calderon–zygmund theory related to´ poincare...
TRANSCRIPT
Function spaces, nonlinear analysis and applications, Lectures notes of the springlectures in analysis. Editors: Jaroslav Lukes and Lubos Pick, Charles University andAcademy of Sciences, (1999), 31-94, ISBN: 80–85863-39-1.
CALDERON–ZYGMUND THEORY RELATED TOPOINCARE-SOBOLEV INEQUALITIES,
FRACTIONAL INTEGRALS AND SINGULARINTEGRAL OPERATORS
Carlos PerezUniversidad de Sevilla
Paseky’s lecture notes
Contents
1 A brief introduction to the Ap theory of weights 4
2 Lp properties of functions with controlled oscillation 9
2.1 BMO and Holder-Lipschitz spaces . . . . . . . . . . . . . . . . . . . . . . . . 10
2.2 Poincare–Sobolev Inequalities . . . . . . . . . . . . . . . . . . . . . . . . . . 10
2.3 Fractional Integrals and Poincare inequalities . . . . . . . . . . . . . . . . . . 12
2.4 Lp self–improving property . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
2.5 Proof of the Poincare-Sobolev inequalities . . . . . . . . . . . . . . . . . . . 22
3 Rearrangements 26
3.1 Rearrangements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
3.2 A basic covering lemma . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
3.3 Proof of the main Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
4 Orlicz maximal functions 34
4.1 Orlicz spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34
4.2 Orlicz maximal functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
5 Fractional integral operators 39
5.1 Weak implies strong . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39
5.2 Step 1: Discretization of the potential operator . . . . . . . . . . . . . . . . 41
5.3 Step 2: Replacing dyadic cubes by dyadic Calderon–Zygmund cubes . . . . . 42
5.4 Step 3: Patching the pieces together . . . . . . . . . . . . . . . . . . . . . . . 45
6 Singular Integral Operators 48
6.1 Introduction: Singular Integral Operators and smoothness of functions . . . 48
6.2 Weighted Lp estimates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49
6.3 Banach function spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54
6.4 Commutators: singular integrals with higher degree of singularity . . . . . . 56
2
7 Appendix I: The Calderon–Zygmund covering lemma 58
8 Appendix II: The A∞ condition of Muckenhoupt 62
9 Appendix III: Exponential self-improving property 67
3
Chapter 1
A brief introduction to the Ap theoryof weights
The Hardy-Littlewood maximal function is the operator defined by
Mf(x) = supx∈Q
1
|Q|
∫Q
|f(y)| dy
where the supremum is taken over all the cubes containing x. By cube we always mean acube with sides parallel to the axes. There are variations of this definition such asreplacing cubes by balls or just considering cubes or balls centered at x but all of them areequivalent with dimensional constants.
Maximal functions arise very naturally in analysis, for proving theorems about theexistence almost everywhere of limits, for controlling pointiwise important objects such asthe Poisson Integrals or for controlling, not pointiwise but at least in average, other basicoperators such as singular integral operators.
The model example of existence almost everywhere of limits is the Lebesguedifferentiation theorem:
f(x) = limr→0
1
|B(x, r)|
∫B(x,r)
f(y) dy
Indeed, as it is well known this follows from the following “weak type” estimate:
Theorem 1.0.1 (Hardy–Littlewood–Wiener) There exists a finite constant C suchthat for each positive λ the following inequality holds,
|x ∈ Rn : Mf(x) > λ| ≤ C
λ
∫Rn
|f(x)| dx (1.1)
As a consequence of (1.1) we can derive:
Corollary 1.0.2 Let 1 < p < ∞ then there exists a constant C such that∫Rn
(Mf)p dx ≤ C
∫Rn
|f |p dx (1.2)
4
We say that w is a weight if w is a a.e. postive locally integrable function in Rn. If E isany measurable set we denote w(E) =
∫E
w.
Theorem 1.0.3 a) There exists a constant C such that for all λ and f
w(x ∈ Rn : Mf(x) > λ) ≤ C
λ
∫Rn
|f(x)|Mw(x)dx. (1.3)
b) As a consequence if 1 < p < ∞ there exists a constant C such that for all f∫Rn
(Mf)p w(x)dx ≤ C
∫Rn
|f |p Mw(x)dx. (1.4)
We may think of this inequality as a kind of duality for the (nonlinear) operator. It canalso be seen as an antecedent of the Ap theory weights. It proved by C. Fefferman and E.Stein in [FS] to derive a vector–valued extension of (1.0.2). Indeed, for a vector offunctions f = fii and for q > 0 the operator M q is defined by
M qf(x) =
(∞∑i=1
(Mfi(x))q
)1/q
.
This operator can also be seen as a generalization of the classical integral of Marcinkiewicz.Then if we denote |f(x)|q = (
∑∞i=1 |fi(x)|q)1/q
= ‖f(x)‖`q we have for 1 < p < ∞ and1 < q ≤ ∞ ∫
Rn
M qf(x)p dx ≤ C
∫Rn
|f(x)|pq dx. (1.5)
A corresponding weak type (1, 1) also holds in the range 1 < p < ∞. These estimates playan important role in Harmonic Analysis, specially in the theory of Littlewood-Paley. Tosome extent these operators behave more as a singular integral (cf. Chapter 6) than as amaximal function. This is very well reflected in the work [RRT] where M qf , as well manyother operators, is seen as a vector–valued singular integral operator.
Proof: The proof of b) follows from the Marcinkiewicz interpolation theorem sinceM : L∞(u) → L∞(v) for arbitrary weights u and v. (see also the proof of (4.6) in Lemma4.2.2).
The proof of a) is based on the classical Vitali covering lemma:
Lemma 1.0.4 (Vitali) Let F = Qii=1,··· ,N be a finite family of cubes (or balls) in Rn.Then we can extract from F a sequence of pairwise disjoint cubes F ′ = Qjj=1,··· ,M suchthat ⋃
Q∈F
Q ⊂m⋃
j=1
5Qj.
5
See [Ma] p.23.
Now if we let Ωλ = x ∈ Rn : Mf(x) > λ for a given λ and let K to be any compactsubset contained in Ωλ. Let x ∈ K then by defintion of the maximal function there is acube Q = Qx containg x such that
1
|Q|
∫Q
|f(y)| dy > λ. (1.6)
Then K ⊂⋃
x∈K Qx and by compactness we can extract a finite family of cubes F = Qsuch that K ⊂
⋃Q∈F Q and where each cube satisfies (1.6). Then by the Vitali lemma we
can extract a pairwise disjoint family of cubes QjMj=1 such that
⋃Q∈F ⊂
⋃Mj=1 5Qj. Then
w(K) ≤M∑
j=1
w(5Qj) ≤1
λ
M∑j=1
w(5Qj)1
|Qj|
∫Qj
|f(y)| dy
≤ C
λ
M∑j=1
w(5Qj)
|5Qj|
∫Qj
|f(y)| dy ≤ C
λ
M∑j=1
∫Qj
|f(y)|Mw(y)dy
≤ C
λ
∫Rn
|f(y)|Mw(y)dy.
2
The main breakthrough in the theory arrived in the early seventies with the discoveryby B. Muckenhoupt in [Mu] of the right class of weights for which the Hardy-Littlewoodmaximal function is bounded on Lp(w), p > 1. This result opened up the possibility ofstudying operators with higher singularity.
Definition 1.0.5 A weight w belongs to the class Ap, 1 < p < ∞, if there is a constant Ksuch that (
1
|Q|
∫Q
w(y) dy
)(1
|Q|
∫Q
w(y)1−p′ dy
)p−1
≤ K (1.7)
for each cube Q. A weight w belongs to the class A1 if there is a constant K such that
1
|Q|
∫Q
w(y) dy ≤ K infQ
w. (1.8)
We will denote the infimum of the constants K by [w]Ap.
Observe that [w]Ap ≥ 1 by Jensen’s inequality.
Sometimes it is much more convenient to use the following equivalent condition for theA1 weight: there is a constant K such that for each x
Mw(x) ≤ K w(x)
6
Observe that it follows from inequality 1.3 that if w is an A1 weight then
w(x ∈ Rn : Mf(x) > λ) ≤ C
λ
∫Rn
|f(x)|w(x)dx (1.9)
It is not hard to see that this condition is also equivalent to saying that w ∈ A1.
Theorem 1.0.6 (Muckenhoupt) Let p > 1. The following conditions are equivalent:
a) w ∈ Ap;
b) there exists a constant K such that for each cube Q and nonnegative function f(1
|Q|
∫Q
f(y) dy
)p
w(Q) ≤ K
∫Q
f(y)p w(y)dy
c) M : Lp(w) → Lp,∞(w)
d) M : Lp(w) → Lp(w)
Conditions a), b) and c) are also equivalent when p = 1.
The main example of A1 weights are provided by the following lemma of Coifman andRochberg (cf. [GCRdF] p. 158).
Lemma 1.0.7 Let µ be a positive Borel measure such that Mµ(x) < ∞, a.e. x ∈ Rn.Then for each 0 ≤ δ < 1 the function (Mµ)δ ∈ A1
In fact it is also shown in that paper that any A1 weight w can be written essentially ofthis form. Furthermore, it is possible to show that
w ∈ Ap if and only if w = w1w1−p2
where w1 and w2 are A1 weights. That the last condition was a Ap was already known byB. Muckenhoupt, but it was P. Jones who proved the necessity of the factorization theoremwith a very difficult proof. The modern approach uses completely different arguments. It isdue to J. L. Rubio de Francia and it is presented in [GCRdF] where we remit the reader formore information about the Ap theory of weights.
Definition 1.0.8 The A∞ class of weights is defined in a natural way by
A∞ = ∪p>1Ap.
7
The A∞ class of weights shares a lot of interesting properties. We present in Appendix 8several characterizations. Some of them will be used along these notes.
Throughout these notes all notation is standard or will be defined as needed. All cubesare assumed to have their sides parallel to the co-ordinate axes. Given a cube Q, l(Q) willdenote the length of its sides and for any r > 0, rQ will denote the cube with the samecenter as Q and such that l(rQ) = rl(Q). We will denote the collection of all dyadic cubesby D and by D(Q) to the collection of all dyadic cubes relative to the (non necessarilydyadic) cube Q. Q(x, r) will denote the cube centered at x and with side–length 2r. Byweights we will always mean a a.e. positive, locally integrable functions which are positiveon a set of positive measure. Given a Lebesgue measurable set E and a weight w, |E| willdenote the Lebesgue measure of E and w(E) =
∫E
w dx. Given 1 < p < ∞, p′ = p/(p− 1)will denote the conjugate exponent of p. Finally, C will denote a positive constant whosevalue may change at each appearance.
8
Chapter 2
Lp properties of functions withcontrolled oscillation
For a locally integrable function f we define the oscillation of f over a cube Q as thenumber
Osc(Q, f) =1
|Q|
∫Q
|f(y)− fQ| dy ≈ infc∈R
1
|Q|
∫Q
|f(y)− c| dy
≈ 1
|Q|2
∫Q
∫Q
|f(y)− f(x)| dy dx.
Osc(Q, f) can be seen as the modern way of checking how smooth is a function. Forinstance, this can be seen in the characterization (2.1) given below of the Holder-Lipschitzspaces and also when considering functions in the Sobolev class, see for instance (2.3) and(2.4). But it goes beyond since R. Coifman and R. Rochberg proved in [CR] that maximalfunctions themselves have some sort of smoothness. They showed that Osc(Q, log Mµ) isbounded by a dimensional constant provided Mµ is finite almost everywhere. What thismeans is that there is a cancellation phenomenon undergoing. Indeed, this result would notmake sense if we remove the average fQ in the definition of Osc(Q, f).
In this chapter we will study properties of classes of functions which have some controlon the oscillation. In particular we will emphasize on Lp self-improving properties.Roughly speaking we will show that higher integrability properties of functions belongingto B.M.O. (or Holder-Lipschitz) and the higher integrability of Sobolev functions areesentially the same phenomenon.
The results and techniques we present in this chapter are taken from [FPW1] and[MP1]. These papers are specially motivated by the work P. Hajlasz and P. Koskela[HaK1], whose roots got back to the theory of Quasiconformal Mappings as can be seen in[HK], but also by the work of L. Saloff–Coste in [SC].
9
2.1 BMO and Holder-Lipschitz spaces
Recall that a function f is said to satisfy the Holder condition with exponent α, 0 < α < 1,if there exists a constant C such that for each x and y
|f(x)− f(y)| ≤ C|x− y|α.
When α = 1 the function is said to satisfy the Lipschitz condition. We will denote them byΛ(α) and it is well–known how relevant is this condition in Analysis. The main examples offunctions in Λ(α) are given by f(x) = |x|α, 0 < α ≤ 1.
From our point view it is more interesting the following characterization due toCampanato and Morrey:
Let 0 < α ≤ 1, then f ∈ Λ(α) if and only if
1
|Q|
∫Q
|f(y)− fQ| dy ≤ C`(Q)α, (2.1)
where C is a constant independent of the cube Q.
Observe that Λ(0) does not yield any interesting space, however if we replace in (2.1) αby zero we obtain the celebrated Bounded Mean Oscillation space introduced by F. Johnand L. Nirenberg in the early sixties in their study of partial differential equations. It isusually denoted by B.M.O.: f ∈ B.M.O. if there exists a constant C such that for eachcube Q
1
|Q|
∫Q
|f − fQ| ≤ C.
The main example is log M(µ)(x). If we take µ to be the point mass at the origin we getlog |x| which is the classical example of unbounded B.M.O. function.
In any case they share the following self–improving property:
Let 0 ≤ α ≤ 1 and let f be a function satisfying (2.1). Then for each1 < p < ∞ there exists a constant c such that(
1
|Q|
∫Q
|f − fQ|p)1/p
≤ c `(Q)α. (2.2)
2.2 Poincare–Sobolev Inequalities
Poincare-Sobolev inequalities play a main role in many places in Analysis. They are crucialtools in many important results for Partial Differential Equations, such as the DeGiorgi-Nash-Moser theorem, Harnack’s inequalty, properties of the Green functions forsecond order elliptic equations and existence results for semilinear equations (cf. for
10
instance some recent books such as [HKM] or citeMaly. It is also important in the theoryof Quasiconformal mappings as can be seen in the recent theory developed by J. Heinonenand P. Koskela [HK]. Very roughly, this theory shows that if X is a proper metric space,i.e. an space where closed balls are compact, endowed with a mesure which isAhlfors–David regular (hence doubling) then we can develop an analogue of the classicaltheory of Quasiconformal mappings if the space “admits’, in an appropriate sense, a (1, p)type Poincare inequality.
• The model example is the classical (1, 1) Poincare inequality:
1
|Q|
∫Q
|f − fQ| ≤ C`(Q)
|Q|
∫Q
|∇f |, (2.3)
where C a constant independent of both Q and f . In fact there is a more generalversion ot this estimate for p ≥ 1(
1
|Q|
∫Q
|f − fQ|p)1/p
≤ C `(Q)
(1
|Q|
∫Q
|∇f |p)1/p
. (2.4)
In fact we will show in Lemma 2.3.1 that they are both equivalent as a first step to amore general result in Theorem 2.4.2
An important observation is that this inequality is not only uniform on cubes butalso on an appropriate class of functions, which may be a Sobolev class of functionsor the Lipschitz class since such functions are differentiable almost everywhere by theRademacher–Stepanov theorem.
• A second example is a sharp version of inequality (2.3) which turns to be equivalentto the isoperimetric inequality: there exists a constant C such that(
1
|Q|
∫Q
|f − fQ|n′)1/n′
≤ C`(Q)
|Q|
∫Q
|∇f |. (2.5)
The version corresponding to p ≥ 1 is(1
|Q|
∫Q
|f − fQ|p∗)1/p∗
≤ C `(Q)
(1
|Q|
∫Q
|∇f |p)1/p
. (2.6)
Here p∗ = pnn−p
denotes the Sobolev exponent. Observe that 1∗ = nn−1
= n′ is theexponent from the isoperimetric inequality.
• We will show below that indeed both inequalities (2.4) and (2.6) are equivalent. Infact we will stress this relationship by introducing weights in these inequalities.These weighted Poincare inequalities are basic in the study of the smoothness of thesolutions of degenerate elliptic equation (see [FKS] and [HKM]) .
11
The main example holds for any w ∈ A2:(1
w(Q)
∫Q
|f − fQ|2w)1/2
≤ C `(Q)
(1
w(Q)
∫Q
|∇f |2w)1/2
. (2.7)
To run the Moser iterative technique the following was derived in [FKS]: there existsa tiny positive constant ε for which(
1
w(Q)
∫Q
|f − fQ|2+εw
)1/(2+ε)
≤ C `(Q)
(1
w(Q)
∫Q
|∇f |2w)1/2
(2.8)
We will show below that these examples are particular cases of a more general situation.
