)ds=uϕ(u).

7.5.3. Let T be as in Lemma 7.5.5. Show that there is a constant Cn<∞such that for all f ∈L^p(Rⁿ)and g∈L^p(Rⁿ)we have

T(f)g−f T^t(g)

H¹(Rⁿ)≤Cf

L^p(Rⁿ)g

L^p (Rⁿ).

In other words, show that the bilinear operator (f,g)→T(f)g−f T^t(g) maps L^p(Rⁿ)×L^p(Rⁿ)to H¹(Rⁿ).

7.5.4. (P´erez [260] ) LetΦ(t) =t log(1+t). Then there exists a positive constant C, depending on the BMO constant of b, such that for any smooth function with

compact support f the following is valid:

αsup>0

1

Φ⁽_α^{1) [b,T](f) >}α ≤C sup

α>0

1

Φ⁽_α^{1) M2(f)>}α ^.

7.5.5. Let R1, R2be the Riesz transforms in R². Show that there is a constant C<∞ such that for all square integrable functions g1, g2on R²the following is valid:

R1(g1)R2(g2)−R1(g2)R2(g1)

H¹ ≤Cpg1L²g2L². Hint: Consider the pairing

g₁,R₂([b,R₁](g₂))−R₁([b,R₂](g₂))

with b∈BMO.

7.5.6. (Coifman, Lions, Meyer, and Semmes [78] ) Use Exercise 7.5.5 to prove that the Jacobian Jf of a map f = (f1,f2): R²→R²,

Jf=det

∂1f1∂2f1

∂1f2∂2f2

,

lies in H¹(R²)whenever f1,f2∈˙L²₁(R²).

Hint: Set gj=Δ^1/2(fj).

7.5.7. LetΦ(t) =t(1+log⁺t)^α, where 0≤α^<∞. Let T be a linear (or sublinear) operator that maps L^p0(Rⁿ)to L^p0,∞(Rⁿ)with norm B for some 1<p0≤∞and also satisfies the following weak type Orlicz estimate: for all functions f inΦ(L),

|{x∈Rⁿ: |T(f)(x)|>λ}| ≤A

RⁿΦ^|f(x)| λ

dx,

for some A<∞and all λ ^>0. Prove that T is bounded from L^p(Rⁿ) to itself, whenever 1<p<p0.

Hint: Set f^λ= fχ_|f|>λand f_λ =f−f^λ. When p0<∞, estimate|{|T(f)|>2λ}|

by|{|T(f^λ)|>λ}|+|{|T(f_λ)|>λ}| ≤A_|f|>λΦ^|f(x)|_λ dx+B^p0_|f|≤λ^|f(x)|_λp0^p0dx.

Multiply by p, integrate with respect to the measureλ^p−1dλfrom 0 to infinity, apply Fubini’s theorem, and use that₀¹Φ(1/λ⁾λ^p−1dλ^<∞to deduce that T maps L^pto L^p,∞. When p0=∞, use that|{|T(f)|>2Bλ}| ≤ |{|T(f^λ)|>Bλ}|and argue as in the case p0<∞. Boundedness from L^pto L^pfollows by applying Theorem 1.3.2.

HISTORICAL NOTES

The space of functions of bounded mean oscillation first appeared in the work of John and Nirenberg [177] in the context of nonlinear partial differential equations that arise in the study of minimal surfaces. Theorem 7.1.6 was obtained by John and Nirenberg [177]. The relationship of BMO functions and Carleson measures is due to Fefferman and Stein [130]. For a variety of issues relating BMO to complex function theory one may consult the book of Garnett [142]. The duality of H¹and BMO (Theorem 7.2.2) was announced by Fefferman in [124], but its first proof appeared

in the article of Fefferman and Stein [130]. This article actually contains two proofs of this result.

The proof of Theorem 7.2.2 is based on the atomic decomposition of H¹, which was obtained subsequently. An alternative proof of the duality between H¹and BMO was given by Carleson [57]. Dyadic BMO (Exercise 7.4.4) in relation to BMO is studied in Garnett and Jones [144]. The same authors studied the distance in BMO to L^∞in [143].

