Harmonic Analysis and Its Applications
In these lectures, we concentrate on the motivations, development and applications of the Calderon-Zygmund operator theory.
Lecture 1. The differential operators with constant coefficients and the first generation of Calderon-Zygmund operators
Consider the following differential operator with constant coefficients: Lu(x) =X
α
aα∂ αu
∂xα. (1.1)
By taking the Fourier transform, d
(Lu)(ξ) =X
α
aα(−2πiξ)αu(ξ).ˆ (1.2)
This suggests one to consider the following more general Fourier multiplier: Definition 1.3: An operator T is said to be the Fourier multiplier if
d
(T f )(ξ) = m(ξ) ˆf (ξ). (1.4)
(1.2) shows that any classical differential operator is a Fourier multiplier. Example 1: Suppose f ∈ L2(R) and F is an analytic extension of f on R2
+ given by F (x + iy) = −i π Z 1 x + iy − tf (t)dt = − i π Z (x − t) − iy (x − t)2+ y2f (t)dt = 1 π Z y (x − t)2 + y2f (t)dt − i π Z (x − t) (x − t)2+ y2f (t)dt. (1.5)
Letting y → 0, then π1 R (x−t)y2+y2f (t)dt → f (x) for a. e. x, and, in general, the second
term above has no limit. However, one can show p.v R 1
x−tf (t)dt exists for a. e. x. Thus
limy→0F (x + iy) = f (x) + iH(f )(x) (1.6)
where H is called the Hilbert transform defined by H(f )(x) = −1
π Z
f (t)
Example 2: Consider the Laplacian 4u = n X j=1 ∂2u ∂x2 j . (1.8)
By taking the Fourier transform, d (− 4 u)(ξ) = n X j=1 (4π|ξj|)2u(ξ) = 4π|ξ|ˆ 2u(ξ).ˆ
Define the Riesz transforms Rj, 1 ≤ j ≤ n, by
d (Rjf )(ξ) = ξj |ξ|f (ξ).ˆ (1.9) Then d ( ∂2u ∂xi∂xj )(ξ) = −4πξiξju(ξ) =ˆ (RiRdj4 u)(ξ). (1.10) Thus, ∂2u ∂xi∂xj = RiRj4 u. (1.11)
Since, it is easy to see that H and Rj, 1 ≤ j ≤ n, are bounded on L2, so we obtain
klimy→0F (x + iy)k2 ≤ Ckf k2 (1.12)
and
k ∂
2u
∂xi∂xj
k2 = kRiRj4 uk2 ≤ Ck 4 uk2. (1.13)
The Hilbert transform, by taking the Fourier transform, can be written as d
H(f )(ξ) = −isign(ξ) ˆf (ξ) (1.14)
and the Riesz transform, by taking the inverse Fourier transform, can be written as Rj(f )(x) = cnp.v.
Z yj
|y|1+nf (x − y)dy, 1 ≤ j ≤ n. (1.15)
In 1952, Calderon and Zygmund introduced the following first generation of singular inte-gral operators: Definition 1.16: T (f )(x) = p.v. Z Ω(y) |y|n f (x − y)dy,
where Ω satisfies the following conditions: Ω(λy) = Ω(y) (1.17) for all λ > 0; Ω ∈ C1(Sn−1); (1.18) Z Ω(y)dσ(y) = 0. (1.19)
First we point out that the first generation of Calderon-Zygmund operators are well defined on S(Rn), the Schwartz test function space. To see this, one can define
p.v. Z
Ω(y)
|y|n f (y)dy = lim²→0
Z |y|>² Ω(y) |y|n f (y)dy = Z |y|<1 Ω(y)
|y|n [f (y) − f (0)]dy +
Z
|y|≥1
Ω(y)
|y|n f (y)dy (1.20)
for all f ∈ S since Ω has zero average. It is then easy to see that both integrals above converge.
Remark 1.21: A necessary condition for p.v.R Ω(y)|y|nf (x − y)dy exists is that Ω has zero
average on Sn−1. In fact, let f ∈ S be such that f (x) = 1 for |x| ≤ 2. Then for |x| < 1,
T (f )(x) = lim²→0 Z ²<|y|<1 Ω(y) |y|n dy + Z |y|≥1 Ω(y) |y|n f (x − y)dy. (1.21)
The second integral is convergent but the first equals lim²→0
R
Sn−1
Ω(y)dσ(y)log(1²). Thus, if this limit is finite, then the integral of Ω on Sn−1 is zero.
Theorem 1.22(Calderon and Zygmund): If T is an operator of the first generation of Calderon-Zygmund operator, then T is bounded on Lp, 1 < p < ∞. Moreover, there is
a constant C such that
kT (f )kp ≤ Ckf kp. (1.23)
The method of the proof of theorem 1.22 is called the real variable method of Calderon and Zygmund. This method includes the following steps.