2.3 Fractional Integrals and Poincare inequalities
It is well known that there is an intimate relationship between smoothness and fractionalintegration. This is best reflected in the following estimate: let f be a, say C1 function, ona cube Q, then there is a universal constant C such that for each x ∈ Q
|f(x)− fQ| ≤ C I1(|∇f |χQ)(x). (2.9)
Here Iα, 0 < α < n, denotes the fractional integral or Riesz potentials of order α and it isdefined by
Iαf(x) =
∫Rn
f(y)
|x− y|n−α dy.
We will prove below the well–known fact that (2.9) implies (2.3). However what it is new isthat in fact these two estimates are equivalent. In fact there is no need to work with thegradient as the following characterization shows.
Lemma 2.3.1 Let α > 0, f a locally integrable function and let g be a nonnegativefunction. Then the following conditions are equivalent:
a) There exists a constant C such that for all cubes Q:
1
|Q|
∫Q
|f − fQ| ≤ C`(Q)α
|Q|
∫Q
g,
b) There exists a constant C such that for all cubes Q:
|f(x)− fQ| ≤ C Iα(gχQ)(x) a.e. x ∈ Q (2.10)
12
c) there is a positive constant C such that for every weight w and for all cubes Q:
1
|Q|
∫Q
|f − fQ|w ≤ C`(Q)α
|Q|
∫Q
g M(wχQ)
d)Let p ≥ 1, then there exists a constant C such that for all cubes Q:(1
|Q|
∫Q
|f − fQ|p)1/p
≤ C `(Q)α
(1
|Q|
∫Q
gp
)1/p
.
The key part is the classical integral representation formula of Sobolev type (2.10). Thisformula is also available in different and more general contexts thanks to the work of[FLW2] which we adapt to our situation.
Proof: a) ⇒ b) It is in this implication where we adapt the ideas from following[FLW2]. Fix Q = Q0 and let x ∈ Q and consider the unique one-way inifinite chain ofdyadic cubes Qk∞k=0, Qk+1 ⊂ Qk such that
x = ∩∞k=0Qk.
See Appendix I. For each k we let fk = 1|Qk|
∫Qk
f . Then by the Lebesgue differentiationtheorem we have a.e. x ∈ Q that
|f(x)− fQ| = | limk→∞ fk − fQ| = |∞∑
k=0
(fk − fk+1)|
≤∞∑
k=0
1
|Qk+1|
∫Qk+1
|f − fQk| ≤ 2n
∞∑k=0
1
|Qk|
∫Qk
|f − fQk|
≤ C 2n
∞∑k=0
`(Qk)α
|Qk|
∫Qk
g = C 2n
∫Q
g(y)∞∑
k=0
`(Qk)α
|Qk|χ
Qk(y) dy.
We now use that Qk ⊂ Q(x, `(Qk)). Observe that the natural estimate
`(Qk)α
|Qk|χ
Qk(y) ≤ `(Qk)
α
|Qk|χ
Q(x,`(Qk))(y) ≤ |y − x|α−nχ
Q(x,`(Qk))(y)
is not sufficient to sum up the series, but we can proceed as follows: let ε be such that0 < ε < n− α, then
∞∑k=0
`(Qk)α
|Qk|χ
Qk(y) =
∞∑k=0
c
`(Qk)n−α−ε
1
`(Qk)εχ
Qk(y) ≤ c
|y − x|n−α−ε
∞∑k=0
1
`(Qk)εχ
Qk(y),
since x, y ∈ Qk. Now pick the integer k0 such that 2k0 ≈ `(Q)|y−x| . Then
c
|y − x|n−α−ε
∞∑k=0
1
`(Qk)εχ
Qk(y) =
c
|y − x|n−α−ε
1
`(Q)ε(1 + 2ε + 22ε + · · ·+ 2k0ε) ≈ 1
|y − x|n−α
13
this shows that a.e. for x ∈ Q
|f(x)− fQ| ≤ Cε,n
∫Q
g(y)
|y − x|n−αdy.
b) ⇒ c) Indeed by assumption and Fubini’s∫Q
|f(x)− fQ|w(x)dx ≤ c
∫Q
∫Q
g(y)
|x− y|n−α dy w(x)dx =
∫Q
∫Q
w(x)
|x− y|n−α dx g(y) dy
For the inner integral we do the following: if the cube Q = Q(z, r) is centered at z andwith radius r we have for each y ∈ Q that Q ⊂ Q(y, 2r). Then if we let r0 = 2r,rk = rk−1/2, and A(y, k) = Q(y, rk) \Q(y, rk−1), k = 1, 2, · · · we have∫
Q(z,r)
w(x)
|x− y|n−α dx ≤∫
Q(y,2r)
w(x)
|x− y|n−α dx =∞∑
k=1
∫A(y,k)
w(x)
|x− y|n−α dx
≤ C∞∑
k=0
rαk
rnk
∫Q(y,rk)
w(x)dx ≤ C Mw(y)∞∑
k=1
rα
2kα= C `(Q)α Mw(y)
c) ⇒ d) The case p = 1 follows by taking w ≡ 1. The case p > 1 follows by a dualityargument combined with a good weighted estimate. This is a very fruitful idea as can beseen in the following Chapters. Indeed, we can write(∫
Q
|f − fQ|p)1/p
= sup|∫
Q
(f − fQ)h|
where the supremum is taken over all functions h such that∫
Q|h|p
′= 1. Taking one of
these h’s we have by hypothesis that
|∫
Q
(f − fQ)h| ≤∫
Q
|f − fQ||h| ≤ C `(Q)α
∫Q
g M(hχQ)
≤ C `(Q)α
(∫Q
gp
)1/p(∫Q
(M(hχQ))p′)1/p′
≤ C `(Q)α
(∫Q
gp
)1/p(∫Q
|h|p′)1/p′
= C `(Q)α
(∫Q
gp
)1/p
d) ⇒ a) Just take p = 12
14
2.4 Lp self–improving property
In this section we will show that Lemma 2.3.1 can be pushed further to get higher Lp
integrability. The methods we will use are completely different and more related to theCalderon-Zygmund theory. We will be working in a more general framework. We consider“functionals” a
a : Q → (0,∞)
Q denotes the family of all cubes from Rn. Our model example is associated to thefractional average
a(Q) =`(Q)α
|Q|
∫Q
dν, (2.11)
where 0 ≤ α and ν is a nonnegative measure. Some interesting cases are provided when themeasure ν is absolutely continous, with dν = g dx, with g ∈ A∞ as we shall see below (forinstance a polynomial weight). Motivated by the theory developed in [HK] we can considera more general case
a(Q) = `(Q)α
(ν(λQ)
|Q|
)1/p
, (2.12)
with λ ≥ 1. Observe that these functionals are not necessarily radial, i.e., a need not be ofthe form a(Q) = ϕ(|Q|) where ϕ(0,∞) → (0,∞) which under certain restrictions on ϕ haveappeared in the literature as generalizations of the Lipschitz spaces [Ja].
Now let f be a locally integrable function on Rn and let a be a functional. Assume thatf satisfies the following estimate
1
|Q|
∫Q
|f(y)− fQ| dy ≤ a(Q) (2.13)
uniformly on Q ∈ Q. Our goal is to find a condition on a such that (2.13) implies a Lq
self–improving property of the form(1
|Q|
∫Q
|f(y)− fQ|q dy
)1/q
≤ C a(Q). (2.14)
We impose the following discrete condition on the functional a relative to a locallyintegrable weight function w.
To get such an improvement we must impose a mild discrete type condition whichreflects the underlying geometry of the space. We expect in the first case that the exponentq will lie on a certain generally open range (1, r) where r = r(a) is an exponent whichdepends upon a mild condition on the functional a. To be more precise we shall assume thefollowing condition.
Definition 2.4.1 Let 0 ≤ r < ∞ and let w be a weight. We say that the functional asatisfies the (weighted) Dr condition if there exists a finite constant c such that for each
15
cube Q and any family ∆ of pairwise disjoint dyadic subcubes of Q,∑P∈∆
a(P )rw(P ) ≤ Cr a(Q)rw(Q). (2.15)
Observe that the condition is of local nature and it reminds a bit the Carlesoncondition. It is convenient to denote the best constant C by ‖a‖. Observe that ‖a‖ ≥ 1and that by Holder’s inequality, the family Dr is decreasing as r increases, that is, ifr < s, Ds ⊂ Dr. Hence for a functional a ∈ Dr we can define the optimal exponent towhich it belogs, namely
ra = supr : a ∈ Dr.
For instance if we consider the fractional functional
a(Q) =`(Q)α
|Q|
∫Q
dν,
we have that ra = nn−α
and furthermore a satisfies the (unweighted) Dra . Indeed if ∆ is anarbitrary family of disjoint cubes we have:∑
P∈∆
a(P )n
n−α |P | =∑P∈∆
(
∫P
dν(y))n
n−α ≤ (∑P∈∆
∫P
dν(y))n
n−α
≤ (
∫Q
dν(y))n
n−α = a(Q)n
n−α |Q| (2.16)
If we consider instead the functional (2.12) with λ > 1 it is possible to show (see [FPW1])that with α > 0, 1 ≤ p < n/α then a ∈ Dt for 1 < t < pn/(n− αp) and hencera = pn/(n− αp). However, it does not satisfy Dra . On the other hand this is good enoughto recover the strong type estimate in part b) of Theorem 2.4.2, but not part a), theoptimal weak type endpoint estimate. Observe that pn/(n− αp) is a more generallySobolev exponent involving the index α.
It can also be shown that for particular ν a is not in D nn−α
+ε for all ε > 0 for a generalmeasure ν. Last observation shows that the class Dr does not share in general aself–improving or openness property of the sort
Dr ⇒ Dr+ε,
which may be of interest to get optimal endpoint estimates (see Theorem 2.4.2). In somecases this openness property holds. For instance consider
a(Q) =|Q|α/n
|Q|
∫Q
w(y) dy, (2.17)
with w ∈ A∞. Then one can shows that a ∈ D nn−α
+ε where ε depends upon the A∞constant of w.
16
On the other hand side there are examples of functionals which satisfy the Dr conditionfor all r. Indeed, this the case of any nondecreasing nonradial case, namely: a(P ) ≤ a(Q),P ⊂ Q. In fact it belongs to
⋂r>1 Dr. An example of this situation is given by
a(Q) =1
|Q|
∫Q
|π(y)| dy, (2.18)
where π is a polynomial or more generality any weight satisfying the reverse Holderproperty of order infinity.
Nondecreasing functionals are special as can be seen in Appendix III. Indeed if fsatisfies (2.13) with nondecraesing a then f is at least of exponential type including theJohn and Nirenberg classical theorem. In fact it is shown in [MP2] (see also Appendix III)that the space B.M.O., or more generally the spaces defined by these nondecreasingfunctionals, can be seen as limiting of certain appropriate generalized Trudingerinequalities.
In next result we will use the following notation
‖g‖Lr,∞(Q) = supt>0
t
(|x ∈ Q : |g(x)| > t|
|Q|
)1/r
,
for the normalized Marcinkiewicz “norm”. Similarly, whenever a weight w is involved wehave
‖g‖Lr,∞(Q,w) = supt>0
t
(w(x ∈ Q : |g(x)| > t)
w(Q)
)1/r
.
Theorem 2.4.2 Let w be an A∞ weight and let a be a functional satisfying the (weighted)Dr condition (2.15) for some r > 0. Let f be a locally integrable function such that
1
|Q|
∫Q
|f − fQ| ≤ a(Q), (2.19)
for every cube Q. Thena) There exists a constant C such that for all the cubes Q in Rn
‖f − fQ‖Lr,∞(Q,w)
≤ C a(Q). (2.20)
b) Let p with 0 < p < r. Then, as a consequence of a) there exists a constant C such thatfor all the cubes Q in Rn we have(
1
w(Q)
∫Q
|f(y)− fQ|p w(y)dy
)1/p
≤ C a(Q). (2.21)
Remark 2.4.3 We emphasize that the natural exponent p = r is not attained in the stronginequality (2.21). In fact it is an open question whether the Dr condition is sufficient toassure the strong Lr result. Under certain extra condition the answer is yes as the nextcorollary shows where a very special selfimproving functional is considered.
17
Remark 2.4.4 It is not hard to see that we can replace the Lebesgue mesure by anydoubling measure µ. However, it is much more interesting the fact that we can go beyondand consider more general measures which are nondoubling. Indeed it has been first shownin [MMNO] how to work with measures that do not “see” hyperplanes parallel to the axes.In particular this is the case of mesures µ satisfying the growth condition µ(Q) ≤ C `(Q)α
for some positive α. These measures have a growing interest since they arise in the studyof the Cauchy transform along curves and its generalizations (see the recent papers [NTV][Tor]). The Ap(µ) theory of weights has been developed in [OP] with applications in thespirit of Theorem 2.4.2 and to singular integrals.
Remark 2.4.5 It is interesting to compare our situation with that in the context of theCampanato-Morrey spaces. These spaces are defined as those functions f satisfying (2.19)with the functional a(Q) = |Q|−δ, where δ is positive. It is well known that this condition isequivalent to requiring that
1
|Q|
∫Q
|f | ≤ C |Q|−δ.
for all cubes Q. This same equivalence holds when the L1 norm is replaced by the Lp norm.This means that the cancellation does not play any role in these spaces. Indeed there areexamples by Piccinini showing that these spaces do not have the self–improving propertiesas derived above.
Corollary 2.4.6 Let f and g be locally integrable functions and let g be an A∞ weight. Let0 < α < n, and suppose that f satisfies the following inequality for all cubes:
1
|Q|
∫Q
|f − fQ| ≤ C`(Q)α
|Q|
∫Q
g. (2.22)
Then with r = nn−α
,
‖f − fQ‖Lr,∞(Q)
≤ C`(Q)α
|Q|
∫Q
g. (2.23)
If furthermore the function g also satisfies the A∞ condition then(1
|Q|
∫Q
|f − fQ|r)1/r
≤ C`(Q)α
|Q|
∫Q
g. (2.24)
Proof of Theorem 2.4.2: Part b) follows from Kolmogorov’s inequality: let 0 < q < rwe have that for nonnegative g(
1
w(Q)
∫Q
g(x)q wdx
)1/q
≤ (r
r − q)1/q ‖g‖Lr,∞(Q,w). (2.25)
See [GCRdF] p. 485 for instance. We just need to apply part a) to this inequality tog = |f − fQ|. To prove part a) we fix a cube Q. Observe that we may assume that fQ = 0.Hence we have to prove that
trw(x ∈ Q : |f(x)| > t)
w(Q)≤ Cr a(Q)r,
18
with C independent of Q, and t.
We will try to get an appropriate good–λ inequality (2.28). Good–λ inequalities is animportant tool discovered by D. L. Burkholder and R. F. Gundy in [BG] (see also [Tor])
Now, for each t > 0, we let Ωt = x ∈ Q : Mf(x) > t where M will denote in this proofthe dyadic Hardy–Littlewood maximal function relative to Q (See Appendix I). Then bythe Lebesgue differentiation theorem
x ∈ Q : |f(x)| > t ⊂ Ωt.
We will asume first that t > a(Q). Hence
t > a(Q) ≥ 1
|Q|
∫Q
|f |
and we can consider the Calderon–Zygmund covering lemma of f relative to Q for thesevalues of t. This yields a collection of dyadic subcubes of Q, Qi, maximal with respect toinclusion, satisfying Ωt = ∪iQi and
t <1
|Qi|
∫Qi
|f | ≤ 2n t (2.26)
for each i.
Now let q > 1 a big enough number that will be chosen in a moment. Since Ωq t ⊂ Ωt,we have
w(Ωq t) = w(Ωq t ∩ Ωt) =∑
i
w(x ∈ Qi : Mf(x) > q t) =
=∑
i
w(x ∈ Qi : M(fχQi
)(x) > qt)
where the last equation follows by the maximality of each of the cubes Qi. Indeed, for anyof these i’s we have
Mf(x) = max supP :x∈P∈D(Q)
P⊆Qi
1
|P |
∫P
|f |, supP :x∈P∈D(Q)
P⊃Qi
1
|P |
∫P
|f |
= supP :x∈P∈D(Q)
P⊂Qi
1
|P |
∫P
|f | = M(fχQi)(x),
since by the maximality of the cubes Qi when P is dyadic (relative to Q) containing Qi
then1
|P |
∫P
|f | ≤ t.
