Uchiyama's lemma and the John-Nirenberg inequality

0.16 MB·English

by  Greg Knese

UCHIYAMA’S LEMMA AND THE JOHN-NIRENBERG
INEQUALITY
2
1 GREG KNESE
0
2
Abstract. UsingintegralformulasbasedonGreen’stheoremand
n
in particular a lemma of Uchiyama, we give simple proofs of com-
a
J parisonsofdifferentBMOnormswithoutusingtheJohn-Nirenberg
inequality while we also give a simple proof of the strong John-
5
2 Nirenberg inequality. Along the way we prove BMOA (H1)∗
and BMO Re(H1)∗. ⊂
] ⊂
V
C
.
h 1. Introduction
t
a
The space of functions of bounded mean oscillation(BMO), initially
m
introduced in the study of PDEs, is most famously known from the
[
Fefferman duality theorem as the dual of the real Hardy space Re(H1)

1
[3]. The John-Nirenberg inequality is the traditional point of entry for
v
4 understanding BMO [4]. BMO on the unit circle T is most naturally
5 defined using the norm
3
5 1
01. ||f||∗ = sIu⊂pT |I| ZI |f −fI|ds
2 where the supremum is over intervals I T and f = 1 fds. A
1 ⊂ I I I
: function f L1 is then in BMO if the above supremu| m| is finite.
v ∈ R
(Here ds is arc length measure.)
i
X
It turns out to be useful to use two other norms on BMO. An-
r
other norm is obtained by using the normalized Poisson kernel P (ζ) =
a z
2π1−1|zz¯|2ζ2 to perform averaging
| − |
f = sup f f(z) P ds
|| ||BMO1 z D T| − | z
∈ Z
where we use the harmonic extension of f, f(z) = fP . The proof
T z
of equivalence is obtained by comparing P to appropriate box kernels
z
R
Date: January 26, 2012.
2010 Mathematics Subject Classification. 30H35, 30H10, 30J99.
Key words and phrases. John-Nirenberg, Uchiyama, BMO, BMOA, Hardy
spaces.
Research supported by National Science Foundation grant DMS 1048775.
1
2 GREG KNESE
1 χ . (See Garnett [2] Chapter 6, Section 1.) Yet another norm is the
I I
G| |arsia norm
f 2 := sup f f(z) 2P ds = sup[( f 2)(z) f(z) 2]
|| ||BMO2 z D T| − | z z D | | −| |
∈ Z ∈
where f 2(z) andf(z) denote values of harmonic extensions of f 2 and
| | | |
f respectively. For this definition we need f L2.
∈
Why are there so many norms? The Garsia norm is the easiest norm
to use when proving that BMO is the dual of the real Hardy space
Re(H1), but the norm and the norm most exemplify the
k·k∗ k·kBMO1
phrase “bounded mean oscillation.” Unfortunately, it is not obvious
that the Garsia norm is equivalent to the earlier norms and indeed
this is one of the main purposes of the John-Nirenberg inequality. The
John-Nirenberg inequality says there exist constants c,C > 0 such that
for any interval I T
⊂
ζ I : f(ζ) f > λ cλ
I
|{ ∈ | − | }| Cexp − .
I ≤ f
| | (cid:18)k k∗(cid:19)
This statement is implied by the strong John-Nirenberg inequality:
there exists c > 0 such that ǫ < c/ f implies
k k∗
1
sup eǫf fI ds < .
| − |
I⊂T |I| ZI ∞
For more background and the traditional approach to all of this mate-
rial, the reader should consult Garnett [2] chapter 6.
Inthispaperwesomewhat turnthingsaroundbyproving theequiva-
lenceofthenorms and withoutusing John-Nirenberg
k·kBMO1 k·kBMO2
andthenproveastrongJohn-Nirenberginequality intermsofthenorm
. All proofs of the John-Nirenberg inequality, of which we are
k·kBMO2
aware, involve some kind of Calderon-Zygmund decomposition and a
stopping-time argument. (More sophisticated variants of these ideas
have been employed in finding sharp versions of the John-Nirenberg in-
equality. See [5], [11], and [10].) In contrast, the proof presented in this
article uses only Green’s theorem and most importantly Uchiyama’s
lemma. This approach owes a great debt to several recent approaches
to traditionally difficult theorems in complex analysis beginning with
the corona theorem as proved by T.Wolff (see [2] Chapter 8), the Hunt-
Muckenhoupt-Wheeden theorem as proved in [6], and the reproducing
kernel thesis for Carleson measures as proved in [9]. The book by
Andersson [1] features many aspects of the present approach as well.