Step 1: T is bounded on L2. To do this, since T is a convolution operator, by the
Plancheral theorem, it suffices to show the Fourier transform of K, the kernel of T , is bounded. In fact, we will show that for ξ ∈ Sn−1,
m(ξ) = ˆK(ξ) = Z Sn−1 Ω(t)[log( 1 |t · ξ|) − i π 2sign(t · ξ)]dσ(t). (1.24)
Indeed, since K is homogeneous of degree −n, so m(ξ) is homogeneous of degree 0. There-fore, we may assume that ξ ∈ Sn−1. Since Ω has zero average,
m(ξ) = lim²→0 Z ²<|y|<1 ² Ω(y) |y|n e −2πiy·ξdy = lim²→0 Z Sn−1 Ω(y)[ 1 Z ² (e−2πity·ξ − 1)dt t + 1 ² Z 1 e−2πity·ξdt t ]dσ(y) = lim²→0 Z Sn−1 Ω(y)[ 1 Z ² (cos(2πty · ξ) − 1)dt t + 1 ² Z 1 cos(2πty · ξdt t ]dσ(y) −ilim²→0 Z Sn−1 Ω(y) 1 ² Z ² sin(2πty · ξ)dt t dσ(y)
Making the change of variables s = 2πty · ξ and assume y · ξ 6= 0, the second term above will be lim²→0 Z Sn−1 Ω(y) 1 ² Z ² sign(2πty · ξ)dt t dσ(y) = Z Sn−1 Ω(y) ∞ Z 0 sign(y · ξ)sin(s) s dsdσ(y) = Z Sn−1 Ω(t)π 2sign(t · ξ)dσ(t). The first, after the change of variables, will be
lim²→0 Z Sn−1 Ω(y)[ 1 Z ² (cos(2πy · ξ) − 1)dt t + 1 ² Z 1 cos(2πy · ξdt t ]dσ(y) = = Z Sn−1 Ω(y)[ 1 Z 2π|y·ξ|² (cos(s) − 1)ds s + 2π|y·ξ|1 ² Z 1 cos(s)ds s − 2π|y·ξ|Z 1 ds s ]dσ(y) = Z Sn−1 Ω(t)[log( 1 |t · ξ|)dσ(t) since Ω has zero average. Now applying the Plancheral yields
Step 2: We show T is of week type (1, 1): There is a constant C such that |{x ∈ Rn: |T (f )(x) > λ}| ≤ Ckf k1
λ (1.25)
for any λ > 0 and f ∈ L1 ∩ L2. To do this, we need the following Calderon-Zygmund
decomposition.
Calderon-Zygmund decomposition 1.26: Given f ∈ L1 and non-negative, and given
a positive λ, there exists a sequence {Qj} of disjoint cubes such that
f (x) ≤ λ (1.27) for x /∈ ∪Qj; | ∪ Qj| ≤ 1 λkf k1; (1.28) λ < 1 |Qj| Z Qj f (x)dx ≤ 2nλ. (1.29)
The proof of this decomposition is to use the so-called stopping time argument. First, choose a large cube Q so that |Q|1 R
Q
f (x)dx ≤ |Q|1 R f (x)dx ≤ λ. Then divide Q to 2n equal
subcubes Q0. Now we use the stopping time argument as follows: if 1 |Q0|
R
Q0
f (x)dx > λ, we keep this subcube Q0. Otherwise, divide this subcube as above and keep this procedure.
Now we get a sequence {Qj}. If x /∈ ∪Qj, this means that there is a sequence {Qn}
with|Qn| → 0 as n → ∞, so that x∈ Qn for all n and |Q1n|
R
Qn
f (x)dx ≤ λ which shows (1.27). To see (1.29), notice that if λ < |Q1j| R
Qj
f (x)dx then λ ≥ |2n1Qj|
R
2nQj
f (x)dx which yields (1.29). Finally, from λ < 1
|Qj| R Qj f (x)dx we obtain |Qj| < λ1 R Qj f (x)dx. Summing up shows | ∪ Qj| ≤ 1 λ X j Z Qj f (x)dx ≤ 1 λkf k1.
Now we apply the Calderon-Zygmund decomposition to show T is of the week type (1, 1). Define g(x) = f (x) for x /∈ ∪Qj and g(x) = |Q1j|
R Qj f (x)dx for x ∈ Qj, and b(x) = f (x) − g(x). Since T is bounded on L2, so |{x ∈ Rn: T (g)(x) > λ 2}| ≤ C kT (g)k2 2 λ2 ≤ C kgk2 2 λ2 ≤ C 1 λ2[ Z (∪Qj)c |f (x)|2dx +X j Z Qj 1 |Qj| Z Qj f (y)dydx] ≤ C1 λ[ Z (∪Qj)c |f (x)|dx + 2n| ∪ Qj|] ≤ C1 λkf k1.