On the other hand for arbitrary x,
|f(x)| ≤ |f(x)− fQi|+ |fQi
| ≤ |f(x)− fQi|+ 1
|Qi|
∫Qi
|f | ≤ |f(x)− fQi|+ 2n t
19
and then for q > 2n
w(Ωq t) ≤∑
i
w(EQi),
where EQi= x ∈ Qi : M((f − fQi
)χQi)(x) > (q − 2n)t.
Let ε > 0 to be chosen in a moment. We split the family Qi in two:(i) i ∈ I if
1
|Qi|
∫Qi
|f − fQi| < ε t
or (ii) i ∈ II if1
|Qi|
∫Qi
|f − fQi| ≥ ε t
Thenw(Ωq t) ≤
∑i
w(EQi) ≤
∑i∈I
w(EQi) +
∑i∈II
w(EQi) = I + II.
For I we use that M is of weak type (1, 1) (with constant one) to control the unweightedpart:
|EQi| ≤ 1
(q − 2n)t
1
|Qi|
∫Qi
|f − fQi| ≤ ε
(q − 2n)|Qi|.
Recall that the A∞ condition is characterized by
w(E)
w(Q)≤ c (
|E||Q|
)ρ,
for some c and ρ and independent of Q and E ⊂ Q. Therefore we get for each i ∈ I
w(EQi) ≤ C εδw(Qi)
for some positive δ. Then
I ≤ C ερ∑i∈I
w(Qi) ≤ C ερ w(∪iQi) = C ερ w(Ωt).
For II we use the hypothesis on a (2.15)
II ≤∑i∈II
w(Qi) ≤∑i∈II
(1
t ε |Qi|
∫Qi
|f − fQi|)r
w(Qi)
≤ 1
tr εr
∑i∈II
a(Qi)rw(Qi) ≤
‖a‖r
tr εra(Q)rw(Q).
Combining all these estimates we have that for q > 2n, t > a(Q)
(qt)r w(Ωqt)
w(Q)≤ ερ qr
(q − 2n)tr
w(Ωt)
w(Q)+‖a‖rqr
εra(Q)r. (2.27)
However, if we choose ε ≤ ‖a‖ we have the same inequality holds for t ≤ a(Q). Inconclusion we have the following inequality:
20
For all q > 2n, t > 0 and 0 < ε ≤ ‖a‖ we have
(qt)r w(Ωqt)
w(Q)≤ ερ qr
(q − 2n)tr
w(Ωt)
w(Q)+‖a‖rqr
εra(Q)r. (2.28)
To conclude we use a standard good–λ method. For N > 0 we let
ϕ(N) = sup0<t<N
trw(Ωt)
w(Q),
which is finite since is bounded by N r. Since ϕ is increasing (2.28) clearly implies
ϕ(N) ≤ ϕ(Nq) ≤ ερ qr
q − 2nϕ(N) +
‖a‖rqr
εra(Q)r.
We now conclude by chosing ε small enough such that ερ qr
(q−2n)< 1, and letting N →∞.
2
We conclude this section by pointing out that as a corollary of the proof of abovetheorem we can deduce a key estimate relating a function and the action of the sharpmaximal operator M# on it. This is perhaps the best way of showing the relationshipbetween a function and its smoothness. Recall that the sharp maximal operator M# is anoperator introduced by C. Fefferman and E. Stein and it is defined as follows:
M#f(x) = supx∈Q
infc
1
|Q|
∫Q
|f(y)− c| dy
which is equivalent to
supx∈Q
1
|Q|
∫Q
|f(y)− fQ| dy ≈ supx∈Q
1
|Q|2
∫Q
∫Q
|f(y)− f(x)| dydx
where as usual the supremum is taken over all the cubes containing x.
In order to derive a global result we need to consider spaces on which we can make theglobal Calderon–Zygmund covering lemma(cf. Appendix I):
CZ = f ∈ L1loc(R
n) : fQ → 0 when Q ↑ Rn
It is easy to see that ∪p≥1Lp ⊂ CZ.
Theorem 2.4.7 Let 0 < p < ∞ and let w be a A∞ weight. Then
a) Suppose that f be an integrable function on a cube Q of Rn. Then there exists aconstant C independent of f such that∫
Q
|f − fQ|p wdx ≤ C [w]pA∞
∫Q
(M#
Qf)p wdx (2.29)
b) Suppose that f ∈ CZ. Then there exists a constant C independent of f such that∫Rn
|f(x)|p w(x)dx ≤ C [w]pA∞
∫Rn
(M#f(x))p w(x)dx. (2.30)
21
2.5 Proof of the Poincare-Sobolev inequalities
The purpose of this section is to prove the Poincare-Sobolev that are in presented Section2.2. The first part of next result is classical (see [EG]). However, our point of view isdifferent and can be extended to other situations such as the case of subelliptic operators[FPW1] [MP1]. The second part is not that well known.
Recall that p∗ = pnn−p
denotes the Sobolev exponent of p.
22
Theorem 2.5.1 Let 1 ≤ p < n.
a) Let f be Lipschitz function, then(1
|Q|
∫Q
|f − fQ|p∗)1/p∗
≤ C `(Q)
(1
|Q|
∫Q
|∇f |p)1/p
. (2.31)
b) Let w ∈ Ap, and let f be a Lipschitz function, then there exists a positive constant δdepending upon the Ap constant of w such that(
1
w(Q)
∫Q
|f − fQ|p(n′+δ)w
)1/p(n′+δ)
≤ C `(Q)
(1
w(Q)
∫Q
|∇f |pw)1/p
(2.32)
Proof/Discussion: a) Define the following functional
a(Q, f) = `(Q)
(1
|Q|
∫Q
|∇f |p)1/p
,
as a function of both Q and f . Observe that a satisfies the (unweighted) Dp∗ condition (seethe calculation 2.16 from previous section) with constant equal one, in particular isuniform on f . Hence, we may apply Theorem 2.4.2 and get the weak type estimate
‖f − fQ‖Lp∗,∞(Q,w)
≤ C a(Q, f),
with constant C independent of Q and f . What we need to do is to pass from this weaktype inequality to the corresponding strong type estimate. To accomplish this, we use someideas taken from R. Long and F. Nie in [LN]. We remit [FPW1] for details andgeneralizations 1. The key point is to have a gradient type functional on the right handside. We will illustrate this in the proof of a (global) sharp two weight isoperimetricinequality Theorem 5.1.1 from Chapter 5.
b) We start with the (1, 1) Poincare inequality
1
|Q|
∫Q
|f − fQ| ≤ C`(Q)
|Q|
∫Q
|∇f |.
By the Ap condition using part b) of Theorem 1.0.6 we have that
1
|Q|
∫Q
|f − fQ| ≤ C `(Q)
(1
w(Q)
∫Q
|∇f |pw)1/p
= a(Q, f).
1P. Koskela informed me that Long-Nie method is somehow hidden in [Maz], although not in the waypresented here; also it has been rediscovered by some other authors [BCLSC]
23
Also by part b) of Theorem 1.0.6 we have that(|P ||Q|
)p
≤ Cw(P )
w(Q)
for any subcube P ⊂ Q or what is the same
`(P )
`(Q)≤ c
(w(P )
w(Q)
)1/np
.
By the openness property of the Ap condition (8.7) there is small ε such that w ∈ Ap−ε.Then w satisfies
`(P )
`(Q)≤ c
(w(P )
w(Q)
)1/n(p−ε)
.
Now let q = λp with λ > 1 such that
1
n(p− ε)=
1
p− 1
q=
1
pλ′
with λ =(
p−εp
n)′
= λ = n′ + δ for some δ > 0. Then we have that
`(P )
`(Q)
(w(P )
w(Q)
)1/q
≤ c
(w(P )
w(Q)
)1/p
.
Using this property it is easy to check that a(Q, f) satisfies the Dq condition with aconstant independent of Q and f . As in part a) we can pass from the weak type estimatewhich follows from Theorem 2.4.2 to obtain the desired result(
1
w(Q)
∫Q
|f − fQ|qw)1/q
≤ C a(Q, f). (2.33)
2
MEJORAR ESTE TROZO QUE SE INCORPORA DESPUES:
These results are not knew, we know give applications to derive new results. Inparticular we improve this theorem at the endpoint p = 1 in two ways. We also improve theresult given in p. 308 [HKM] since we are able to reach both endpoints: p = 1 and for theSobolev exponent by p∗ = pn
n−p, 1 ≤ p < n. The result is even new in the euclidean setting.
Theorem 2.5.2 Also let w be a weight function such that w ∈ A1. Then, we have(1
w(Q)
∫Q
|f − fQ|p∗wdx
)1/p∗
≤ C `(Q)
(1
w(Q)
∫Q
|∇f |p wdx
)1/p
(2.34)
with C independent of f , Q.
24
Observe that this result sharpen the previous theorem when p = 1.
Proof: We follow here [LP2] where the result is given in a more general setting and forhigher order derivatives.
Let w ∈ A1 and we begin again by using the (1, 1) Poincare inequality
1
|Q|
∫Q
|f − fQ| ≤ C `(Q)1
|Q|
∫Q
|∇f |
Now, since A1 ⊂ Ap we have that the right hand side is less or equal than
`(Q)
(1
w(Q)
∫Q
|∇f |p w
)1/p
= b(Q, f) = b(Q)
It will enough to check the Dp∗ condition and for which it is enough to check
`(Q1)
`(Q2)
(w(Q1)
w(Q2)
)1/p∗
≤ C
(w(Q1)
w(Q2)
)1/p
for all cubes Q1, Q2 such that Q1 ⊂ Q2. But this is equivalent to(`(Q1)
`(Q2)
)n
≤ Cw(Q1)
w(Q2)
since 1n
= 1p− 1
p∗. Now by the doubling property it will be enough to check
|Q1||Q2|
≤ Cw(Q1)
w(Q2)
and this follows from the fact that w ∈ A1.
25
Chapter 3
Rearrangements
The main purpose of this chapter to improve the main result in the previous chapter bymeans of a new technique based on rearrangements. This method yields not only yieldsstronger but also provide new direct proofs avoiding the use the classical good-λ inequalityof Burkholder and Gundy as in described in the prevoius chapter.
The natural question that we consider is the following: Is it possible to relax the L1
norm in (2.19) to derive the same self-improving property? In this Chapter, we study thisissue and weaken the hypothesis to obtain such self-improving property. More precisely, wewill show that the L1 norm can be replaced by any Lq quasi-norm with 0 < q < 1 (SeeCorollaries 3.0.4 and 3.0.5 below). Even further, we will show that the left hand side of(2.19) can be replaced by a weaker expression which is defined in terms of non-increasingrearrangements (see below for the precise definition).
Our main result is the following.
Theorem 3.0.3 Let ω be an A∞ weight and let a be a functional satisfying the Dr(ω)condition (2.15) for some r > 0. Suppose that f is a measurable function such that for anycube Q
infα
((f − α)χ
Q
)∗(λ|Q|
)≤ cλ a(Q), 0 < λ < 1. (3.1)
Then the same consequences of Theorem 2.4.2 holds namely, there exists a constant c suchthat for any cube Q
‖f − fQ‖Lr,∞(Q,ω)
≤ c a(Q). (3.2)
Recall that the non-increasing rearrangement f ∗ of a measurable function f is defined by
f ∗(t) = infλ > 0 : µf (λ) < t
, t > 0,
where
µf (λ) = |x ∈ Rn : |f(x)| > λ|, λ > 0,
is the distribution function of f .
26
An important fact is that ∫Rn
|f |p dx =
∫ ∞
0
f ∗(t)p dt,
or more generally if E is any measurable function:∫E
|f |p dx =
∫ |E|
0
f ∗(t)p dt,
If f is only a measurable function and if Q is a cube then we define the followingquantity:
(fχQ)∗(λ|Q|), 0 < λ ≤ 1.
We can think of this expression as another way of averaging the function. Indeed forany p > 0, and 0 < λ ≤ 1
(fχQ)∗(λ|Q|) ≤
(1
λ|Q|
∫Q
|f |p dx
)1/p
.
Hence we have the following.
Corollary 3.0.4 Let f , a and ω as in Theorem 3.0.3. Suppose that for some 0 < p < 1and for any cube Q
infα
( 1
|Q|
∫Q
|f − α|pdx)1/p
≤ a(Q)
Then there is a geometric constant c such that for any cube Q
‖f − fQ‖Lr,∞(Q,ω)
≤ c a(Q).
Using that for ay measurable function f , and any 0 < δ < 1(1
|Q|
∫Q
|f |p dx
)1/p
≤ cp ‖f‖L1,∞(Q).
Corollary 3.0.5 If as above we suppose that for any cube Q
‖f − fQ‖L1,∞(Q)
≤ a(Q),
then there is a geometric constant c such that for any cube Q
‖f − fQ‖Lr,∞(Q,ω)
≤ c a(Q).
27
Our proof of Theorem 3.0.3 is essentially based on the following two ingredients. Thefirst one is a relation between rearrangements and oscillations of a function f . Thistechnique goes back to [BDVS], and it was further developed in [L1, L2]. We also use someideas from the works [J, S]. The second ingredient is an appropriate covering lemma ofCalderon-Zygmund type. In the context of the standard Euclidean space Rn with doublingmeasure such covering lemmas can be obtained simply by application of the usualCalderon-Zygmund lemma to characteristic functions (see, e.g., [BDVS, L1]. In the case ofRn with non-doubling measure a covering lemma of such type has been recently proved in[MMNO]. Our covering lemma, in the context of spaces of homogeneous type, is presentedin Section 3 below.
The main result of this paper can be applied to improve the results of [MP2, OP]. In[MP2], an exponential self-improving property was established assuming (2.19) with afunctional a satisfying the so-called Tp condition, stronger than the Dr condition. InChapter 9, we will improve this result by relaxing the initial assumption (2.19) as inTheorem 3.0.3. In [OP], a non-homogeneous variant of Theorem 2.4.2 was proved in thecontext of Rn with any (non-doubling) measure. We will show that this result can be alsoproved under similar relaxed assumptions.
We also obtain an analogue of the main results of [L1, L2] concerning the so-called localsharp maximal function, in the context of spaces of homogeneous type.
3.1 Rearrangements
We mention here some simple properties of rearrangements. It follows easily from thedefinition that
µx ∈ Rn : |f(x)| > f ∗µ(t)
≤ t and µ
x ∈ Rn : |f(x)| ≥ f ∗µ(t)
≥ t.
Next, for any 0 < λ < 1,
(f + g)∗µ(t) ≤ f ∗µ(λt) + g∗µ((1− λ)t). (3.3)
For any measurable set E with finite positive measure µ we have,
infE|f | ≤ (fχ
E)∗µ(µ(E)). (3.4)
Finally we recall that the (weighted normalized) weak Lr norm is defined by
‖g‖Lr,∞(E,ω)
= supλ>0
λ
(ω(x ∈ E : |g(x)| > λ)
ω(E)
)1/r
or, equivalently,
‖g‖Lr,∞(E,ω)
= sup0<t≤ω(E)
(gχE)∗ω(t)
( t
ω(E)
)1/r
.
We will need the following two propositions about local rearrangements.
28
Proposition 3.1.1 For any measurable function f , any weight ω, and each measurable setE ⊂ Rn with 0 < ω(E) < ∞, and for any 0 < λ < 1:(
fχE
)∗ω
(λω(E)
)≤ 2 inf
c
((f − c)χ
E
)∗ω
(λω(E)
)+(fχ
E
)∗ω
((1− λ)ω(E)
).
Proof:
Essentially the same was proved in [L1] in a less general context. The proof of thisproposition follows the same lines, and we shall recall it only for the sake of completeness.
Using (3.3) and (3.4), for any constant α we get
|α| ≤ infx∈E
(|f(x)− α|+ |f(x)|
)≤
((|f − α|+ |f |)χE
)∗ω
(ω(E)
)≤
((f − α)χE
)∗ω
(λω(E)
)+(fχE
)∗ω
((1− λ)ω(E)
).
From this and from the estimate(fχE
)∗ω
(λω(E)
)≤((f − α)χE
)∗ω
(λω(E)
)+ |α|
we get immediately the required inequality.2
Proposition 3.1.2 Let ω ∈ A∞. For any cube Q and any measurable function f , and forany 0 < λ < 1: (
fχQ
)∗ω
(λω(Q)
)≤(fχ
Q
)∗((λ
c)1/ε|Q|
),
where c and ε are A∞-constants of ω.