Carleson measures first appeared in the work of Carleson [53] and [54]. Corollary 7.3.6 was first proved by Carleson, but the proof given here is due to Stein. The characterization of Carleson measures in Theorem 7.3.8 was obtained by Carleson [53]. A theory of balayage for studying BMO was developed by Varopoulos [323]. The space BMO can also be characterized in terms Carleson measures via Theorem 7.3.8. The converse of Theorem 7.3.8 (see Fefferman and Stein [130]) states that if the functionΨsatisfies a nondegeneracy condition and|f∗Ψt|^{2 dx dt}t is a Carleson measure, then f must be a BMO function. We refer to Stein [292] (page 159) for a proof of this fact, which uses a duality idea related to tent spaces. The latter were introduced by Coifman, Meyer, and Stein [83] to systematically study the connection between square functions and Carleson measures.

The sharp maximal function was introduced by Fefferman and Stein [130], who first used it to prove Theorem 7.4.5 and derive interpolation for analytic families of operators when one endpoint space is BMO. Theorem 7.4.7 provides the main idea why L^∞can be replaced by BMO in this con- text. The fact that L²-bounded singular integrals also map L^∞to BMO was independently obtained by Peetre [254], Spanne [286], and Stein [290]. Peetre [254] also observed that translation-invariant singular integrals (such as the ones in Corollary 7.4.10) actually map BMO to itself. Another inter- esting property of BMO is that it is preserved under the action of the Hardy–Littlewood maximal operator. This was proved by Bennett, DeVore, and Sharpley [19]; see also the almost simultaneous proof of Chiarenza and Frasca [60]. The decomposition of open sets given in Proposition 7.3.4 is due to Whitney [331].

An alternative characterization of BMO can be obtained in terms of commutators of singular integrals. Precisely, we have that the commutator[b,T](f)is L^pbounded for 1<p<∞if and only if the function b is in BMO. The sufficiency of this result (Theorem 7.5.6) is due to Coifman, Rochberg, and Weiss [85], who used it to extend the classical theory of H^pspaces to higher di- mensions. The necessity was obtained by Janson [176], who also obtained a simpler proof of the sufficiency. The exposition in Section 7.5 is based on the article of P´erez [260]. This approach is not the shortest available, but the information derived in Lemma 7.5.5 is often useful; for instance, it is used in the substitute of the weak type(1,1)estimate of Exercise 7.5.4. The inequality (7.5.3) in Lemma 7.5.4 can be reversed as shown by P´erez and Wheeden [263]. Weighted L^pestimates for the commutator in terms of the double iteration of the Hardy–Littlewood maximal operator can be deduced as a consequence of Lemma 7.5.5; see the article of P´erez [261].

Orlicz spaces were introduced by Birbaum and Orlicz [26] and furher elaborated by Orlicz [251], [252]. For a modern treatment one may consult the book of Rao and Ren [269]. Bounded mean oscillation with Orlicz norms was considered by Str¨omberg [297].

The space of functions of vanishing mean oscillation (V MO) was introduced by Sarason [277]

as the set of integrable functions f on T¹satisfying lim_δ→0sup_I:|I|≤δ|I|⁻¹I|f−Avg_If|dx=0.

This space is the closure in the BMO norm of the subspace of BMO(T¹)consisting of all uniformly continuous functions on T¹. One may define V MO(Rⁿ)as the space of functions on Rⁿthat satisfy lim_δ→0sup_Q:|Q|≤δ|Q|⁻¹Q|f−Avg_Qf|dx=0, limN→∞sup_Q:(Q)≥N|Q|⁻¹Q|f−Avg_Qf|dx=0, and limR→∞sup_{Q: Q∩B(0,R)=/0}|Q|⁻¹Q|f−AvgQf|dx=0; here I denotes intervals in T¹and Q cubes in Rⁿ. Then V MO(Rⁿ)is the closure of the the space of continuous functions that vanish at infinity in the BMO(Rⁿ)norm. One of the imporant features of V MO(Rⁿ)is that it is the predual of H¹(Rⁿ), as was shown by Coifman and Weiss [86]. As a companion to Corollary 7.4.10, singular integral operators can be shown to map the space of continuous functions that vanish at infinity into V MO. We refer to the article of Dafni [101] for a short and elegant exposition of these results as well as for a local version of the V MO-H¹duality.