To estimate T (b)(x), it is easy to see
|{x ∈ Rn : |T (b)(x)| > λ
2}| ≤ | ∪ 2Qj| + |{x ∈ (∪2Qj)
c : |T (b)(x)| > λ
2}|.
By (1.28), it suffices to show |{x ∈ (∪2Qj)c : |T (b)(x)| > λ2}| ≤ Cλ1kf k1. Rewrite b(x) =
P
j
bj(x), where bj(x) = [f (x) −|Q1j|
R
Qj
f (x)dx]χQj(x) and the sum converges in L
2, as well
as pointwise. Ifx ∈ (∪2Qj)c, then, by the fact that bj has zero average,
T (bj)(x) =
Z
K(x − y)bj(y)dy =
Z
[K(x − y) − K(x − xQj)]bj(y)dy
where xQj is the center of Qj. Thus, by the fact thatx ∈ (∪2Qj)
c and the smoothness of
K, |T (bj)(x)| ≤ C Z Qj (Qj) |x − xQj|n+1 |bj(y)|dy
We now estimate the integral Z (∪2Qj)c |T (b)(x)|dx ≤ CX j Z (∪2Qj)c Z Qj (Qj) |x − xQj|n+1 |bj(y)|dydx.
We apply Fubini’s theorem to change the order of integration in each of the double integrals and it gives CP
j
kbjk1. This, in turn, can be estimated by using (1.28) and (1.29). So we
get Z
(∪2Qj)c
|T (b)(x)|dx ≤ Ckf k1
which shows |{x ∈ (∪2Qj)c : |T (b)(x)| > λ2}| ≤ Ckf kλ1. This fact together with (1.28)
shows |{x ∈ Rn : |T (b)(x)| > λ 2}| ≤ C kf k1 λ . Noting |{x ∈ Rn : |T (f )(x)| > λ}| ≤ |{x ∈ Rn : |T (g)(x)| > λ 2}| + |{x ∈ Rn : |T (b)(x)| > λ
2}| yields the proof of (1.25) and the proof of step 2 is complete.
Step 3: By Marcinkiewicz’s interpolation theorem, T is bounded on Lp for 1 < p ≤ 2,
and
kT (f )kp ≤ Ckf kp.
Step 4: We observe that the adjoint T∗ of T satisfies the same conditions as T , so T∗ is
Example: Let f (x, y) be the density of a mass distribution in the plane. Then its New-tonian potential in the half-space R3
+ is g(x, y, z) = Z R2 f (u, v) [(x − u)2+ (y − v)2+ z2]12 dudv. Formally, we have limz→0∂g ∂x(x, y, z) = − Z R2 f (u, v)(x − u) [(x − u)2+ (y − v)2]32 dudv.
However, this integral does not converge in general. But it exists as a principle value if f is smooth. This is an operator of the first generation of Calderon-Zygmund operators by considering Ω(x, y) = − x
x2+y2.
We can consider the first generation of Calderon-Zygmund operators as the Fourier multipliers.
Theorem 1.30: If m ∈ C∞(Rn\{0}) is a homogeneous function of degree 0, and T m is
the Fourier multiplier defined by d(Tmf ) = m ˆf , then there exist a, a complex number and
Ω ∈ C∞(Sn−1) with zero average such that for any f ∈ S,
Tmf = af + p.v.
Z Ω(y)
|y|n f (x − y)dy. (1.31)
Since any homogeneous function of degree 0 is the sum of a constant and a homogeneous function of degree 0 with zero average on Sn−1, theorem 1.30 is a consequence of the
following lemma.
Lemma 1.32: Let m ∈ C∞(Rn\{0}) be a homogeneous function of degree 0, and T m is
the Fourier multiplier defined by d(Tmf ) = m ˆf , then there exist a, a complex number and
Ω ∈ C∞(Sn−1) with zero average such that ˆm(y) = p.v.Ω(y) |y|n.
Proof: Since m is a tempered distribution, ˆm exists. Thus, d (∂nm ∂xn i )(ξ) = Cξn i m(ξ),ˆ
where C is a constant. The function ∂∂xnmn
i is homogeneous of degree -n, in C
∞(Rn\{0})
and has zero average on Sn−1. Moreover,
∂nm ∂xn i = p.v.∂ nm ∂xn i + X |α|≤k CαDαδ, (1.33)
where δ is the Dirac measure at the origin, since the difference between ∂∂xnmn
i and p.v.