Proof: It follows from the definitions of A∞ and of the rearrangement that for anyδ > 0,
ωx ∈ Q : |f(x)| >
(fχQ
)∗µ
((λ
c)1/ε|Q|
)+ δ
≤ c
(µx ∈ Q : |f(x)| >
(fχQ
)∗µ
((λ
c)1/ε|Q|
)+ δ
|Q|
)ε
ω(Q) < λω(Q),
which is equivalent to that(fχQ
)∗ω
(λω(Q)
)≤(fχQ
)∗µ
((λ
c)1/ε|Q|
)+ δ.
Letting δ → 0 yields the required inequality.2
A relevant class of weights is given by the A∞ class of Muckenhoupt: if there arepositive constants c, ε such that
ω(E) ≤ c
(|E||Q|
)ε
ω(Q) (3.5)
for every cube Q and every measurable set E ⊂ Q.
29
3.2 A basic covering lemma
In this section we prove a covering lemma of Calderon-Zygmund which is nothing but aspecial case of the usual Calderon-Zygmund classical lemma for a cube.
We assume that ω is dyadic doubling:
ω(Q′) ≤ dω(Q)
where Q′ is the ancestor of Q.
Lemma 3.2.1 Let ω be a doubling weight with doubling dyadic constant d ≥ 1. Then forany 0 < λ < 1, any cube Q and any set E ⊂ Q with ω(E) ≤ λω(Q), there is a countablefamily of pairwise disjoint dyadic cubes Qii from Q with E ⊂ ∪iQi, except for a ω-nullset such that
(i) ω(Qi ∩ E) ≤ dλω(Qi);
(ii) ω(Qi ∩ E) ≥ λω(Qi);
(iii) ω(E) ≤ d λ∑
i ω(Qi).
Proof:
Items (i) and (ii) follow from the weighted version of the Calderon-Zygmund for cubes(see 7.0.5): If
1
ω(Q)
∫Q
|f |wdx < λ
there exists mutually disjoint dyadic cubes Qii (dyadic with respect to Q) contained inQ such that
λ1
ω(Q)
∫Q
|f |ωdx ≤ dλ
and |f(x)| ≤ λ for almost x ∈ Q \ ∪iQi.
Letting f = χE
we get (i) and (ii). Observe that since 0 < λ < 1, w(Q \ ∪iQi) = 0 andhence E ⊂ ∪iQi, except for a ω-null set. Part (iii) is immediate.
2
3.3 Proof of the main Theorem
Adapting ideas from [L1, L2] we prove in this section the main theorem of this ChapterTheorem 3.0.3.
Proof:
30
The goal is to prove the following estimate: there are constants c′, λ with0 < λ < c′ < ∞ such that for any cube Q and for any 0 < t < ω(Q)
(fχQ)∗ω(t) ≤ c′ a(Q)
(ω(Q)
t
) 1r
+ (fχQ)∗(λ|Q|). (3.6)
First we will prove this estimate for small values of t.
The key result in the proof is the following estimate for the rearrangement of f : thereexists a constant λ′, 0 < λ′ < 1 such that for any 0 < t < λ′
2ω(Q)
(fχQ)∗ω(t) ≤ c a(Q)
(ω(Q)
t
) 1r
+ (fχQ)∗ω(2t). (3.7)
Throughout the proof d is doubling constant and c and ε are the constants from theA∞-condition (3.5) of ω.
For any t we consider the set E = Et
E = x ∈ Q : |f(x)| ≥ (fχQ)∗ω(t).
We can assume that (fχQ)∗ω(t) > (fχ
Q)∗ω(2t), since otherwise (3.7) is trivial. Observe that
t ≤ ω(E) ≤ 2t.
Set λ′ = 14d
and λ = (λ′
c)1/ε and let 0 < t < λ′
2ω(Q). Then we have that
ω(E) ≤ λ′ω(Q),
and we can apply Lemma 3.2.1 to the set E and number λ′. Hence, we get a countablefamily of pairwise disjoint balls Qi satisfying (i)-(iii) of Lemma 3.2.1. Thus combiningitem (ii) and Proposition 3.1.1 we have,
(fχQ)∗ω(t) ≤ inf
x∈E|f(x)| ≤ inf
iinf
x∈E∩Qi
|f(x)|
≤ infi
(fχ
E∩Qi
)∗ω
(ω(E ∩Qi)
)≤ inf
i
(fχ
Qi
)∗ω
(λ′ω(Qi)
)≤ inf
i
[2 inf
α
((f − α)χ
Qi
)∗ω
(λ′ω(Qi)
)+
(fχ
Qi
)∗ω
((1− λ′)ω(Qi)
)].
Applying item (i) together with Proposition 3.1.2 we obtain for each i
infα
((f − α)χ
Qi
)∗ω
(λ′ω(Qi)
)≤ inf
α
((f − α)χ
Qi
)∗(λ|Qi|
)≤ cλa(Qi),
where in the last step we have used the basic initial hypothesis (3.1) to the ball Qi. Hence,
(fχQ)∗ω(t) ≤ inf
i
(2cλa(Qi) +
(fχ
Qi
)∗ω
((1− λ′)ω(Qi)
)).
31
Since the balls Qi are dyadic, disjoint and contained in Q we have by the Dr condition(2.15) ∑
i
a(Qi)rω(Qi) ≤ ‖a‖r a(Q)rω(Q).
Let M a big enough number to be chosen soon. We split the family of cubes as follows:i ∈ I if
a(Qi) ≤ M1/r‖a‖ a(Q)(ω(Q)
t
) 1r
(3.8)
and i ∈ II if it satisfies the opposite inequality. The proof below shows that the family I isnot empty. We claim that ∑
i∈I
ω(Qi) ≥ (1
dλ′− 1
M)t, (3.9)
where we choose M > dλ′. Indeed, by the Dr condition∑i∈II
ω(Qi) ≤t
M,
and hence (3.9) follows from the item (iii) of Lemma 3.2.1 and our choice of λ′.
Therefore we have
(fχQ)∗ω(t) ≤ 2cλM
1/r‖a‖ a(Q)(ω(Q)
t
) 1r
+ infi∈I
(fχ
Qi
)∗ω
((1− λ′)ω(Qi)
).
Set now for i ∈ I
Ei =x ∈ Qi : |f(x)| ≥
(fχ
Qi
)∗ω
((1− λ′)ω(Qi)
).
By (3.9),
ω(∪i∈IEi) =∑i∈I
ω(Ei) ≥ (1− λ′)∑i∈I
ω(Qi) ≥ (1− λ′)(1
dλ′− 1
M)t > 2t,
if we choose for instance M > 2dλ′ and 0 < λ′ < 11+4d
.
Thus,
infi∈I
(fχ
Qi
)∗ω
((1− λ′)ω(Qi)
)≤ inf
i∈Iinf
x∈Ei
|f(x)| = infx∈∪i∈IEi
|f(x)|
≤ (fχQ)∗ω(ω(∪i∈IEi)
)≤ (fχ
Q)∗ω(2t),
and hence (3.7) is proved for the values 0 < t < λ′
2ω(Q).
Now, for one of these values of t, there is k = 1, · · · , such that
λ′
2k+1ω(Q) ≤ t <
λ′
2kω(Q),
32
and iterating (3.7) yields
(fχQ)∗ω(t) ≤ ca(Q)
(ω(Q)
t
) 1r
k∑j=0
(2−j)1r + (fχ
Q)∗ω(2k+1ω(Q))
≤ c′ a(Q)(ω(Q)
t
) 1r
+ (fχQ)∗ω(λ′ω(Q)). (3.10)
Next, we observe that (3.10) trivially holds for t ≥ λ′ω(Q), and hence this formula holdsfor any 0 < t < ω(Q).
Applying Proposition 3.1.2 to the second term on the right-hand side of (3.10), weobtain (3.6).
To finish the proof we observe that if α is any number, f − α also satisfies the initialassumption (3.1) and hence (3.6) holds for f − α as well:
((f − α)χQ)∗ω(t) ≤ c′ a(Q)
(ω(Q)
t
) 1r
+ ((f − α)χQ)∗(λ′|Q|).
Recalling that
‖g‖Lr,∞(Q,ω) = sup0<t≤ω(Q)
(gχQ)∗ω(t)( t
ω(Q)
)1/r
,
we have by multiplying(
tω(Q)
) 1r
and taking the supremum over 0 < t < ω(Q) that
‖f − α‖Lr,∞(Q,ω)
≤ c′ a(Q) + ((f − α)χQ)∗(λ′|Q|).
Taking the infimum over all α and using again (3.1) combined with the Dr condition wehave
infα‖f − α‖
Lr,∞(Q,ω)≤ c′ a(Q) + inf
α((f − α)χ
Q)∗(λ′|Q|)
≤ c′ a(Q) + cλa(Q)
≤ c′′ a(Q).
The proof is now complete.2
33
Chapter 4
Orlicz maximal functions
4.1 Orlicz spaces
To derive some sharp weighted estimates in Chapters (5) and (6) we need to considercertain maximal functions defined in terms of Orlicz spaces. In this section we recall somebasic definitions and facts about Orlicz spaces, referring to [RR] and [KJF] for a completeaccount.
A function B : [0,∞) → [0,∞) is called a Young function if it is continuous, convex,increasing and satisfies B(0) = 0 and B(t) →∞ as t →∞. It follows that B(t)/t isincreasing. Some relevant examples are B(t) = tp(log(1 + t))α, 1 ≤ p < ∞, α > 0 orB(t) = Φ(t) = exp tp − 1, 1 ≤ p < ∞.
For Orlicz norms we are usually only concerned about the behavior of Young functionsfor t large. Given two functions B and C, we write B(t) ≈ C(t) if B(t)/C(t) is boundedand bounded below for t ≥ c > 0.
By definition, the Orlicz space LB consists of all measurable functions f such that∫Rn
B(|f |λ
) dµ < ∞
for some positive λ. The space LB is a Banach function space with the Luxemburg normdefined by
‖f‖B = infλ > 0 :
∫Rn
B(|f |λ
) dµ ≤ 1.
Each Young function B has an associated complementary Young function B satisfying
t ≤ B−1(t)B−1(t) ≤ 2 t (4.1)
for all t > 0. The function B is called the conjugate of B, and the space LB is called theconjugate space of LB. For example, if B(t) = tp for 1 < p < ∞, thenB(t) = tp
′, p′ = p/(p− 1), and the conjugate space of Lp(µ) is Lp′(µ). Another example
34
that will be used frecuently is B(t) ≈ tp(log t)−1−ε, 1 < p < ∞, ε > 0 with complementaryfunction B(t) ≈ tp
′(log t)(p′−1)(1+ε) (cf. [O]).
A very important property of Orlicz (or, more generally, Banach function spaces) is thegeneralized Holder inequality ∫
Rn
|fg| dµ ≤ ‖f‖B‖g‖B. (4.2)
We will always assume that the Young function B satisfies the doubling conditionB(2t) ≤ C B(t). If B is doubling then B′(t) ≈ B(t)/t almost everywhere.
LB is a rearrangement–invariant space with fundamental function (see [BS]) given by
ϕB(t) = ϕ
BLB(µ)(t) =
1
B−1(1t). (4.3)
In particular if E is a measurable subset of X, then∥∥χE
∥∥B,µ
=1
B−1( 1µ(E)
). (4.4)
4.2 Orlicz maximal functions
We begin the section by defining a class of Young functions that will play a key role inChapters 5 and 6. We need to define an appropriate maximal in terms of Orlicz norm.Given a Young function B and a cube Q, the mean Luxembourg norm of a measurablefunction f on Q is defined by
‖f‖B,Q = inf
λ > 0 :
1
|Q|
∫Q
B
(|f |λ
)dx ≤ 1
. (4.5)
When B(t) = t log(1 + t)p−1+δ, this norm is also denoted by ‖ · ‖L(log L)p−1+δ ,Q.
Given a Young function B, define the maximal function
MBf(x) = sup
Q3x‖f‖B,Q.
Definition 4.2.1 Let 1 ≤ p < ∞. A nonnegative function B(t), t > 0, satisfies the Bp
condition if there is a constant c > 0 such that∫ ∞
c
B(t)
tpdt
t≈∫ ∞
c
(tp′
B(t)
)p−1dt
t< ∞.
Simple examples of functions which satisfy Bp are tp−δ and tp(log(1 + t))−1−δ, both whenδ > 0.
The interest of this class of Young functions can be seen from the following lemma takenfrom [P1]. It is used in this paper to derive sharp two–weight norm inequalities for theHardy-Littlewood maximal function.
35
Lemma 4.2.2 Let 1 < p < ∞. Then the following statements are equivalent.
a) B is a Young doubling function satisfying the Bp condition;
b) there is a constant c such that∫Rn
MBf(y)p dy ≤ c
∫Rn
f(y)p dy (4.6)
for all nonnegative, locally integrable functions f ;
c) there is a constant c such that∫Rn
Mf(y)p 1
[MB(u)(y)]p
dy ≤ c
∫Rn
f(y)p
u(y)pdy, (4.7)
for all nonnegative functions f and u ;
Proof: To prove b) we will use the classical approach (cf. for instance [GCRdF] Ch.2.). Observe that we may assume that f ≥ 0. For t > 0 we letΩt = x ∈ Rn : M
Bf(x) > t. We claim that
|Ωt| ≤ C
∫Rn
B
(f(y)
t
)dy (4.8)
Assume the claim for the moment. Using (4.8) together with the fact that MB
is boundedon L∞ we write f as f = f1 + f2, where f1(x) = f(x) if f(x) > t
2, and f1(x) = 0 otherwise.
Then MBf(x) ≤ M
Bf1(x) + M
Bf2(x) ≤ M
Bf1(x) + t
2. and it follows immediately that
|Ωt| ≤ C
∫x:f(x)>t/2
B
(f(x)
t
)dx
Hence, combining this with the change of variable t = f(y)s
yield∫Rn
MBf(y)p dy = p
∫ ∞
0
tp|y ∈ Rn : MBf(y) > t| dt
t≤
≤ C
∫ ∞
0
tp∫y∈Rn:f(y)>t/2
B
(f(y)
t
)dy
dt
t= C
∫Rn
∫ 2f(y)
0
tpB
(f(y)
t
)dt
tdy =
= C
∫Rn
f(y)p dy
∫ ∞
1/2
B(t)
tpdt
t= C
∫Rn
f(y)p dy,
since B ∈ Bp.
36
The proof of the claim follows easily from the Vitali covering lemma 1.0.4. Indeed,assuming that Ωt is not empty let K be any compact contained in Ωt. Let x ∈ K, then bydefintion of the maximal function there is a cube Q = Qx containg x such that
‖f‖B,Q
> t, (4.9)
or what is the same1
|Q|
∫Q
B(f
t) > 1.
Then K ⊂⋃
x∈K Qx and by compactness we can extract a finite family of cubes F = Qsuch that K ⊂
⋃Q∈F Q and where each cube satisfies (4.9). Then by Vitali’s lemma we can
extract a pairwise disjoint cubes QjMj=1 such that
⋃Q∈F ⊂
⋃Mj=1 5Qj. Then by definition
of ‖f‖B,Q
|K| ≤ C
M∑j=1
|Qj| ≤ C
M∑j=1
∫Qj
B(f
t) ≤ C
∫Rn
B(f
t).
This proves that a) implies b).
Let us assume that c) holds. Observe that (4.7) is equivalent to∫Rn
M(fg)(y)p 1
[MB(g)(y)]p
dy ≤ c
∫Rn
f(y)p dy,
for all nonnegative functions f , g. Then c) follows immediately from (4.6) after anapplication of the generalized Holder’s inequality (4.2)
M(fg)(y) ≤ MBf(y)M
Bg(y) y ∈ Rn.
To prove that c) implies a) we test (4.7) with f = u = χQ(0,1)
where Q(0, 1) is the cube
centered at the origin and with radius 1:∫Rn
Mf(y)p 1
[MB(f)(y)]p
dy ≤ C. (4.10)
On the other hand a computation using (4.4) shows that for large x that
MB(f)(x) ≈ 1
B−1( 1
|x|n )
and we can continue (4.10) with
C ≥∫
Rn
Mf(y)p 1
[MB(f)(y)]p
dy ≥ C
∫|y|>c
1
|y|np
1
B−1( 1
|y|n )pdy =
= C
∫ ∞
c
1
rnp
1
B−1( 1rn )p
rn dr
r≈∫ ∞
c
B(t)
tpdt
t.
37
This shows that c) ⇒ a).2
We conclude the Chapter by proving a weighted inequality “dual” to the classicalFefferman–Stein inequality∫
Rn
Mf(y)p w(y)dy ≤ c
∫Rn
f(y)p Mw(y)dy.