Acknowledgments. Thanks to John McCarthy, Kabe Moen, and Ta-
van Trent for useful comments and conversations.
UCHIYAMA’S LEMMA AND THE JOHN-NIRENBERG INEQUALITY 3
2. Main Results
For definiteness, we shall say a real valued function f L2(T) mod-
∈
ulo constant functions is in BMO if the norm f as above is
finite. An analytic function f in the Hardy spacke HkB2M(TO)2 modulo the
constant functions is in BMOA if f as above is finite.
k kBMO2
The John-Nirenberg inequality is typically required to prove the
norms f and f are equivalent (or even to show that
k kBMO1 k kBMO2
f L2 with finite norm is in L1). We get around this fact in
∈ k·kBMO2
the setting of BMOA, and we are able to prove the following theorem
without using the John-Nirenberg theorem.
Theorem 2.1. If F BMOA, then
∈
F F 2√e F .
|| ||BMO1 ≤ || ||BMO2 ≤ || ||BMO1
The first inequality is just Cauchy-Schwarz. Certain aspects of our
approach become more technical in the case of real BMO. Never-
theless, we are still able to prove the following comparison without
John-Nirenberg.
Theorem 2.2. If u BMO, then
∈
u u (21) u .
k kBMO1 ≤ k kBMO2 ≤ k kBMO1
The explanation for the non-sharp constant 21 will have to wait until
Section 7. Finally, we prove the following version of the strong John-
Nirenberg inequality.
Theorem 2.3. Let F BMOA. For any ǫ < 2 , we have
∈ √ekFkBMO2
3
eǫF F(z)P ds < .
| − | z
T (1 ǫ√e F )3/2
Z − 2 k kBMO2
Let us point out a couple of direct consequences. If u L2(T) is
harmonically extended into the unit disk D and F = u+ i∈u˜, where u˜
is the harmonic conjugate of u, then we can prove
3
eǫu u(z)P ds <
| − | z
ZT (1−ǫ√2ekukBMO2)3/2
using the fact 2 u 2 = F 2 (Remark 3.2).
k kBMO2 k kBMO2
Using Theorem 2.1, we also have
3
eǫF F(z)P ds <
| − | z (1 ǫe F )3/2
ZT − k kBMO1
which shows this integral is finite so long as ǫ < 1/(e F ).
k kBMO1
4 GREG KNESE
3. Definitions, Green’s theorem, and Hardy-Stein
identities
We use ds to denote arc length measure on the unit circle T or the
circle rT of radius r. The measure dA denotes area measure in the
complex plane C, and D and rD refer to the open unit disk and the
disk of radius r, respectively. We use the following notations
1 ∂ ∂ 1 ∂ ∂
¯ ¯
∂ = ( i ),∂ = ( +i ),∆ = 4∂∂.
2 ∂x − ∂y 2 ∂x ∂y
One form of Green’s theorem is
fP(r)ds f(z) = ∆fg(r)dA
z − z
ZrT ZrD
where z < r 1,
| | ≤
1
1 z ζ − 1 z/r 2
g(r)(ζ) = log − and P(r)(ζ) = −| | ,
z 2π (cid:12)r z¯ζ(cid:12) z 2π 1 z¯ζ/r2 2
(cid:12) − r (cid:12) | − |
(cid:12) (cid:12)
assuming f C2(rD)(cid:12). Write(cid:12) P = P(1) and g = g(1). It is worth
(cid:12) (cid:12) z z z z
∈
noting that
g(r) g
z ր z
as r 1, and
ր
P(r)(rζ) P (ζ) 0
| z − z | →
uniformly for ζ T as r 1.
∈ ր
Applying Green’s theorem to ( f 2+ǫ)p/2 or (u2+ǫ)p/2 and carefully
| |
making sure we can send ǫ 0 and then r 1, is one way to prove
ց ր
the following classical Hardy-Stein identities. See [8].
Lemma 3.1. For f Hp(D), 0 < p < , z D
∈ ∞ ∈
f pP ds f(z) p = p2 ∂f 2 f p 2g dA.
z − z
| | −| | | | | |
T D
Z ZZ
For 1 < p < , u Lp(T) extended harmonically into D, and z D
∞ ∈ ∈
we have
u pP ds u(z) p = p(p 1) u p 2 u 2g dA.
z − z
| | −| | − | | |∇ |
T D
Z ZZ
Here Hp(D) is the Hardy space on the unit disk with exponent p.