∂nm
∂xn
both sizes of (1.33) and note that the left-hand side and the first term on the right-hand size are homogeneous distributions of degree 0, so the polynomial, the Fourier transform of the second term on the right-hand size, is a constant. Thus the right-hand side of (1.33) is a homogeneous function of degree 0 which is in C∞(Rn\{0}). Since this valid for
1 ≤ i ≤ n, ˆm coincides on Rn\{0} with a homogeneous function of degree -n. We denote
its restriction to Sn−1 by Ω. To see that Ω has zero average, fix a radial function φ ∈ S,
which is supported on 1 ≤ |x| ≤ 2 and positive for 1 < |x| < 2. Then ˆ m(φ) = Z Ω(x) |x|n φ(x)dx = c Z Ω(x)dσ(x), (1.34)
where c > 0. On the other hand, since bφ is radial and m is homogeneous, ˆ
m(φ) = m(bφ) = c0 Z
m(x)dσ(x) = 0
which together with (1.34) shows R Ω(x)dσ(x) = 0. Finally, to see that bm is identical to p.v. Ω(x)|x|n, consider their difference bm−p.v.
Ω(x)
|x|n, which is supported at the origin. By taking
the Fourier transform of this difference, we get a polynomial which must be a constant because both m and(p.v.dΩ(x)|x|n) are bounded. Furthermore, this constant must be zero since
both m and Ω have zero average on Sn−1.
Theorem 1.35: The set A of operators defined by theorem 1.30 is a commutive algebra. An element of A is invertible if and only if m is never zero on Sn−1.
Proof: If Tm1 and Tm2 are in A, then Tm1Tm2 = Tm1m2.Tm is invertible if and only if
T1
m ∈ A, and hence, 1/m ∈ C
∞(Rn\{0}) which shows m(ξ) 6= 0 for any ξ ∈ Sn−1. We
now return to study Lu(x) = P
|α|=m aα∂ αu ∂xα. Define d (Λf )(ξ) = 2π|ξ| ˆf (ξ). (1.36) Then Lu = T Λmu, (1.37)
where the operator T is defined by d
(T u)(ξ) = imP (ξ)
|ξ|mu(ξ)ˆ (1.38)
where P (ξ) = P
|α|=m
aα(2πiξ)α. The multiplier P (ξ)|ξ|m is homogeneous of degree zero and it
is a C∞ function on Sn−1, and hence, T is an operator in A. To solve
Lu(x) = X
|α|=m
aα∂ αu
∂xα = f,
Lecture 2. Differential operators with variable coefficients, the second generation of Calderon-Zygmund operators, and pseudo-differential operators
We are interested in studying the following differential operator with variable coeffi-cients: Lu(x) = X |α|=m aα(x)∂ αu ∂xα (2.1) where aα(x) ∈ C∞(Rn).
Formally, by taking the Fourier transform and then the inverse Fourier transform, we get
Lu(x) = X
|α|=m
aα(x)
Z
(−2πiξ)αu(ξ)eˆ 2πiξ·xdξ
= Z
[ X
|α|=m
aα(x)(−2πiξ)α]ˆu(ξ)e2πiξ·xdξ. (2.2)
(2.2) can be written by more general form: σ(x, D)f (x) =
Z
σ(x, ξ) ˆf (ξ)e2πiξ·xdξ. (2.3) Calderon and Zygmund wanted to keep what they did for the first generation of Calderon-Zygmund operators and rewrite (2.3) by
T f (x) = Z L(x, x − y)f (y)dy (2.4) where p.v. Z L(x, y)e−2πiξ·ydy = σ(x, ξ). (2.5) The relationship (2.5), discovered by Calderon and Zygmund in the 1950s, opened the way to all later developments in which the pseudo-differential operators were defined using algebras of symbols, without reference to any kernels. After the golden age just described, the two points of view diverged: Kohn and Nirenberg, for their part, and Hormander, for his, systematically favored the definition of pseudo-differential operators by symbols. Research on kernels remained very active in the school of Calderon and Zygmund and led to what we will introduce the third generation of Calderon-Zygmund operators in next lecture.
To see (2.4) and (2.5), formally, by taking the inverse Fourier transform, we obtain T f (x) =
Z
σ(x, ξ) ˆf (ξ)e2πiξ·xdξ = Z Z
σ(x, ξ)e2πiξ·xf (y)e−2πiy·ξdydξ
= Z
[ Z
σ(x, ξ)e2πiξ·(x−y)dξ]f (y)dy = Z
Using the operator Λ again, Calderon and Zygmund rewrote (2.2) as Lu(x) = Z ([ X |α|=m aα(x)(−2πiξ)α 1 |ξ|m]|ξ| mu(ξ)eˆ 2πiξ·xdξ = T (Λmu) (2.6) where T is defined by T f (x) = Z σ(x, ξ) ˆf (ξ)e2πiξ·xdξ (2.7) with σ(x, ξ) = P |α|=m aα(x)(−2πiξ)α |ξ|m .
Thus, σ is homogeneous of degree 0 in the variable ξ. By (2.5), T f (x) =
Z
K(x, x − y)f (y)dy (2.8)
where K(x, y) =R σ(x, ξ)e2πiy·ξdξ.