The main interest follows from the fact that its natural “dual” inequality, thinking as if Mwere linear and selfadjoint, would be∫
Rn
Mf(y)p′ Mw(y)1−p′dy ≤ c
∫Rn
f(y)p′ w(y)1−p′dy.
However this estimate is false in general as can be shown by taking f = w positive andintegrable. However we the following sharp result.
Corollary 4.2.3∫Rn
Mf(y)p [M [p′]+1(w)(y)]1−pdy ≤ c
∫Rn
|f(y)|pw(y)1−pdy. (4.11)
The proof is a consequence of (4.7), choosing B such that B ≈ tp′(log t)p′−1+δ, combined
with Lemma 5.4.1 below. See [P1] for details.
38
Chapter 5
Fractional integral operators
We have already discussed in Chapter 2 the relationship of a function f and its gradientvia the fractional integral. Indeed, recall that if f is a sufficiently smooth function, thenthere is a universal constant C such that for each cube and for each x ∈ Q
|f(x)− fQ| ≤ C I1(|∇f |χQ)(x) (5.1)
If we further assume that f has compact support then this inequality clearly shows that foreach x ∈ Rn
|f(x)| ≤ C I1(|∇f |)(x). (5.2)
Recall that Iα, 0 < α < n, denotes the fractional integral of order α on Rn or Rieszpotentials defined by
Iαf(x) =
∫Rn
f(y)
|x− y|n−α dy.
5.1 Weak implies strong
The main point of this section is to illustrate the fact that a weak type inequality with agradient (or gradient like) operator “on the right hand side” implies the correspondingstrong type inequality. This is never the case in the standard interpolation theory. As wementioned in Section 2.5 the basic ideas are taken from [LN].
Theorem 5.1.1 (weighted isoperimetric inequality) Let n > 1 and let µ be anynonnegative measure. Then there is a constant C such that for any Lipschitz function fwith compact support(∫
Rn
|f(x)|n′dµ
)1/n′
≤ C
∫Rn
|∇f(x)| (Mµ(x))1/n′dx. (5.3)
This result and generalizations can be found in [FPW2].
39
Proof: The key point is to prove a corresponding weak type estimate which in this caseis
‖f‖Ln′,∞(µ) ≤ C
∫Rn
|∇f(x)| (Mµ(x))1/n′dx, (5.4)
where we recall that
‖f‖Lq,∞(w) = supt>0
t w(x ∈ Rn : |f(x)| > t)1/q,
is the standard notation for the Marcinkiewicz “norm”. When q > 1, ‖‖Lq,∞(w) is equivalentto a norm, and therefore using (5.2) and Minkowski’s inequality we obtain
‖f‖Ln′,∞(w) ≤ C
∫Rn
|∇f(x)| ‖| · −x|1−n‖Ln′,∞(w)dx.
Now observe that
‖| · −x|1−n‖Ln′,∞(µ) = sup λµ(y ∈ Rn : |y − x|n > λ)1/n′
=
(sup λµ(y ∈ Rn : |y − x|n <
1
λ))1/n′
=
(supt>0
1
tn
∫B(x,t)
µ
)1/n′
≈ (Mµ(x))1/n′ ,
from which we get the desired inequality (5.5).
Now that we have the weak type estimate we proceed as follows. Given a nonnegativefunction g and a positive number λ, we denote by τλ(g) the truncation of g
τλ(g) = ming, 2 λ −ming, λ =
0 if g(x) ≤ λg(x)− λ if λ ≤ g(x) < 2 λλ if g(x) ≥ 2 λ
A very nice fact about Lispchitz functions is that they are preseved by absolute values andby truncations. Hence if f ∈ Λ(1) ⇒ τλ(|f |) ∈ Λ(1), λ > 0. Now, if we define for eachinteger k, Fk = x ∈ Rn : 2k < |f(x)| ≤ 2k+1, we have∫
Rn
|f(x)|n′ w(x)dx =∞∑
k=−∞
∫x∈Rn:2k+1<|f(x)|≤2k+2
|f(x)|n′ w(x)dx
≈∞∑
k=−∞
2kn′w(Fk+1)
Now, observe that for x ∈ Fk+1, 2k = τ2k(|f |)(x). Also observe that the support of thegradient of τ2k(|f |) equals Fk. Then by the weak type estimate(∫
Rn
|f(x)|n′ dµ(x)
)1/n′
≤ C∞∑
k=−∞
2kµ(Fk+1)1/n′
40
≤ C∞∑
k=−∞
2kµ(x ∈ Rn : τ2k(|f |)(x) > 2k−1)1/n′ ≤ C
∞∑k=−∞
∫Rn
|∇τ2k(|f |)(x)|(Mµ(x))1/n′dx
= C∞∑
k=−∞
∫Fk
|∇f(x)|(Mµ(x))1/n′dx = C
∫Rn
|∇f(x)|(Mµ(x))1/n′dx
since the sets Fk are pairwise disjoint.2
Remark 5.1.2 In fact the following estimate holds
‖f‖Ln′,1(µ) ≤ C
∫Rn
|∇f(x)| (Mµ(x))1/n′dx, (5.5)
To prove sharp estimates for fractional integrals are much more difficult as we will seenow.
Theorem 5.1.3 Suppose that 0 < α < n and that v is a nonnegative measurable functionon Rn.
Let 1 < p < ∞. Then there exists a constant C = Cn,p such that∫Rn
|Iαf(x)|p v(x) dx ≤ C
∫Rn
|f(x)|p Mαp(M[p]v)(x) dx. (5.6)
This result shows the Mαp(M[p]v) is the operator that “controls” fractional integrals in
the Lp norm. It should be compared with (1.4).
This result can be found in [P4] and can be seen as a sharpening version of theChang-Wilson-Wolff estimate derived in [CWW] which in turn is sharper than thecondition of Fefferman–Phong related to the the theory of Schrodinger operators (see [P4]for more details).
Proof: By the duality between the spaces Lp(Rn) and Lp′(Rn), it is enough to showthat there exists a positive constant C such that
(I) =
∫Rn
Iαf(x) v(x)1p g(x)dx ≤ C
[∫Rn
f(x)p Mαp(M[p]v)(x)dx
]1/p [∫Rn
g(x)p′ dx
]1/p′
.
Furthermore, by density, we can assume that both functions f and g are nonnegative,bounded and have compact support.
5.2 Step 1: Discretization of the potential operator
In this section we will show a procedure to discretize the operator which is interesting onits own right. In fact the method can be used to give a different proof which avoids
41
good–lambda inequalities of a well–known result of Muckenhoupt and Wheeden [MW]. Seealso [PW] for further results related to subelliptic vector fields.
Observe first that Iαf(x) < ∞ for all x ∈ Rn so that the manipulations that we areabout to start make sense.
We discretize the operator Iαf as follows:
Iαf(x) =∑ν∈Z
∫2−ν−1<|x− y|≤2−ν
f(y)
|x− y|n−α dy
=∑ν∈Z
∑Q∈D
`(Q)=2−ν
χQ(x)
∫`(Q)/2<|x− y|≤`(Q)
f(y)
|x− y|n−α dy
≤ C∑Q∈D
|Q|α/n
|Q|
∫|y − x|≤`(Q)
f(y) dy χQ(x).
D denotes the family of dyadic cubes on Rn. Since the ball B(x, `(Q)) is contained in thecube 3Q when x ∈ Q we have
Iαf(x) ≤∑Q∈D
|Q|α/n
|Q|
∫3Q
f(y) dy χQ(x),
and then ∫Rn
Iαf(x) g(x)dµ(x) ≤ c∑Q∈D
|Q|α/n
|Q|
∫3Q
f(y) dy
∫Q
g(x) dµ(x). (5.7)
5.3 Step 2: Replacing dyadic cubes by dyadic
Calderon–Zygmund cubes
Observe that the sum in equation (5.7) runs over all the dyadic cubes from Rn and this istoo much to get a good control of, therefore the next task is to replace the family D by asubclass that satisfies certain good properties. This family is formed by consideringappropriate Calderon-Zygmund cubes at each level ak where a is a large enough constant.To be more precise we have the following lemma.
Lemma 5.3.1 Let a > 2n, then there exists a family of cubes Qk,j and a family ofpairwise disjoint subsets Ek,j, with Ek,j ⊂ Qk,j such that for each k, j,
|Qk,j| <1
1− 2n
a
|Ek,j| (5.8)
and such that∫Rn
Iαf(x) g(x)dµ(x) ≤ C∑k,j
|Qk,j|α/n
|Qk,j|
∫3Qk,j
f(y) dy1
|Qk,j|
∫Qk,j
g(y) dµ(y) |Ek,j|. (5.9)
42
Proof: Recall that a > 2n and consider for each integer k the set
Dk = x ∈ Rn : Md(g dµ)(x) > ak,
where Md is the usual dyadic Hardy–Littlewood maximal operator. Then we apply theCalderon–Zygmund covering lemma(see Appendix I) it follows that if Dk is not emptythere exists some dyadic cube Q with
ak <1
|Q|
∫Q
g(y) dµ(y), (5.10)
then Q is contained in one dyadic cube satisfying this condition and maximal with respectto inclusion. Thus for each k we can write Dk = ∪jQk,j where the cubes Qk,j arenonoverlapping, they satisfy (5.10), and by maximality we also get
ak <1
|Qk,j|
∫Qk,j
g(y) dµ(y) ≤ 2n ak. (5.11)
We also need the following property. For all integers k, j we let Ek,j = Qk,j \Dk+1.Then Ek,j is a disjoint family of sets which satisfy
|Qk,j ∩Dk+1| <2n
a|Qk,j|, (5.12)
and therefore
|Qk,j| <1
1− 2n
a
|Ek,j|. (5.13)
Indeed, by standard properties of the dyadic cubes we can compute what portion of Qk,j iscovered by Dk+1 as in [GCRdF] p. 398:
|Qk,j ∩Dk+1||Qk,j|
=∑
i
|Qk,j ∩Qk+1,i||Qk,j|
=∑
i:Qk+1,i⊂Qk,j
|Qk+1,i||Qk,j|
<ak
ak+1
∑i:Qk+1,i⊂Qk,j
1
|Qk,j|
∫Qk+1,i
g(y) dµ(y)
≤ ak
ak+1
1
|Qk,j|
∫Qk,j∩∪iQk+1,i
g(y) dµ(y) ≤ ak
ak+1
1
|Qk,j|
∫Qk,j
g(y) dµ(y) ≤ 2n
a,
by the right hand side of (5.11). This gives (5.12).
We continue with the proof of the Lemma by adapting some ideas from [SW]. For eachinteger k we let
Ck = Q ∈ D : ak <1
|Q|
∫Q
g(y) dµ(y) ≤ ak+1.
43
Every dyadic cube Q for which∫
Qg(y) dµ(y) 6= 0 belongs to exactly one Ck. Furthermore,
if Q ∈ Ck it follows that Q ⊂ Qk,j for some j. Then∫Rn
Iαf(x) g(x)dµ(x)
≤∑k∈Z
∑Q∈Ck
|Q|α/n
|Q|
∫3Q
f(y) dy
∫Q
g(y) dµ(y)
≤ a∑k∈Z
ak∑j∈Z
∑Q∈Ck
Q⊂Qk,j
|Q|α/n
∫3Q
f(y) dy
≤ C∑k,j
1
|Qk,j|
∫Qk,j
g(y) dµ(y) |Qk,j|α/n
∫3Qk,j
f(y) dy
The last inequality follows from the fact that for each dyadic cube P∑Q∈DQ⊂P
|Q|α/n
∫3Q
f(y) dy ≤ C |P |α/n
∫3P
f(y) dy
Indeed, if `(P ) = 2−ν0∑Q∈DQ⊂P
|Q|α/n
∫3Q
f(y) dy = C∑ν≥ν0
∑Q∈D:Q⊂P
`(Q)=2−ν
2−ν α
∫3Q
f(y) dy
= C∑ν≥ν0
2−ν α∑
Q∈D:Q⊂P
`(Q)=2−ν
∫3Q
f(y) dy
≤ C |P |α/n∑
Q∈D:Q⊂P
`(Q)=2−ν
∫3Q
f(y) dy ≤ C |P |α/n
∫3P
f(y) dy
since the overlap is finite. Hence ∫Rn
Iαf(x) g(x) dµ(x)
≤ C∑k,j
|3Qk,j|α/n
|3Qk,j|
∫3Qk,j
f(y) dy1
|Qk,j|
∫Qk,j
g(y) dµ(y) |Qk,j|
≤ C∑k,j
|3Qk,j|α/n
|3Qk,j|
∫3Qk,j
f(y) dy1
|Qk,j|
∫Qk,j
g(y) dµ(y) |Ek,j|,
by (5.13).2
44
5.4 Step 3: Patching the pieces together
Now, by Lemma 5.3.1 with dµ(x) = v(x)1/p dx we have that
(I) ≤ C∑k,j
|3Qk,j|α/n
|3Qk,j|
∫3Qk,j
f(y) dy1
|Qk,j|
∫Qk,j
v(y)1/pg(y) dy |Ek,j|.
Next we apply the generalized Holder’s inequality (4.2) with respect to the associated thespaces Lp(log L)(p−1)(1+ε) and Lp′(log L)−1−ε, ε > 0, (see [O] for instance). After that weapply Holder’s inequality at the discrete level with exponents p and p′. We can follow theestimate with
C∑k,j
|3Qk,j|α/n
|3Qk,j|
∫3Qk,j
f(y) dy∥∥v1/p
∥∥Lp(log L)(p−1)(1+ε)
|Ek,j|1/p‖g‖Lp′ (log L)−1−ε
|Ek,j|1/p′
≤ C
[∑k,j
(|Qk,j|α/n
|3Qk,j|
∫3Qk,j
f(y) dy
)p ∥∥v1/p∥∥p
Lp(log L)(p−1)(1+ε),
|Ek,j|
]1/p
×
×
[∑k,j
‖g‖p′
Lp′ (log L)−1−ε|Ek,j|
]1/p′
.
Now, if ε = [p]p−1
− 1 > 0, then∥∥v1/p∥∥p
Lp(log L)(p−1)(1+ε)= ‖v‖
L(log L)[p](5.14)
Now, we need the following Lemma
Lemma 5.4.1 If k = 1, 2, 3, · · · , then there exists a constant C = Cn such that for allbounded functions f with support contained in Q
‖f‖L(log L)k,Q
≤ C
|Q|
∫Q
Mkf(y) dy. (5.15)
Proof: By definition of Luxemburg norm (5.15) is equivalent to showing that for somepositive constant c
1
|Q|
∫Q
w
c λQ
logk(1 +w
c λQ
) ≤ 1.
where we denote λQ = 1|Q|
∫Q
Mkw.
We first recall the following Wiener estimate which can be seen as a “reverse” weak type(1, 1) estimate (cf. (4.8 with B(t)=t) which can be deduced from Appendix 7. For eacht > fQ
1
t
∫x∈Q:f(x)>t
f(x)dx ≤ 2n |x ∈ Q : MdQ(f)(x) > t| (5.16)
45
To prove the lemma we use induction. We prove (5.15) with k=1 and let f = w2nλQ
.
Now, using the formula∫X
ϕ(f) dν =
∫ ∞
0
ϕ′(t) ν(x ∈ X : f(x) > t) dt
we have
1
|Q|
∫Q
f(x) log(1 + f(x))dx =1
|Q|
∫ ∞
0
1
1 + tf(x ∈ Q : f(x) > t) dt
=1
|Q|
∫ 1
0
1
1 + tf(x ∈ Q : f(x) > t) dt +
1
|Q|
∫ ∞
1
1
1 + tf(x ∈ Q : f(x) > t) dt = I + II
Now, recalling that λQ = 1|Q|
∫Q
Mw we observe that I ≤ 1 since
f(Q)
|Q|=
1
|Q|
∫Q
wc|Q|
∫Q
Mw≤ 1
by the of M and since c ≥ 1. For II we use estimate (5.16):
II ≤ 1
|Q|
∫ ∞
1
1
1 + tf(x ∈ Q : f(x) > t) dt ≤ 2n
|Q|
∫ ∞
1
t
1 + t|x ∈ Q : Mf(x) > t| dt
≤ 2n
|Q|
∫ ∞
0
|x ∈ Q : Mf(x) > t| dt =2n
|Q|
∫Q
Mf(x) dx =2n
|Q|
∫Q
Mw(x)dx =1
2nλQ
= 1.
We now assume that the estimate holds for certain k. Then with f = wc λQ
and
λQ = 1|Q|
∫Q
Mk+1w.