Remark 3.2. Notice that
f f(z) 2P ds = 4 ∂f 2g dA.
z z
| − | | |
T D
Z ZZ
UCHIYAMA’S LEMMA AND THE JOHN-NIRENBERG INEQUALITY 5
If u = Re(f) then 2 ∂f 2 = u 2 and we see that
| | |∇ |
f f(z) 2P ds = 2 (u u(z))2P ds
z z
| − | −
T T
Z Z
which implies f 2 = 2 u 2 , a fact we use several times.
k kBMO2 k kBMO2
4. Uchiyama’s lemma
Uchiyama’s lemma is our most important tool. (See Nikolskii [7]
page 290 and the notes on page 296.)
Lemma 4.1. If φ C2(D) and f is holomorphic in D, then for 0
∈ ≤
z < r < 1
| |
f eφP(r)ds ∆φeφ f g(r)dA
| | z ≥ | | z
ZrT ZZrD
Proof. For any ψ C2(D)
∈
∆eψ = eψ(∆ψ + ψ 2) eψ∆ψ.
|∇ | ≥
Applying this to ψ = φ+ 1 log( f 2+ǫ) we have
2 | |
∆(eφ( f 2+ǫ)1/2) eφ( f 2+ǫ)1/2∆φ
| | ≥ | |
after noticing that
1 2ǫ ∂f 2
∆ log( f 2+ǫ) = | | 0.
2 | | ( f 2+ǫ)2 ≥
| |
By Green’s theorem,
eφ( f 2+ǫ)1/2P(r)ds eφ(z)( f(z) 2 +ǫ)1/2
| | z − | |
ZrT
= ∆(eψ)g(r)dA
z
ZZrD
eφ( f 2 +ǫ)1/2∆φg(r)dA
≥ | | z
ZZrD
Letting ǫ 0, we get
→
eφ f P(r)ds eφ f ∆φg(r)dA.
| | z ≥ | | z
ZrT ZZrD
(cid:3)
Lemma 4.2. Let F BMOA,f H1. Then,
∈ ∈
e
|∂F|2|f|gzdA ≤ 4||fPz||L1||F||2BMO2
D
ZZ
6 GREG KNESE
Let u BMO,f H1. Then,
∈ ∈
e
|∇u|2|f|gzdA ≤ 2kfPzkL1kuk2BMO2
D
ZZ
Proof. Let z < r < 1. Apply the previous lemma to φ = ( F(ζ) 2
| | | | −
( F 2)(ζ))/ F 2 which is non-positive, bounded below by 1, and
| | || ||BMO2 −
subharmonic since
∆φ = 4( ∂F 2)/ F 2 .
| | || ||BMO2
We arrive at
f P(r)ds eφ f P(r)ds
| | z ≥ | | z
ZrT ZrT
4
eφ ∂F 2 f g(r)dA
≥ F 2 | | | | z
ZZrD k kBMO2
4e 1
− ∂F 2 f g(r)dA
≥ F 2 | | | | z
k kBMO2 ZZrD
After doing so let r 1 and the first part of the lemma is proved.
→
For the second part, set φ = ((u(ζ))2 u2(ζ))/ u 2 1 and
− k kBMO2 ≥ −
notice that
u 2
∆φ = 2 |∇ | .
u 2
k kBMO2
Then,
2e 1
f P(r)ds − u 2 f g(r)dA.
| | z ≥ u 2 |∇ | | | z
ZrT k kBMO2 ZZrD
After letting r 1, the second inequality follows. (cid:3)
→
5. Theorem 2.1: norm comparison for BMOA
Along the way to proving Theorem 2.1 (the norm comparison for
BMOA), it is useful to prove the key estimate for proving BMOA
⊂
(H1) or BMO Re(H1) . The approach is similar to Andersson
∗ ∗
⊂
[1] (see chapters 8 and 9) with a stricter accounting of the constants
involved.
Theorem 5.1. Let F BMOA,h H2. Then,
∈ ∈
Fh¯Pzds ≤ 2√e||F||BMO2||hPz||L1.