For fixed x, K(x, ·) is the inverse Fourier transform of σ(x, ·). By theorem 1.30, for each fixed x there is a constant a(x) and a function Ω(x, ·) ∈ C∞(Sn−1) with zero average
on Sn−1 such that
K(x, y) = a(x)δ(y) + p.v.Ω(x, y) |y|n .
This was the motivation for Calderon and Zygmund to introduce the second generation of Calderon-Zygmund operators.
Theorem 2.9: Suppose that Ω(x, y) is a function satisfying the following conditions:
Ω(x, y) = Ω(x, λy) (2.10)
for all x ∈ Rn and all λ > 0;
Ω(x, y) ∈ C∞(Rn× Sn−1); (2.11)
Z
Sn−1
Ω(x, y)dσ(y) = 0 (2.12)
for all x ∈ Rn.
The second generation of Calderon-Zygmund operators is the set of T defined by T f (x) = p.v.
Z
Ω(x, y)
|y|n f (x − y)dy. (2.13)
Then the operator T given by (2.13) is bounded on Lp, 1 < p < ∞.
To show this theorem, by the Calderon-Zygmund real variable method, it suffices to prove the L2 boundedness of T . To see this, we follow Calderon and Zygmund, by
performing a spherical harmonic expansion of σ(x, ξ) on the Sn−1 for fixed x ∈ Rn, where
σ(x, ξ) is the Fourier transform of p.v. Ω(x,y)|y|n with the variable y whenever x is fixed. The
proof of lemma 1.32 shows that σ satisfies (2.10), (2.11) and (2.13) with Ω replaced by σ. By the regularity with respect to ξ, this gives a norm-convergent sequence
σ(x, ξ) = ∞ X 0 mk(x)hk(ξ) with P∞ 0 kmkk∞khkk∞ < ∞.
We extend hk(ξ) as a homogeneous function of degree 0, which is the symbol of an
operator we denote by Hk, and we write Mk for the operator of pointwise multiplication
by mk. This yields T = ∞ X 0 MkHk
and the series of operators is convergent in the L2− norm. It is easy to see that the
sec-ond generation of Zygmund operators includes the first generation of Calderon-Zygmund operators.
Now we consider formally a pseudo-differential operator defined by Tσf (x) =
Z
σ(x, ξ) ˆf (ξ)e2πix·ξdξ.
If σ is independent of ξ, σ(x, ξ) = a(x), then T is a multiplication operator: T f (x) = a(x)f (x). When σ is independent of x, σ(x, ξ) = m(ξ), then T is a Fourier multiplier oper-ator: d(T f )(ξ) = m(ξ) ˆf (ξ) which shows that pseudo-differential operators are genelizations of the Fourier multiplier operators. We shall consider the standard symbol class, denoted by Sm, which is most common and useful of the general symbol classes.
Definition 2.16: A function σ belongs to Sm (and is said to be of order m) if σ(x, ξ) is
a C∞ function of (x, ξ) ∈ Rn× Rn and satisfies the differential inequalities
|∂xβ∂ξασ(x, ξ)| ≤ Cα,β(1 + |ξ|)m−|α|, (2.17)
for all multi-indices α and β.
It is easy to see that if σ is a polynomial in ξ and independent of x, then (2.17) is satisfied. Roughly speaking, the conditions (2.17) mean that the behavior of σ(x, ξ) looks like a polynomial of order m.
Given a symbol in Sm, the operator T
σ will initially be defined on the Schwartz class
of testing functions S. In fact, the integral (2.15) converges absolutely and is infinitely differentiable. An integration by parts argument shows that Tσ(f ) is a rapidly decreasing
function. Indeed, note that
and define the operator
Lξ = (1 + 4π2|x|2)−1(I − 4ξ),
then (Lξ)Ne2πix·ξ = e2πix·ξ. Inserting this in (2.15) and carrying out the repeated
integra-tions by parts gives
Tσf (x) =
Z
(Lξ)N[σ(x, ξ) ˆf (ξ)]e2πix·ξdξ
which shows Tσf (x) is rapidly decreasing. Since this argument works for any partial
derivative of Tσ, and, hence, Tσ maps S to S, and this mapping is continuous. It is worth
to pointing out that if {σk} is a pointwise convergent sequence of symbols in Sm that
satisfy the conditions (2.17) uniformly in k, then Tσk(f ) → Tσ(f ) in S for f ∈ S. An
alternative way of writing Tσ defined in (2.15) is as a repeated integral
Tσf (x) =
Z Z
σ(x, ξ)e2πi(x−y)·ξf (y)dydξ. (2.18)
However, the integral in (2.18) does not necessarily converge absolutely, even when f ∈ S. To with with this integral, fix a function γ ∈ C∞
0 (Rn× Rn) with γ(0, 0) = 1. Set σ²(x, ξ) =
σ(x, ξ)γ(²x, ²ξ). Notice that if σ ∈ Sm, then σ
² ∈ Sm and they satisfy the condition (2.17)
uniformly in ², for 0 < ², 1. As mentioned above, Tσ²(f ) → Tσ(f ) in S when f ∈ S, as
² → 0. Moreover, since the operator Tσ defined in (2.18) converges when σ has compact
support, we get that
Tσf (x) = lim²→0
Z Z
σ²(x, ξ)e2πi(x−y)·ξf (y)dydξ. (2.19)
By the duality relation < Tσf, g >=< f, Tσ∗g > when f, g ∈ S. The same proof shows Tσ∗
also maps S to S.