1
|Q|
∫Q
f logk+1(1 + f)dx =k + 1
|Q|
∫ ∞
0
logk(1 + t)
1 + tf(x ∈ Q : f(x) > t) dt
=k + 1
|Q|
(∫ 1
0
+
∫ ∞
1
)logk(1 + t)
1 + tf(x ∈ Q : f(x) > t) dt = I + II
Again by the Lebesgue differentiation theorem we can chose c = ck ≥ 1 such that I ≤ 1.
Let ϕ(t) = t logk(1 + t). Then ϕ′(t) = logk(1 + t) + kt logk−1(1+t)1+t
and again for II we useestimate (5.16):
II ≤ k + 1
|Q|
∫ ∞
1
logk(1 + t)
1 + tf(x ∈ Q : f(x) > t) dt
≤ 2n(1 + k)
|Q|
∫ ∞
1
t logk(1 + t)
1 + t|x ∈ Q : Mf(x) > t| dt
46
≤ 2n(1 + k)
|Q|
∫ ∞
1
ϕ′(t)|x ∈ B : Mf(x) > t| dt
≤ 2n(1 + k)
|Q|
∫Q
Mf(x) logk(1 + Mf(x))dx ≤ 1.
In the last inequality we have used the induction hypothesis since f = wc λQ
and since
λQ = 1|Q|
∫Q
Mk+1w ≥ 1|Q|
∫Q
Mkw, for appropriate constant c. This completes the proof ofthe lemma and hence the proof of Theorem 5.1.3.
2
Then using (5.14) and (5.15) we get
(I) ≤ C
[∑k,j
(|Qk,j|α/n
|3Qk,j|
∫3Qk,j
f(y) dy
)p1
|Qk,j|
∫Qk,j
M [p]v(y) dy |Ek,j|
]1/p
×
×
[∑k,j
‖g‖p′
Lp′ (log L)−1−ε|Ek,j|
]1/p′
≤ C
[∑k,j
∫Ek,j
(1
|3Qk,j|
∫3Qk,j
f(y) Mαp(M[p]v)(y)1/p dy
)p
dx
]1/p
×
×
[∑k,j
∫Ek,j
MLp′ (log L)−1−ε
g(x)p′ dx
]1/p′
≤ C
[∑k,j
∫Ek,j
M(f Mαp(M[p]v)1/p)(x)p dx
]1/p
×[∫
Rn
MLp′ (log L)−1−ε
g(x)p′ dx
]1/p′
≤ C
[∫Rn
M(f Mαp(M[p]v)1/p)(x)p dx
]1/p
×[∫
Rn
g(x)p′ dx
]1/p′
≈[∫
Rn
f(x)p Mαp(M[p]v))(x) dx
]1/p
×[∫
Rn
g(x)p′ dx
]1/p′
.
We used first that the sets in the family Ek,j are pairwise disjoint. Also it is used that M isa bounded operator on Lp(Rn) and that the maximal operator M
Lp′ (log L)−1−εis bounded on
Lp′(Rn) by Lemma 4.2.2 since the Young function B(t) ≈ tp′(log t)−1−ε satisfies the Bp′
condition.
47
Chapter 6
Singular Integral Operators
6.1 Introduction: Singular Integral Operators and
smoothness of functions
What follows can be seen as a motivation to the study of weighted inequalities for singularintegrals.
We fix a pair of weights u, v and consider the weighted inequality(∫Rn
|f |qu)1/q
≤ C
(∫Rn
|∇f |pv)1/p
. (6.1)
It is easy to see that (6.1) is implied by(∫Rn
|I1f |qu)1/q
≤ C
(∫Rn
|f |pv)1/p
(6.2)
since for appropriate functions (see the introduction to Chapter 5) we have
|f(x)| ≤ C I1(|∇f |)(x).
Inequality (6.2) should be, in principle, be more flexible to be treated due to all theinformation we have about fractional integrals. The question is whether these inequalitiesare equivalent. The answer is no in general. Indeed, assuming that (6.1) holds we test itwith f = I1(g) were g is any smooth function with compact support. Then we are led toconsider ∇(I1(g)) and it is not hard to show that
∇(I1(f)) = c Rg
where R = Rjgnj=1 is the classical vector–valued Riesz transform where each component
is defined by
Rjg(x) = limε→0
∫|x−y|>ε
xj − yj
|x− y|n+1g(y) dy
48
Therefore we are led to study of weighted norm inequalities for the Riesz Transforms,namely (∫
Rn
|Rf |pv)1/p
≤ C
(∫Rn
|f |pv)1/p
.
It is possible to show that a necessary and sufficient condition is the Ap condition. In factwe will show in Corollary 6.2.4 that the Ap condition is also a sufficient condition for awider class of operators. To simplify the presentation we can think of the case of regularconvolution singular integrals. These operators are defined for smooth functions f withcompact support by
Tf(x) = limε→0
∫|x−y|>ε
K(x− y)f(y) dy.
We will assume that:a) T is bounded on L2(Rn) or what is equivalent the Fourier transform of K, as a
principal value distribution, is a bounded functionb) There is a constant c such that for every x different from zero:
|K(x)| ≤ c
|x|n
c) K satisfies regularity condition: there is a constant C such that for |x| > 2|y|
|K(x− y)−K(x)| ≤ C|y||x|n+1
.
This condition is satisfied whenever
|∇K(x)| ≤ c
|x|n+1
when x is different from zero.
We remit the reader to the books [Ch], [Jo] or [D] for further information andgeneralizations about singular integrals.
6.2 Weighted Lp estimates
The basic tool we are going to use is a modification of the sharp maximal operator M# ofC. Fefferman and E. Stein: for δ > 0 we define the δ–sharp maximal operator M#
δ as
M#δ (f) = M#(|f |δ)1/δ.
Recall that M# is defined by
M#f(x) = supx∈Q
infc
1
|Q|
∫Q
|f(y)− c| dy ≈ supx∈Q
1
|Q|
∫Q
|f(y)− fQ| dy
Our first result concerns operators that are controlled in some sense by theHardy–Litllewood maximal function. To be more precise we give the following definition.
49
Definition 6.2.1 We say that T an operator satisfies the condition (D) if for any0 < δ < 1 there is a constant c such that for all f ,
M#δ (Tf)(x) ≤ cMf(x), (D).
In the most classical situation this pointwise estimate is derived for Calderon-Zygmundoperators with δ = 1 and Mf is replaced by M(f r)1/r for all r > 1. However this estimateis not sharp enough for the results we will present below.
Condition (D) is used in [AP] to derive a different proof of Coifman’s theorem [C]relating the (weighted) Lp norm of Calderon-Zygmund operators and (weighted) Lp normof the Hardy–Littlewood maximal function. See Theorem 6.2.3 below.
Some examples of operators satisfying condition (D) are:
• Calderon-Zygmund operators These operators are generalization of the regularsingular integral operators as defined above. See [Jo] [Ch] or [D] for appropriatedefinitions. They are the model examples of operators satisfying condition (D). Thiswas observed in [AP] (cf Lemma 6.2.2 below).
• weakly strongly singular integral operators These operators were considered byC. Fefferman in [F].
• pseudo-differential operators. To be more precise, the pseudo-differentialoperators satisfying condition (D) are those that belong to the Hormander class (seeHormander [H]).
• oscillatory integral operators These operators wrere by introduced by Phong andStein [PS].
The proof of the last three cases can be found in [AHP].
We illustrate how to verify condition (D) in the case of regular singular integraloperators.
Lemma 6.2.2 Let T be any regular singular integral operator or more generally anyCalderon-Zygmund operator. Let 0 < δ < 1. Then, there exists a positive constant c = cδ
such that,M#
δ (Tf)(x) ≤ cMf(x)
for all smooth functions f .
Proof: Let Q = Q(x, r) be an arbitrary cube. Since 0 < δ < 1 implies||α|δ − |β|δ| ≤ |α− β|δ for α, β ∈ R it is enough to show for some constant c = cQ thatthere exists C = Cδ > 0 such that,(
1
|Q|
∫Q
|Tf(y)− c|δ dy
)1/δ
≤ C Mf(x). (6.3)
50
Let f = f1 + f2, where f1 = f χ2Q. If we pick c = (T (f2))Q , we can estimate the lefthand side of (6.3) by a multiple of(
1
|Q|
∫Q
|T (f1)(y)|δ dy
)1/δ
+
(1
|Q|
∫Q
|T (f2)− (T (f2))Q|δ dy
)1/δ
= I + II.
Since T : L1(Rn) → L1,∞(Rn) ([Ch] or [Jo]) and 0 < δ < 1, Kolmogorov’s inequality(2.25) yields
I ≤ C
|Q|
∫Rn
|f1(y)| dy =C
|2Q|
∫2Q
|f(y)| dy ≤ C M(f)(x).
To take care of II we use the regularity of the kernel. Indeed, by Jensen’s inequality andFubini’s theorem we have
II ≤ 1
|Q|
∫Q
|T (f2)(y)− (T (f2))Q| dy ≤
≤ 1
|Q|2
∫Q
∫Q
∫Rn−2Q
|k(y − w)− k(z − w)||f(w)|dw dz dy ≤
≤ 1
|Q|2
∫Q
∫Q
∞∑j=1
∫2jr≤|w − x|<2j+1r
|y − z||x− w|n+1 |f(w)|dw dz dy ≤
≤ C∞∑
j=1
2−j
(2j+1r)n
∫2j+1Q
|f(w)| dw ≤ C∞∑
j=1
2−jMf(x) = C Mf(x).
2
The following a priori estimate for the operator has many applications as we will seebelow.
Theorem 6.2.3 Let T be a operator satisfying condition (D). Also we let 0 < p < ∞ andsuppose that w ∈ A∞. Then the following inequalities hold∫
Rn
|Tf(x)|p w(x)dx ≤ C
∫Rn
Mf(x)p w(x)dx, (6.4)
and
supλ>0
λpw(x ∈ Rn : |Tf(x)| > λ) ≤ C [w]pA∞ supλ>0
λpw(x ∈ Rn : Mf(x) > λ), (6.5)
whenever f is a function for which the left hand side is finite, where the constant C alwaysdepend upon the A∞ constant of w.
51
Proof: By the Lebesgue differentiation theorem and Fefferman-Stein inequality (2.30)∫Rn
|Tf |p w ≤∫
Rn
M(|Tf |δ)p/δ w ≤ C
∫Rn
M#(|Tf |δ)p/δ w
= C
∫Rn
M#δ (Tf)p w ≤ C
∫Rn
(Mf)p w
since T satisfies the condition (D).
The other inequality is proved in the same way.2
Corollary 6.2.4 Let T be a operator satisfying condition (D).
a) Let 1 < p < ∞ and suppose that w ∈ Ap. Then, there exists a constant C such that∫Rn
|Tf(x)|p w(x)dx ≤ C
∫Rn
|f(x)|p w(x)dx. (6.6)
b) Suppose that w ∈ A1. Then, there exists a constant C such that for all functions λ
w(x ∈ Rn : |Tf(x)| > λ) ≤ C
λ
∫Rn
|f(x)|w(x)dx (6.7)
Observe that we have imposed appropriate conditions on the weights to get the desiredestimates. In next theorem we don’t impose any condition on the weight w.
Corollary 6.2.5 Let T be a linear operator such that its conjugate operator satisfiescondition (D). Let 1 < p < ∞ and let w be a weight. Then∫
Rn
|Tf(x)|p w(x)dx ≤ C
∫Rn
|f(x)|p M [p]+1w(x)dx. (6.8)
Furthermore:
a) the inequality does not hold if M [p]+1 is replaced by the smaller operator M [p].
b) the operator M [p]+1 can be replaced by the smaller and sharp operators
ML(log L)p−1+ε
ε > 0, being false the result for the case ε = 0.
Remark 6.2.6 We remark that the unweighted theory or the one weight Ap theorycoincides for both the Hardy–Littlewood maximal function and singular integrals. Howeverthis is not the case when considering the two weight theory. This phenomenon is wellreflected in last corollary, result that can be found in [P2].
52
It should be compared estimate (6.8) with the one related to the maximal function (1.4)observaing that by the Lebesgue differentiation theorem we always have
w ≤ Mw ≤ M2w ≤ M3w ≤ · · ·
We just point out in applications most of the operators (such as those listed above areeither selfadjoint or the adjoint is a constant multiple of the operator).
The proof of the corollary is just a combination of Theorem 6.2.4, Corollary 4.2.3 andthe easy part of the factorization theorem. Indeed, by duality (6.8) is equivalent to∫
Rn
|Tf(x)|p′[M [p]+1(w)(x)]1−p′dx ≤ c
∫Rn
|f(x)|p′w(x)1−p′dx.
We claim that [Mf(x)]1−q, q > 1, in a A∞. In fact We claim that [Mf(x)]1−q ∈ Aq+ε,ε > 0, but is not in Aq. Indeed, we write
(Mf(x))1−q =[(Mf(x))
q−1p−1
]1−p
which by Lemma 1.0.7 and the easy part of the factorization theorem yields the claim.Therefore by Theorem 6.2.4 and Corollary 4.2.3 we have∫
Rn
|Tf(x)|p′ [M [p]+1(w)(x)]1−p′dx ≤ c
∫Rn
(Mf(x))p′ [M [p]+1(w)(x)]1−p′dx
≤ c
∫Rn
|f(x)|p′ w(x)1−p′dx.
In the case p = 1 we have the following result from [P2]. We will use the followingYoung function Aε(t) = t log(1 + t)ε, ε > 0.
Theorem 6.2.7 Let T be a linear operator such that its conjugate operator satisfiescondition (D). Then for any weight w, ε > 0 and function f ,
w(x ∈ Rn : |Tf(x)| > t) ≤ Cε
t
∫Rn
|f |ML(log L)εw dx. (6.9)
We conjecture that this result must hold for ε = 0 but we don’t know how to prove it. Thisresult must also be compared with (1.3).
To conclude this section we just mention that (6.9) is the key inequality in some recentwork [CP1] where some sharp two weight, weak-type norm inequalities for singular integraloperators are derived. We just state the main result.
Theorem 6.2.8 Let T be a Calderon-Zygmund operator. Given a pair of weights (u, v)and p, 1 < p < ∞, suppose that for some ε > 0 and K > 0 and for all cubes Q,
‖u‖L(log L)p−1+ε,Q
(1
|Q|
∫Q
v1−p′ dx
)p−1
≤ K. (6.10)
53
Then for every t > 0 and f ∈ Lp(v),
u(x ∈ Rn : |Tf(x)| > t) ≤ C
tp
∫Rn
|f |p vdx. (6.11)
Condition (6.10) is stronger than the Ap conditions for two weights, namely(1
|Q|
∫Q
u(y) dy
)(1
|Q|
∫Q
v(y)1−p′ dy
)p−1
≤ K
which is a neccesary and sufficient condition for the corresponding two weight problem forthe Hardy–Littlewood maximal function, namely
u(x ∈ Rn : Mf(x) > t) ≤ C
tp
∫Rn
|f |p vdx.
The condition of the theorem is sharp. However, for other operators such as fractionalintegrals or commutators, we are not able to derive similar estimates using correspondingsharp conditions as we would expect. See [CP2].
6.3 Banach function spaces
We have seen in the previous section the relevance of Coifman’s theorem (6.4). We will seein this section that it can also be applied to other situations, for instance when estimatingoperators within the context of Banach function spaces (BFS). These are interestinggeneralizations of the Lebesgue spaces as well other spaces such as Lorentz and Orliczspaces. We refer to [BS] for more information about the subject.
Next result provides a different proof of the classical theorem of Boyd (see [BS] p. 154).Boyd’s proof only works for the one dimensional Hilbert transform. Our method is moreflexible and allows to consider any operator satisfying condition (D). The key points are:
a) a good a priori weighted inequality (1, 1) for the operator; to be precise we needestimate (6.4) with p = 1 but only for any weight w in the A1 class.
b) We need to adapt “Rubio de Francia’s algorithm” to the context of BFS.
c) To adapt this algorithm we need to use Lorentz-Shimogaki’s theorem (valid this timeon Rn) which characterizes the rearrengement invariant BFS for which theHardy–Littlewood maximal is bounded (see [BS] p. 154).
Rearrangement invariant BFS are special cases of BFS. The basic property is that if uand v are equidistributed and u belongs to a BFS then v belongs to the space and theirnorm coincide.
Theorem 6.3.1 Let T be a operator satisfying condition (D) and let X be a BFS such thatthe Hardy–Littlewood maximal function is bounded on X ′, the associated space to X, then
‖Tf‖X ≤ c ‖Mf‖X .