T
(cid:12)Z (cid:12)
(cid:12) (cid:12)
Let u BMO,h(cid:12) H2. The(cid:12)n,
∈ (cid:12)∈ (cid:12)
uRe(h)Pzds ≤ √2e||u||BMO2||hPz||L1.
T
(cid:12)Z (cid:12)
(cid:12) (cid:12)
(cid:12) (cid:12)
(cid:12) (cid:12)
UCHIYAMA’S LEMMA AND THE JOHN-NIRENBERG INEQUALITY 7
The reason for having h H2 as opposed to H1 is that the integrals
∈
may not converge absolutely for h H1. However, since H2 is dense in
∈
H1, the estimates imply that integration against a function in BMOA
extends to a bounded linear functional on H1.
Proof of Theorem 5.1. WemayassumeF(z) = 0sinceF isonlyafunc-
tion modulo constant functions. By Green’s theorem (or a polarized
Hardy-Stein identity for p = 2),
¯
FhP ds = 4 ∂F∂hg dA
z z
T D
Z ZZ
By Cauchy-Schwarz, in modulus this is less than or equal to
1/2 ∂h 2 1/2
4 ∂F 2 h g dA | | g dA
z z
| | | | h
D D
(cid:18)ZZ (cid:19) (cid:18)ZZ | | (cid:19)
e 1/2
≤ 4 4||hPz||L1||F||2BMO2 ||hPz||1L/12
≤ 2√(cid:16)e||hPz||L1||F||BMO2(cid:17)
wherethefirstinequalityfollowsfromLemmas4.2and3.1(withp = 1).
Similarly,
¯
uRe(h)P ds = 4Re ∂h∂ug dA
z z
T D
Z ZZ
and again by Cauchy-Schwarz this is bounded by
1/2 ∂h 2 1/2
2 u 2 h g dA | | g dA
z z
|∇ | | | h
D D
(cid:18)ZZ (cid:19) (cid:18)ZZ | | (cid:19)
≤ √2e||hPz||L1||u||BMO2
(after being careful with using 4 ∂u 2 = u 2).
| | |∇ | (cid:3)
Remark 5.2. If f L2(T) or f L1(T) is harmonically extended into
D, then defining f∈(ζ) := f(rζ)∈we have
r
f f as r 1 for j = 1,2
k rkBMOj ր k kBMOj ր
and this holds even if one of the norms is infinite. (We leave the proof
of this fact to the reader.) Because of this it suffices to prove Theorem
2.1 for F or Theorem 2.2 for u .
r r
Indeed, if we assume that f L1(T) and f < , and if we
∈ k kBMO1 ∞
have proven
f C f ,
k rkBMO2 ≤ k rkBMO1
8 GREG KNESE
then in particular sup f 2ds < and so f L2(T) by stan-
0<r<1 T| r| ∞ ∈
dard approximate identity properties for the Poisson kernel. It then
R
follows that f C f .
k kBMO2 ≤ k kBMO1
Proof of Theorem 2.1: As remarked above, we can replace F with F .
r
In this case, Theorem 2.1 is an immediate corollary of Theorem 5.1 if
we now replace F with F F(z) and h with F F(z) in the statement
− −
of Theorem 5.1. This gives
F F(z) 2P ds 2√e F F F(z) P ds
| − | z ≤ || ||BMO2 | − | z
T T
Z Z
and taking a supremum over z yields Theorem 2.1. (cid:3)
Theorem 2.2 (the norm comparison for real BMO) is seemingly not
so easy to deduce from Theorem 5.1 as the best it gives is the estimate
u √2e u+iu˜
k kBMO2 ≤ k kBMO1
where u˜ is the harmonic conjugate of u. As there is no direct compar-
ison of u and u˜ in terms of L1 norms (unlike in the L2 situation), it
seems the BMO condition needs to play a more active role in the com-
parison of u and u . One of the lemmas in the proof of
k kBMO1 k kBMO2
the strong John-Nirenberg inequality is used in proving Theorem 2.2,
so we postpone the proof to Section 7.
6. Theorem 2.3: The strong John-Nirenberg inequality
Lemma 6.1. For F Hk,
∈
F F(z) kP ds = k2 ∂F 2 F F(z) k 2g dA
z − z
| − | | | | − |
T D
Z ZZ
Proof. This is lemma 3.1 with f = F F(z) and p = k. (cid:3)
−
Lemma 6.2. For F BMOA,
∈
F F(z) kP ds e(k/2)2 F F(z) k 2P ds F 2
| − | z ≤ | − | − z k kBMO2
T T
Z Z
so that inductively we have
F F(z) 2kP ds ekk!2 F 2k
| − | z ≤ k kBMO2
T
Z
and
2
(2k +1)!