Theorem 2.20: If σ ∈ S0, then the operator T
σ initially defined on S, extends to a
bounded operator on L2.
Proof: It suffices to show that
kTσf k2 ≤ Ckf k2, (2.21)
whenever f ∈ S, with C independent of f . Indeed, suppose f ∈ L2 and let {f
n} ∈ S so that fn → f in L2. Then, by (2.21), Tσ(fn)
converges in L2 norm, and hence, T
σ(fn) converges to Tσ(f ) in the sense of distributions.
We return to the proof of (2.21). First, we assume that σ(x, ξ) has compact support in x. We then write
σ(x, ξ) = Z
b
σ(µ, ξ)e2πiµ·xdµ
since σ has compact support in x variable. An integration by parts shows for each multi-index α,
(2πiµ)ασ(µ, ξ) =b Z
and
|(2πiµ)αbσ(µ, ξ)| ≤ cα,
uniformly in ξ. As a result, we obtain
supξ|bσ(µ, ξ)| ≤ CN(1 + |µ|)−N
for arbitrary N ≥ 0. Now Tσf (x) =
Z
σ(x, ξ) ˆf (ξ)e2πix·ξdξ = Z Z
b
σ(µ, ξ)e2πiµ·xf (ξ)eˆ 2πix·ξdξdµ
= Z (Tµf )(x)dµ, where (Tµf )(x) = e2πix·µ (Tbσ(µ,ξ)f )(x).
Since for each µ, Tbσ(µ,ξ) is a Fourier multiplier operator on the Fourier side, by Plancherel’s
theorem we have that
kTbσ(µ,ξ)f k2 ≤ sup ξ |bσ(µ, ξ)|k ˆf k2 = supξ |bσ(µ, ξ)|kf k2 (2.22). By (2.22), kTµk ≤ C N(1 + |µ|)−N, which yields kTσk ≤ CN Z (1 + |µ|)−Ndµ < ∞
if we choose N > n. Thus (2.21) is proved when σ has compact support in x.
The proof for general symbols needs to use the singular integral realization of the operator Tσ. That is, we shall write
Tσf (x) =
Z
k(x, z)f (x − z)dz (2.23)
where for each x, k(x, ·) is the distribution whose Fourier transform is the function σ(x, ·), formally,
σ(x, ξ) = Z
k(x, z)e−2πiz·ξdz.
Thus, Tσ can be interpreted as the convolution of the distribution k(x, ·) with the function
f ∈ S, evaluated at the point x. We first have the following estimate on k(x, z). |k(x, z)| ≤ CN|z|−N
for all |z| ≥ 1 and all N > 0, uniformly in x.
To see this, note that Tσ(f )(x) equals [k(x, ·) ∗ f ](x), where k(x, ·) is the
(−2πiz)αk(x, ·), with the distribution k(x, ·) thought of as acting on functions of z, equals
the inverse Fourier transform of ∂α
ξσ(x, ξ); by (2.17), ∂ξασ(x, ξ) is integrable in ξ whenever
|α| ≥ n + 1. This shows that k(x, ·) equals a function away from the origin, and that |z|N|k(x, z)| ≤ C
N for N > n, and from this, (2.24) follows.
We return to the proof of theorem 2.20, without assuming that σ(x, ξ) has compact support in x. To begin with, we will show that, for each x0 ∈ Rn,
Z |x−x0|≤1 |Tσf (x)|2dx ≤ CN Z Rn |f (x)|2 (1 + |x − x0|)Ndx (2.25) for all N≥ 0.
We prove (2.25) first when x0 = 0. To do this, we split f by f = f1 + f2, with f1
supported in B(0, 3), f2 supported outside B(0, 2), f1 and f2 smooth, and with |f1|, |f2| ≤
|f |. We fix η ∈ C∞
0 so that η = 1 in B(0, 1). Then ηTσ(f1) = Tησ(f1), and the symbol
η(x)σ(x, ξ) has compact support in x, so the previous result applies. Hence Z B(0,1) |Tσf1|2 ≤ Z Rn |Tησf1|2 ≤ C Z Rn |f1|2 ≤ C Z Rn |f |2 (2.26).