54
Therefore, if the Hardy–Littlewood maximal function is also bounded on X then T isbounded on X.
Corollary 6.3.2 Let T be a operator satisfying condition (D) and let X be arearrangement invariant BFS such that the Boyd indices αX and αX satisfy
0 < αX ≤ αX < 1.
Then T is bounded on X.
Proof of the Theorem:. The ‖Tf‖X can be writen as
‖Tf‖X = sup |∫
Rn
Tf g|
where the supremum is taken over all functions g ∈ X ′. Fix one of these g we have∫Rn
|Tf | |g|
We now adapt Rubio de Francia’s algorithm to this context: consider
G =∞∑
k=0
Mk(g)
(2A)k
where Mk is the operator M iterated k times and A is the norm of M as bounded operatoron X ′. It is immediate to seee that:
a) g ≤ G
b) ‖G‖X′ ≤ 2 ‖g‖X′
c) G ∈ A1, in fact MG ≤ 2A G
In particular since G ∈ A∞ we can apply Theorem 6.4∫Rn
|Tf | |g| ≤∫
Rn
|Tf |G ≤ C
∫Rn
Mf G ≤ C ‖Mf‖X‖G‖X′ ≤ C ‖Mf‖X‖g‖X′ .
Then, taking the supremum over all g ∈ X ′ we deduce the theorem.2
The corollary follows directly from Lorentz-Shimogaki’s characterization of therearrengement invariant BFS for which the Hardy–Littlewood maximal is bounded as canbe found in [BS].
55
6.4 Commutators: singular integrals with higher
degree of singularity
The purpose of this section is to illustrate that the basic ideas developed in the previoussections of this chapter can be used to treat other operators of interest such as commutatorof singular integral operators with BMO functions. We want to make emphasize that themethod is sharp since the results obtained show that these commutators share a highersingularity as compared with Calderon–Zymund operators. This is not reflected in theknown results from the literature.
Recall that commutators are defined by
Tmb f(x) =
∫(b(x)− b(y))mK(x− y)f(y) dy,
where m = 0, 1, 2, · · · and K is a Calderon–Zygmund kernel. Observe that T 0b is the
singular integral operator. T 1b = [b, T ] is the classical commutator introduced by R.
Coifman, R. Rochberg and G. Weiss in [CRW] where they showed that it is a boundedoperator on Lp(Rn), 1 < p < ∞, when b is a BMO. They also showed that BMO is also anecesary condition when T is the vector–valued Riesz transform.
It is shown in [P3] the following pointwise estimate which can be seen as a version ofcondition (D) for commutators.
Lemma 6.4.1 Let b ∈ BMO, and let 0 < δ < ε < 1 Then, there exists C = Cδ,ε > 0 suchthat,
M#δ (Tm
b f)(x) ≤ Cm−1∑j=0
‖b‖jBMO Mε(T
jb f)(x) + ‖b‖m
BMO Mm+1f(x) (6.12)
for all smooth functions f , where Mε(f) = M(|f |ε)1/ε.
Using this lemma we can give a version for commutators of Coifman’s theorem 6.4. See[P5].
Theorem 6.4.2 Let 0 < p < ∞ and let w ∈ A∞ and b ∈ BMO. Then, there exists aconstant C depending upon the A∞ constant of w, such that∫
Rn
|Tmb f(x)|p w(x)dx ≤ C ‖b‖mp
BMO
∫Rn
Mm+1f(x)p w(x)dx, (6.13)
for any function f with finite left hand side. Furthermore, the inequality does not hold ifMm+1 is replaced by the smaller operator Mm.
As above we can give can give the following version of Boyd’s theorem for commutators.
56
Theorem 6.4.3 Let Tmb be a commutator operator as above with b ∈ BMO. As in
Theorem 6.3.1 we let X be a BFS such that the Hardy–Littlewood maximal function isbounded on X ′. Then
‖Tmb f‖X ≤ c ‖Mm+1f‖X .
Therefore, if the Hardy–Littlewood maximal function is also bounded on X then Tmb is
bounded on X.
In particular, if X is any Rearrangement Invariant Function Space such that0 < αX ≤ αX < 1 we have
Tmb : X → X.
Another application of this lemma was derived in [P3] where it is shown that thecommutators [b, T ] are not in general of weak type (1, 1) but they satisfy the following“L log L” type estimate:
Theorem 6.4.4 Let b ∈ BMO and let Φm(t) = t logm(e + t). Then there exists a constantC > 0 depending upon the BMO constant of b such that for all t > 0
|y ∈ Rn : |Tmb f(y)| > t| ≤ C
∫Rn
Φm(|f(y)|
t) dy, (6.14)
for all bounded functions f with compact support.
A different proof of this result which allows to include non A∞ weights was given in [PP]:
w(x ∈ Rn : |[b, T ]f(x)| > λ) ≤ C
∫Rn
Φ(|f(y)|
t) M
L(log L)1+ε(w)(x)dx. (6.15)
Comparing these results with similar estimates for Calderon-Zygmund operators, theyshow that these commutators carry a higher degree of singularity.
These results have been further exploited in [PT] to the case of “multilinear”commutators of the form
T~b(f)(x) =
∫Rn
[m∏
j=1
(bj(x)− bj(y))
]K(x, y)f(y)dy,
where K is a Calderon-Zygmund kernel, with vector symbol ~b = (b1, ··, bm).
57
Chapter 7
Appendix I: The Calderon–Zygmundcovering lemma
In this Appendix we recall the definition and basic properties of the (dyadic)Calderon–Zygmund cubes which is also the basis of the Calderon–Zygmund decomposition.
First we recall how to build the family D of (left open) dyadic cubes on Rn. For anyinteger k we consider the lattice Λk = 2−kZn. The mesh Dk is the collection of all left opencubes determined by Λk. The cubes in Dk are congruent, each have sides length 2k andvolume 2kn. The mesh Dk is a disjoint cover of Rn. It is also a refinement of Dk−1, becauseevery cube Q in Dk is contained in exactly one cube Q′ ∈ Dk−1 (the “ancestor” of Q). As amatter of fact Q is one of the 2n congruent cubes obtained by bisecting the sides ofQ′ ∈ Dk−1.
We will denote by
D =⋃k∈Z
Dk
as the family of dyadic cubes of Rn. Dyadic cubes are very useful for constructions ofdisjoint covers. Here there are some very useful properties of the dyadic cubes which canbe verified without any difficulty.
a) Every two dyadic cubes are either disjoint or one of them contains the other.
b) Every family F ⊂M of dyadic cubes contains a disjoint subfamily F ′ ⊂ F such that
∪Q∈FQ = ∪Q∈F ′Q
c) Each point x ∈ Rn belongs to one and only one cube in Dk, k ∈ Z,. Therefore xgenerates a unique two-way inifinite chain Qk∞k=−∞, Qk+1 ⊂ Qk such that
x = ∩∞k=0Qk.
In a similar way, given a cube Q we can construct its family of dyadic cubes D(Q).Indeed, by bisecting each side of the cube Q we obtain 2n “descendent” from Q which is
58
the first generation D1; then we keep dividing in this way each descendent. ThenD(Q) =
⋃∞k=0 Dk is the family of dyadic cubes relative to Q. They share the same
properties as the dyadic cubes D. In particular each x ∈ Q generates a unique one-wayinifinite chain Qk∞k=0, Qk+1 ⊂ Qk, such that x = ∩∞k=0Qk.
This construction is intimately to dyadic maximal function. In the local case is thedefined by
Md
Qf(x) = M
Qf(x) = sup
P∈D(Q)
x∈P
1
|P |
∫P
|f(y)| dy.
The global dyadic maximal function can be defined in a very similar way by replacingD(Q) by D.
For a nonnegative function f in L1(Q) we define
Ωt = x ∈ Q : MQf(x) > t.
The we have the following local version of the Calderon–Zygmund covering lemma.
Lemma 7.0.5 (local Calderon–Zygmund covering lemma) Let f be a nonegativefunction integrable on a cube Q and let t ≥ 1
|Q|
∫Q
f . Then, if Ωt is not empty, there is a
family of pairwise disjoint dyadic cubes Qi (the Calderon–Zygmund cubes of f at level t)such that
Ωt = ∪iQi,
with
t <1
|Qi|
∫Qi
f(x) dx ≤ 2n t (7.1)
for each i. As a consequence we have the weak type (1, 1) property with constant 1:
|Ωt| ≤1
t
∫Q
f(x) dx. (7.2)
We also have thatf(x) ≤ t for a.e. x /∈ ∪iQi (7.3)
and we can further refine (7.2) with
1
2n
∫x∈Q:f(x)>t
f(x) dx ≤ t|Ωt| ≤∫
x∈Q:f(x)>t/2
f(x) dx. (7.4)
Inequality (7.4) seems to be due to Wiener.
Proof: For each x ∈ Ωt, there is a dyadic cube P containing x and such that
1
|P |
∫P
f(y) dy > t.
59
Take the (unique) maximal (under inclusion) of these P ’s and denote it by P x. Observethat P x is strictly contained in Q since we assume t ≥ 1
|Q|
∫Q
f . We let Qi denote thiscollection of maximal dyadic cubes which are pairwise disjoint. Then it is clear that
Ωt = ∪iQi,
as well as the left hand side of (7.1). The right hand side of (7.1) follows by the maximalityof each of the cubes Qi and by the doubling property of the Lebesgue measure. Theweak type (1, 1) property follows immediately from the left hand side of (7.1).
The weak type (1, 1) property together with a standard method (see [GCRdF] p. 139)implies the dyadic version of the Lebesgue defferentiation theorem, namely that a.e. x ∈ Q
f(x) = limk→∞
1
|Qk|
∫Qk
f(y) dy,
where Qk∞k=0 is the unique chain of dyadic cubes such that x = ∩∞k=0Qk. Thenf(x) ≤ M
Qf(x) and therefore f(x) ≤ t if x /∈ ∪iQi = Ωt.
To prove the left hand side of (7.4) we use that by the Lebesgue defferentiation theoremwe have f(x) ≤ Mf(x) and∫
x∈Q:f(x)>tf(x)dx = f(x : f(x) > t) ≤ f(Ωt) = f(∪iQi) =
∑i
f(Qi)
≤ 2n t∑
i
|Qi| ≤ 2n t |Ωt|.
To prove the right hand side we use that M is bounded on L∞. Indeed, we write f asf = f1 + f2, where f1(x) = f(x) if f(x) > t
2, and f1(x) = 0 otherwise. Then
Mf(x) ≤ Mf1(x) + Mf2(x) ≤ Mf1(x) + t2. and it follows immediately that
|Ωt| ≤∫
x:f(x)>t/2
f(x)
tdx
by the weak type (1, 1) estimate.
2
There is a global version of this Calderon–Zygmund covering lemma. The minimalassumption on f to construct the Calderon–Zygmund cubes is that it must belong to thefollowing class of functions:
CZ = f ∈ L1loc(R
n) :1
|Q|
∫Q
f → 0 when Q ↑ Rn.
Observe that ∪p≥1Lp ⊂ CZ, in fact there are functions f ∈ CZ \ ∪p≥1L
p.
60
Then if f ∈ CZ and t > 0 (compare with the local situation where we must assumet ≥ 1
|Q|
∫Q
f) there is a family of pairwise disjoint dyadic cubes Qi verifying all theproperties listed in the local Calderon–Zygmund covering lemmawhere this timeΩt = x ∈ Q : Mf(x) > t and
Mdf(x) = Mf(x) = supP∈Dx∈P
1
|P |
∫P
|f(y)| dy. (7.5)
61
Chapter 8
Appendix II: The A∞ condition ofMuckenhoupt
Recall that the A∞ class of weights is defined naturally by
A∞ = ∪p>1Ap,
since the Ap class of weights are increasing on p. This condition was also introduced byMuckenhoupt in [Mu] and was further studied in the classical paper by R. Coifman and C.Fefferman [CF]. We have the following characterization; and specially condition f) isspecially interesting in applications.
Lemma 8.0.6 For a weight w the following conditions are equivalent:
a) w ∈ A∞;
b) there exists a positive constant C such that
1
|Q|
∫Q
w dx ≤ C exp
(1
|Q|
∫Q
log w dx
)(8.1)
c) there are positive constants α and β such that
|x ∈ Q : w(x) ≤ β wQ| ≤ α |Q| (8.2)
d) There exists a constant C such that for λ > wQ
w(x ∈ Q : w(x) > λ) ≤ Cλ |x ∈ Q : w(x) > β λ|. (8.3)
e) w satisfies a reverse Holder inequality, namely there are positive constants c and δ suchthat (
1
|Q|
∫Q
w1+δ dx
) 11+δ
≤ c
|Q|
∫Q
w dx
62
f) there are positive constants c and ρ such that for any cube Q and any measurable set Econtained in Q then
w(E)
w(Q)≤ c (
|E||Q|
)ρ. (8.4)
g) w satisfies the following condition: there are positive constants α, β < 1 such thatwhenever E is a measurable set of a cube Q
|E||Q|
< α impliesw(E)
w(Q)< β. (8.5)
Proof: Here we combine the methods from [CF] and [GCRdF].
a) ⇒ b)
By Jensen’s inequality it is enough to show that
1
|Q|
∫Q
w dx ≤ C exp
(1
|Q|
∫Q
log w dx
)Since the Ap classes are increasing on p if w ∈ ∪p>1Ap there exists some p0 > 1 such
that w ∈ Ap for p ≥ p0. Then, there exists a constant K such that for p ≥ p0(1
|Q|
∫Q
w dx
)(1
|Q|
∫Q
w1−p′ dx
)p−1
≤ K.
Letting p tend to ∞ we obtain b).
b) ⇒ c)
Dividing w by an appropriate constant (to be precise exp(
1|Q|
∫Q
log w dx)) we may
assume that∫
Qlog w dx = 0 and, consequently the hypothesis (8.1) becomes wQ ≤ C.
|x ∈ Q : w(x) ≤ β wQ| ≤ |x ∈ Q : w(x) ≤ β C|
= |x ∈ Q : log(1 +1
w(x)≥ log(1 +
1
β C)|
≤ 1
log(1 + 1β C
)
∫Q
log(1 +1
w) dx =
1
log(1 + 1β C
)
∫Q
log(1 + w) dx
since∫
Qlog w dx = 0. Now, since log(1 + t) ≤ t, t ≥ 0, we get:
|x ∈ Q : w(x) ≤ β wQ| ≤1
log(1 + 1β C
)
∫Q
w dx ≤ C |Q|log(1 + 1
β C)≤ 1
2|Q|,
if we choose β small enough.
c) ⇒ d)
63
Since we assume that λ > wQ we may consider the local Calderon–Zygmund coveringlemmaof w (see Appendix I) and we find a family of disjoint cubes Qj satisfying
λ <1
|Qj|
∫Qj
w dx ≤ 2n λ
for each j. By the properties of the cubes combined with (8.2) we have
w(x ∈ Q : w(x) > λ) ≤∑
j
w(Qj)
≤ 2λ∑
j
|Qj| ≤2 λ
1− α
∑j
|x ∈ Qj : w(x) > β wQj|
≤ 2 λ
1− α|x ∈ Q : w(x) > β wQ|
since wQj> λ > wQ.
d) ⇒ e)
We will be using the formula∫X
f(x)p dν = p
∫ ∞
0
λpν(x ∈ X : f(x) > λ) dλ
λ
which holds for every nonnegative measurable function f and in any arbitrary measurespace (X, ν) with nonnegative measure ν. Then for arbitrary positive δ we have
1
|Q|
∫Q
w1+δ dx =δ
|Q|
∫ ∞
0
λδw(x ∈ Q : w(x) > λ) dλ
λ
=δ
|Q|
∫ wQ
0
λδw(x ∈ Q : w(x) > λ) dλ
λ+
δ
|Q|
∫ ∞
wQ
λδw(x ∈ Q : w(x) > λ) dλ
λ
≤ (wQ)1+δ +δ
|Q|
∫ ∞
wQ
λδw(x ∈ Q : w(x) > λ) dλ
λ
≤ (wQ)1+δ +C δ
|Q|
∫ ∞
wQ
λδ+1|x ∈ Q : w(x) > β λ| dλ
λ
≤ (wQ)1+δ +C δ
β1+δ
1
|Q|
∫ ∞
wQ β
λδ+1|x ∈ Q : w(x) > λ| dλ
λ
≤ (wQ)1+δ +C δ
β1+δ
1
|Q|
∫Q
w1+δ dx.