F F(z) 2k+1P ds (e/4)k F 2k+1 .
| − | z ≤ 2kk! k kBMO2
T
Z (cid:18) (cid:19)
UCHIYAMA’S LEMMA AND THE JOHN-NIRENBERG INEQUALITY 9
Proof. IfweapplyLemma4.2withF BMOAandf = (F F(z))k 2,
−
∈ −
then
∂F 2 F F(z) k 2g dA (e/4) F F(z) k 2P ds F 2 .
| | | − | − z ≤ | − | − z k kBMO2
D T
ZZ Z
Coupled with Lemma 6.1,
F F(z) kP ds (e/4)k2 F F(z) k 2P ds F 2
| − | z ≤ | − | − z k kBMO2
T T
Z Z
(cid:3)
and the rest follows by iterating this inequality.
Proof of Theorem 2.3. Observe that by Lemma 6.2
ǫk
eǫF F(z)P ds = F F(z) kP ds
| − | z z
k! | − |
T T
Z k 0 Z
X≥
ǫ2k ǫ2k+1
= F F(z) 2kP ds+ F F(z) 2k+1P ds
z z
(2k)! | − | (2k +1)! | − |
T T
k 0 Z Z
X≥
ǫ2k ǫ2k+1 (2k +1)! 2
ek(k!)2 F 2k + (e/4)k F 2k+1
≤ (2k)! k kBMO2 (2k +1)! 2kk! k kBMO2
k 0 (cid:18) (cid:19)
X≥
(k!)2 2 (2k +1)!
= (2x)2k + x x2k
(2k)! √e 4k(k!)2
! !
k 0 k 0
X≥ X≥
where x = (1/2)ǫ√e F . The last expression can be explicitly
k kBMO2
computed. Whenever x < 1 it is equal to
1 xarcsin(x)+ 2 x π + 2
√e 2 √e
+
1 x2 (1 x2)3/2 ≤ (1 x)3/2
− − −
and since π + 2 < 3
2 √e
3
eǫF F(z)P ds .
| − | z
T ≤ (1 ǫ√e F )3/2
Z − 2 k kBMO2
(cid:3)
7. Theorem 2.2, norm comparison for real BMO
The Hardy-Stein identity for harmonic functions fails for p = 1 and
so we do not have a nice Green’s theorem formula for the expression
u u(z) P ds.
z
| − |
T
Z
A replacement is in the following lemma.
10 GREG KNESE
Lemma 7.1. If u L1(T) (and extended harmonically into D), then
∈
u 2
|∇ | g dA u u(z) P ds
((u u(z))2 +1)3/2 z ≤ | − | z
D T
ZZ − Z
Proof. By Green’s theorem, for z < r < 1
| |
(7.1)
u 2
|∇ | g(r)dA = ((u u(z))2 +1)1/2P(r)ds 1.
((u u(z))2 +1)3/2 z − z −
ZZrD − ZrT
Setting v = u u(z), this follows from
−
∆(v2 +1)1/2 = v 2(v2 +1) 3/2.
−
|∇ |
Since u converges to u in L1(T) as r 1 (recall this only uses basic
r
ր
approximate identity properties of the Poisson kernel), it can be shown
that
lim ((u u(z))2 +1)1/2P(r)ds = ((u u(z))2 +1)1/2P ds
rր1ZrT − z ZT − z
since √x2 +1 y2 +1 C x y . On the other hand, the left
| − | ≤ | − |
hand side of (7.1) converges monotonically to
p
u 2
|∇ | g dA = ((u u(z))2 +1)1/2P ds 1.
((u u(z))2 +1)3/2 z − z −
D T
ZZ − Z
The desired inequality
((u u(z))2 +1)1/2P ds 1 u u(z) P ds
z z
− − ≤ | − |
T T
Z Z
follows from the inequality √1+x2 1+ x . (cid:3)
≤ | |
Theorem 2.2 follows from the next result since e2/351/335/3 21.
≤
Theorem 7.2. Let u L1(T) with u < . Then,
∈ k kBMO1 ∞
u e2/351/335/3 u
k kBMO2 ≤ k kBMO1