If x ∈ B(0, 1), since f2 is supported away from B(0, 2), the representation (2.23) holds:
Tσf2(x) =
Z
(B(0,2))c
k(x, x − z)f2(z)dz. (2.27)
Since |x − z| ≥ 1 when x ∈ B(0, 1) and z /∈ B(0, 2), using the estimate (2.24) yields |Tσf2(x)| ≤ CN
Z
(B(0,2))c
|z|−N|f (z)|dz.
By Schwartz inequality, we obtain, for N > n, Z
B(0,1)
|Tσf2(x)|2dx ≤ CN
Z
(1 + |x|)−N|f (x)|2dx. (2.28)
Now Tσ(f ) = Tσ(f1) + Tσ(f2), and combining (2.26) and (2.28) shows (2.25) when x0 = 0.
The passage to (2.25) for general x0 can be achieved by noting that, while an individual
pseudo-differential operator is not(in general) translation-invariant, the class Sm, in fact,
is. To see this, let τh, h ∈ Rn, denote the unitary translation operator given by (τhf )(x) =
f (x − h). Then, τhTστ−h = Tσh, where σh(x, ξ) = σ(x − h, ξ). Note that the symbol σh
satisfies the same estimates that σ does, uniformly in h. Hence, (2.25) holds for x0 = 0,
with σ replaced by σh, with a bound independent of h. If we set h = x0, we see that (2.25)
is established. Finally, it is only a matter of integrating (2.25) with respect to x0, choosing
By combining the L2 result with the Calderon-Zygmund real variable method, we can
also prove the Lp boundedness of these operators. However, we need to realize
pseudo-differential operators in the class S0 as singular integrals. To be precise, we shall prove
the following result.
Proposition 2.29: Suppose σ ∈ Sm. Then the kernel k(x, z) is in C∞(Rn × (Rn\{0})),
and satisfies
|∂xβ∂zαk(x, z)| ≤ Cα,β,N|z|−n−m−|α|−N, z 6= 0, (2.30)
for all multi-indices α and β, and all N ≥ 0 so that n + m + |α| + N > 0.
Proof: The proof uses the so-called dyadic decomposition. We begin by fixing η ∈ C∞ 0
with the properties that η(ξ) = 1 for |ξ| ≤ 1, and η(ξ) = 0 for |ξ| ≥ 2. We also define another function δ, by δ(ξ) = η(ξ) − η(2ξ). Then we have the following two partitions of unity of the ξ − space :
1 = η(ξ) +
∞
X
j=1
δ(2−jξ), (2.31)
for all ξ, and
1 = ∞ X j=−∞ δ(2−jξ), (2.32) for all ξ 6= 0.
It is worth to pointing out that for each ξ there are at most two nonzero terms in the sums (2.31) and (2.32).
Let φ be the inverse Fourier transform of η, i.e., bφ(ξ) = η(ξ). Then φ ∈ SandR φdx = 1. Define ψ by bψ(ξ) = δ(ξ). Then R ψdx = 0. Writing φt(x) = t−nφ(xt) and ψj = ψ2−j, we
then have ψj = φj− φj−1, while d(φj)(ξ) = η(2−jξ) and d(ψj)(ξ) = δ(2−jξ). We now define
the operator Sj by Sj(f ) = f ∗ φj and 4j(f ) = Sj(f ) − Sj−1(f ) = f ∗ ψj. In parallel with
(2.31) and (2.32), we have the operator identities I = S0+ ∞ X j=1 4j (2.33) and I = ∞ X j=−∞ 4j. (2.34)
Note that if f is a tempered distribution, Sj(f ) is well defined, and
S0(f ) + N
X
j=1
4j(f ) = SN(f ) → f,
as N → ∞, in the sense of distribution. However, it is not true that SM(f ) → 0 as
M → −∞, for arbitrary f . It fails when f = 1. Thus (2.34) holds only under some restriction on f .
We now return to the operator Tσ. Using (2.33), we write Tσ = T S0+ ∞ X j=1 T 4j = ∞ X j=0 Tσj,
where σ0(x, ξ) = σ(x, ξ)η(ξ), and σj(x, ξ) = σ(x, ξ)δ(2−jξ) for j ≥ 1.
Each of the pseudo-differential operators Tσj will be written in its singular integral
form by
Tσjf (x) =
Z
kj(x, z)f (x − z)dz.
Since σj have compact ξ − support and are smooth, the kernel kj will also be smooth, and
the integrals above will converge for all x. The kernels kj are given by
kj(x, z) =
Z
σj(x, ξ)e2πiξ·zdz.
We claim the following estimate: If σ ∈ Sm, then
|∂xβ∂zαkj(x, z)| ≤ Cα,β,N|z|−N2j[n+m−N +|α|], (2.35)
for all multi-indices α, β, and N ≥ 0, where Cα,β,N is independent of j ≥ 0. In fact, observe
that
(−2πiz)γ∂xβ∂zαkj(x, z) ≤
Z
∂ξγ[(2πiξ)α∂xβσj(x, ξ)]e2πiξ·zdξ.