If we choose δ small enough such that C δβ1+δ < 1 the last term can be absorbed by the first
term of the string of inequalities.
e) ⇒ f)
64
This is just Holder inequality with r = 1 + δ. Indeed if E ⊂ Q
w(E)
|Q|=
1
|Q|
∫Q
χEw dx ≤
(1
|Q|
∫Q
wr dx
)1/r ( |E||Q|
)1/r′
≤
C
|Q|
∫Q
w dx
(|E||Q|
)1/r′
and this implies the A∞ condition (8.4) with ρ = 1/r′.
f) ⇒ g)
This is immediate.
g) ⇒ c)
First observe that condition (8.5) is equivalent to saying that there are positiveconstants α, β < 1 such that whenever E is a measurable set of a cube Q
w(E)
w(Q)< α implies
|E||Q|
< β. (8.6)
Then let E = x ∈ Q : w(x) > bwQ where b ∈ (0, 1) is going to be chosen now and letE ′ = Q \ E = x ∈ Q : w(x) ≤ bwQ. Then w(E ′) ≤ b wQ|E ′| ≤ b w(Q). Then if we takeb = β we have that |E ′| ≤ α |Q| and hence |E| ≥ (1− α) |Q|. This yields (8.2).
Therefore we have shown that c) ⇔ d) ⇔ e) ⇔ f) ⇔ g)
c) ⇒ a)
We use again that condition e) is symmetric, namely that condition (8.6) holds. We alsouse that the measure w dx is doubling. Now, if we write dx = w−1wdx and since c) ⇔ e)we have that there are positive constants c and δ such that(
1
w(Q)
∫Q
(w−1)1+δ wdx
) 11+δ
≤ c
w(Q)
∫Q
w−1 wdx.
Hence
w(Q)
|Q|
(1
|Q|
∫Q
w−δ dx
)1/δ
≤ C.
Then, if we let δ = 1p−1
, that is, p = 1δ
+ 1 > 1 we have that w ∈ Ap.
The proof of the Lemma is now complete.
2
It is now very easy to see the following important openness property shared by the Ap
classes.
65
Corollary 8.0.7 Let p > 1, then
w ∈ Ap implies w ∈ Ap−ε, (8.7)
for some ε small enough depending upon the Ap constant.
66
Chapter 9
Appendix III: Exponentialself-improving property
• Generalized John-Nirenberg Inequalities
In the first part of this appendix we will prove the John–Nirenberg theorem in a bitof more generality. Our starting point is
1
|Q|
∫Q
|f(y)− fQ| dy ≤ a(Q),
where we assume that the funtional a is nondecraesing, namely if P ⊂ Qa(P ) ≤ a(Q).
We recall the normalized norm
‖g‖expL(Q,w)
= infλ > 0 :1
w(Q)
∫Q
(exp(
|f(y)|λ
)− 1
)w(y)dy ≤ 1.
Theorem 9.0.8 Let f be a locally integrable function such that
infα
((f − α)χ
Q
)∗(λ|Q|
)≤ cλ a(Q). (9.1)
where a is nondecreasing. Let w ∈ A∞. Then, there exists a constant C such that forall the cubes Q in Rn
‖f − fQ‖exp(L)(Q,w) ≤ C a(Q). (9.2)
As a sample consider in the one dimensional case the generalized Poincare inequality
1
|I|
∫I
|f(y)− fI | dy ≤∫
I
g(y) dy (9.3)
for some g locally integrable, then f has an exponential decay, even more we have theuniform decay
67
‖f − fI‖exp(L)(I,w) ≤ C
∫I
g(y)dy.
Remark 9.0.9 We want to remark it is possible to show that this generalizedJohn-Nirenberg can be seen as a limit of certain appropriate generalized Trudinger’sinequality. For this point of view see [MP2] or below .
Proof of the Theorem:
By Theorem (3.0.3) and since a : Q → (0,∞) is nondecreasing it is easy to see thatwe can assume that f is locally integrable and that for some constant c
1
|Q|
∫Q
|f(y)− fQ| dy ≤ c a(Q) (9.4)
for all the cubes Q. It is enough to show that there is a constant C such that for eachcube Q
1
|Q|
∫Q
exp(|f(y)− fQ|
C a(Q)) dy ≤ 1.
which will follow by standard arguments from the inequality
eβ t
w(Q)w(x ∈ B :
|f(x)− fQ|a(Q)
> t) ≤ 1.
where β is an appropriate constant independent of t > 0.
It is clear that we may assume that t > t0 for a large enough universal constant t0 .
For each t > 0 we let E(Q, t) = x ∈ Q : |f(x)− fQ| > t a(Q) and
ϕ(t) = supQ∈Q
w(E(Q, t))
w(Q),
By hypothesis ϕ(t) ≤ min1t, 1.
Recall since w is a A∞ weight there are positive constants c and ρ such that for anycube Q and any measurable set E contained in Q then
w(E)
w(Q)≤ c (
|E||Q|
)ρ.
Lemma 9.0.10 There exists γ > 1 such that for every t > γ 2n
ϕ(t) ≤ 1
eϕ(t− γ 2n). (9.5)
Reiterating this inequality [ tγ 2n ] times we get for t > γ 2n
ϕ(t) ≤ C e−Ct
68
We can consider the local Calderon–Zygmund covering lemma(see Appendix I) off−fQ
a(Q)relative to Q at level γ, γ > 1, since
1
|Q|
∫Q
|f(y)− fQ|C a(Q)
dy ≤ 1 < γ.
This yields a collection of dyadic subcubes of Q, Qi, maximal with respect toinclusion, satisfying
γ <1
|Qi|
∫Qi
|f(y)− fQ|a(Q)
dy ≤ γ2n (9.6)
for each integer i. Also by the Lebesgue diferentiation theorem
E(Q, γ) ⊂ ∪iQi.
Now, for t > γ2n and using the monotonicity of a together with
|f(x)− fQ| ≤ |f(x)− fQi|+ |fQi
− fQ| ≤ |f(x)− fQi|+ γ2na(Q)
we havew(E(Q, t)) = w(E(Q, γ) ∩ E(Q, t))
=∑
i
w(x ∈ Qi : |f(x)− fQ| > a(Q) t)
=∑
i
w(x ∈ Qi : |f(x)− fQi| > a(Q) (t− γ2n))
≤∑
i
w(x ∈ Qi : |f(x)− fQi| > a(Qi) (t− γ2n))
≤∑
i
w(Qi) ϕ(t− γ2n) = ϕ(t− γ2n) w(∪iQi).
Now by the A∞ condition
w(∪iQi) ≤ c
(| ∪i Qi||Q|
)δ
w(Q),
but by (9.6) and by dijointness of the cubes
| ∪i Qi| =∑
i
|Qi| ≤1
γ
∫Q
|f(y)− fQ|a(Q)
dy ≤ 1
γ|Q|
and hence we conclude for t > γ2n:
ϕ(t) ≤ c
γρϕ(t− γ2n).
To conclude it is enough to choose γ such that cγρ < 1
e.
2
69
• Generalized Trudinger Inequalities
Recall that the Sobolev Embedding Theorem says that functions in W 1,qloc (Rn)
actually lie in Lrloc for r = nq/(n− q) when 1 ≤ q < n. We have already mentioned
that a more precise version of this statement is the following local inequality:(1
|Q|
∫Q
|f − fQ|r) 1
r
≤ C `(Q)(1
|Q|
∫Q
|∇f |q)1q (9.7)
for any cube Q. When q tends to n the constant C on the righthand side blows upand so the limiting case with q = n, i.e., W 1,n
loc ⊆ L∞loc, is false. The correct result inthis instance is that W 1,n
loc lies locally in the class exp Ln′ . The correspondinginequality, called Trudinger’s Inequality, is
‖f − fQ‖exp Ln′ (Q) ≤ C (
∫Q
|∇f |n)1n . (9.8)
It should be mentioned that it was also derived by Yudovich in [Y]. See [GT], forinstance, for a proof in the case that f has compact support. The norm on the lefthand side is the Luxemburg (or Orlicz) norm associated to the functionΦ(t) = exp tn
′ − 1 (see Section 4.1). Trudinger’s inequality has proved to be one ofthe key results in the study of parabolic and elliptic equations at the critical index(which is n in the case of Rn). In addition, the sharp value for the constant Cappearing on the righthand side (see Moser [Mo], and Adams [Ad] for the higherdimensional version) plays a major role in geometry, especially in the problem ofprescribing Gaussian curvatures on spheres.
Motivated by the work [HaK2] we consider in [MP2] the following more generalsituation. As above we will consider locally integrable functions such that for eaverycube Q
1
|Q|
∫Q
|f(y)− fQ| dy ≤ a(Q),
where a : Q → [0,∞). The question is: What kind of geometric condition, in thespirit of (2.15), do we have to impose to get a exponential estimate similar to (9.8)?
Definition 9.0.11 . Let 1 < r < ∞. We say that the functional a satisfies the Tr
condition if there exists a finite constant c such that for each cube Q and any family∆ of pairwise disjoint subcubes of Q,∑
P∈∆
a(P )r ≤ Cr a(Q)r. (9.9)
We will use ‖a‖ to denote the smallest constant C for which (9.9) holds. We always
have ‖a‖ ≥ 1. The main examples are a(Q) =(∫
Qgr) 1
rwhere g ∈ Lr
loc(Rn) and,
more generally,a(Q) = ν(Q)
1r
70
where ν is a locally finite measure. Observe that these conditions are increasing, inthe sense that r < s ⇒ Tr ⊂ Ts. Also observe that this condition is much strongerthan the Dr condition (2.15) since in particular a is increasing.
Theorem 9.0.12 Assume that the functional a satisfies the Tr condition for some1 < r < ∞, and that w is a doubling weight. Let f is a locally integrable functionsuch that for all cubes Q in Rn
infα
((f − α)χ
Q
)∗(λ|Q|
)≤ cλ a(Q). (9.10)
Then there exists a constant C independent of f such that
‖f − fQ‖exp Lr′ (Q,w)
≤ C a(Q). (9.11)
for any cubes Q.
Note the important fact that the functional a need not depend on f , in contrast to
the classical case where the functional is given by a(Q) =(∫
Q|∇f |n
) 1n
As a matter
of fact we could work in spaces with no differential structure at all. See [MP2].
Remark 9.0.13 A point of interest here is that the limiting “r = ∞” case of thisTheorem is just above generalized John–Nirenberg theorem 9.0.8. Indeed, let T∞ bedefined as those functionals a “essentially” in the sense that for some finite constantc and for arbitrary cubes Q, P with P ⊂ Q then a(P ) ≤ c a(Q) (Theoren 9.0.8 alsoholds in this case). Observe that Tr ⊂ T∞ and that in fact the condition T∞ is simplythe limit as r →∞ of the condition Tr. Recall that the functional a(Q) ≡ 1 is thefunctional associated to the space BMO, satisfies T∞.
Proof: To simplify the proof we just consider the unweighted case w ≡ 1.
As above, by Theorem (3.0.3) and since a : Q → (0,∞) is in particular nondecreasingit is easy to see that we can assume that f is locally integrable and that for someconstant c
1
|Q|
∫Q
|f(y)− fQ| dy ≤ c a(Q) (9.12)
for all the cubes Q.
We claim that for a fixed cube Q and for a.e. x ∈ Q
|f(x)− fQ| ≤ C a(Q)((log A(x))1/r′ + 1
). (9.13)
Indeed as in the proof of Lemma 2.3.1, if x ∈ Q consider the unique one-way inifinitechain of dyadic cubes starting from Q = Q0, Qk∞k=0, such that
x = ∩∞k=−∞Qk.
71
See Appendix I. For each k we let fk = 1|Qk|
∫Qk
f . Then by the Lebesguedifferentiation theorem we have a.e. x ∈ Q that
|f(x)− fQ| = | limk→∞ fk − f0| = |f0 − fN +∞∑
k=N+1
(fk−1 − fk)|
≤ |f0 − fN |+∞∑
k=N
|fk − fk+1| = IN
+ IIN
To estimate IIN
we define the following (normalized) maximal function associated toa by
A(x) = AQ(x) = supP∈D(Q)
x∈P
a(P )
|P |1/r
|Q|1/r
a(Q).
Then we can estimate IIN
as follows
IIN≤
∞∑k=N
1
|Qk+1|
∫Qk+1
|f − fQk| ≤ 2n
∞∑k=N
a(Qk)
≤ 2n A(x)a(Q)
|Q|1/r
∞∑k=N
|Qk|1/r ≈ A(x) a(Q) 2−Nn/r.
To estimate IN
we are going to link the cubes QN and Q by means of a special chainof disjoint cubes Pii=0,··· ,M contained in Q where P0 = QN (in the picture is thesmallest nondash cube). We also define PM+1 = Q.
1
In the picture below, ε = 2−N`(Q) denotes the sidelenght of the first cube. First, weselect the face of Q whose distance to QN is furthest and build a “stairs” of cubesfollowing this direction. To define P1 we take the adjacent cube to P0 with sidelenght2`(P0) = 2ε as in the picture (again is nondash cube). P2 is defined in a similar wayas shown in the picture. We continue in this way until we reach QM with QM ⊂ Qand with M ≈ N .
1I wish to thank J. Orobitg for showing me this argument.
72
ε2ε
4ε
8ε...
......
Now recalling the notation fE = 1|E|
∫E
f , we have
IN
= |fQN− fQ| ≤ |fQ − fQM
|+ |fQM− fQN
|
≤ 1
|QM |
∫QM
|f − fQ|+M∑i=1
|fPi− fPi+1
|.
The first term is immediate:
1
|QM |
∫QM
|f − fQ| ≤C
|Q|
∫Q
|f − fQ| ≤ C a(Q).
As for the second, for each i = 1, · · · , M we compare Pi and Pi+1 with a third cubePi which can be defined as the minimal cube containing Pi and Pi+1 following the“direction” of the “stairs” (in the picture these are the dash cubes). Then
|fPi− fPi+1
| ≤ |fPi− fPi
|+ |fPi− fPi+1
| ≤ C a(Pi)
since `(Pi) ≈ `(Pi) ≈ `(Pi+1). Hence, combining these estimates we have
IN≤ C a(Q) +
M∑i=1
a(Pi) ≤ C a(Q) +
(M∑i=1
a(Pi)r
)1/r
M1/r′
73
Now, if we want to use the Tr condition we must have disjoint cubes. However,observe from the picture that the overlap is finite, namely two. Then we can split thefamily Pi in two (the odd and even index for instance) and each family is formedby pairwise disjoint cubes. Then
IN≤ C a(Q) + C M1/r′ a(Q) ≈ N1/r′ a(Q).
Combining the estimates for IN
and IIN
we have
|f(x)− fQ| ≤ C a(Q)(N1/r′ + A(x) 2−N/r.
).
We optimize N by choosing N = N(x) to be an integer approximately equal tolog A(x). Then
|f(x)− fQ| ≤ Ca(Q)((log A(x))1/r′ + 1
)and the claim (9.13) follows.
Now if we let s > 0, then a simple manipulation of (9.13) yields
exp
(s
(|f(x)− fQ|
C a(Q)
)r′)≤ exp (s (1 + log A(x))) ≤ CA(x)s
for a.e. x ∈ Q. Averaging over Q we have
1
|Q|
∫Q
exp
(s
(|f(x)− fQ|
C a(Q)
)r′)
dx ≤ C
|Q|
∫Q
A(x)s dx.
Now if we choose s < r and apply again Kolmogorov’s inequality 2.25 we have
1
|Q|
∫Q
exp
(s
(|f(x)− fQ|
C a(Q)
)r′)
dx ≤ C ‖A‖sLr,∞(Q)
To conclude with the proof of the Theorem we are left with showing the followinglemma.
Lemma 9.0.14‖A‖Lr,∞(Q) ≤ C ‖a‖
Proof: The proof of
|x ∈ Q : A(x) > λ| ≤ C ‖a‖r
λr|Q|
is by a standard (dyadic type) covering lemma. If A(x) > λ then we can take themaximal dyadic subcube P of Q with x ∈ P such that
a(P )
|P |1/r
|Q|1/r
a(Q)> λ.
74
Denote this family by Pi which by maximality they are disjoint. Since each Pi
satisfies last inequality we raise it to the power r to get
|x ∈ Q : A(x) > λ| =∑
i
|Pi| ≤|Q|
λra(Q)r
∑i
|Pi|a(Pi)
r
|Pi|≤ 1
λr‖a‖r|Q|
by the Tr condition. This means that ‖A‖Lr,∞(Q) ≤ ‖a‖.2
The proof of the Theorem is now complete.
2
75
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