Noting that the integrand is supported in 2j−1 ≤ |ξ| ≤ 2j+1 and estimates on σ(x, ξ) and
δ(2−jξ), we obtain that when |γ| = M,
|zγ∂xβ∂zαkj(x, z)| ≤ Cα,β,N2j[n+m−M +|α|].
Taking the supremum over all γ with |γ| = M, gives (2.35). Since k(x, z) = P∞
j=0
kj(x, z), it
suffices to show P∞
j=0
|∂β
x∂zαkj(x, z)| satisfies the estimate given by the right side of (2.29).
Consider the case when 0 < |z| ≤ 1, first. We break the above sum into two parts: the first where 2j ≤ |z|, the second where 2j > |z|. For the first sum we use the estimate
(2.35) with M = 0, which is then majorized by a multiple of P
2j≤|z|
2j(n+m+|α|). This in
turn is O(|z|−n−m−|α|) when n + m + |α| > 0, or O(log(|z|−1) + 1) when n + m + |α| ≤ 0. In
either case we get the estimate O(|z|−n−m−|α|−M), under restriction that |z| ≤ 1, M ≥ 0
and n + m + |α| + M ≥ 0. Next, for the second sum, we choose M > n + m + |α|. By (2.35), we get the estimate O(|z|−M) P
2j>|z|−1
2j[n+m+|α|−M ] = O(|z|−n−m−|α|). The last term is
Finally, when |z| ≥ 1, and if M > n + m + |α| + N , then (2.35) shows that the sum is majorized by O(|z|−M), which is O(|z|−n−m−|α|−N) for every N , since |z| ≥ 1. The proof
of the proposition is therefore concluded.
Now the L2 boundeness and the estimates of kernel of T
σ allow us to apply the
Calderon-Zygmund real variable method, and hence, the Lp, 1 < p < ∞, boundedness
follows.
As mentioned before, after Calderon and Zygmund introduced the second generation of Calderon-Zygmund operators, they seek to compose the operators in order to obtain an algebra with a precise symbolic calculus. To get this, it remains to be seen whether the operators in question can be defined by symbols satisfying simple conditions of regularity and rate of growth at infinity. There are two such algebras A∞ and B∞. The algebra
A∞ is the set of all operators T , where the symbols of T and its adjoint T∗ satisfy the
following illicit estimates: |∂β
x∂ξασ(x, ξ)| ≤ Cα,β|ξ|−(|α|−|β|). (2.36)
The algebra B∞ is the set of all operators T , where the symbols of T and its adjoint T∗
satisfy the following illicit estimates:
|∂xβ∂ξασ(x, ξ)| ≤ Cα,β(1 + |ξ|)−(|β|−|α|). (2.37)
Several years later(1965), Calderon took another look at the problem of the symbolic calculus when he sought conditions of minimal regularity with respect to x. In applications to partial differential equations, the regularity with respect to x is given by that of the coefficients aα(x) of the differential operators
P
aα(x)∂α. This problem reduces to the
study of the commutators [A, T ], where A is the operator of pointwise multiplication by a function a(x), and where T = ( ∂
∂x1)T1 + · · · + (
∂
∂xn)Tn, the Tj, 1 ≤ j ≤ n, being first
generation Calderon-Zygmund operators.
In dimension 1, T is replaced by DH, where D = −i d
dx and H is the Hilbert transform.
In 1965, Calderon showed that the commutator [A, DH] was bounded on L2(R) if and only
if the function a(x) was Lipschitz, that is, there is a constant C such that |a(x) − a(y)| ≤ C|x − y|, for all x, y ∈ R. It is easy to see that this condition is necessary. The reverse implication is deep, and Calderon’s proof relies on the characterization, established by Calderon for this purpose, of complex Hardy space H1 by the integrability of Lusin’s area
function. This remarkable result lead to the third generation Calderon-Zygmund operators which do not belong to the pseudo-differential operators. A further operator belonging to Calderon’s program is related the classical method of using a double layer potential to solve the Dirichlet and the Neumann problem in a Lipschitz domain. The operator involved, similar to the commutator [A, DH], is given in local coordinates by
T f (x) = 1 ωn p.v. Z K(x, y)f (y)dy where
K(x, y) = a(x) − a(y) − (x, y) · 5a(y) (|x − y|2+ (a(x) − a(y))2)(n+1)/2,
and f ∈ L2(Rn). The Lipschitz domain is defined (locally) by t > a(x), where x ∈ Rn, t ∈
R, and the function a(x) is Lipschitz.
If n = 1, this kernel is precisely the real part of the kernel of the Cauchy integral on a Lipschitz curve. All these new operators are non-convolution operators. The method to obtain the L2 boundedness for first and second generation Calderon-Zygmund operators
Lecture 3. Littlewood-Paley Theory and Function Spaces
There is a number of ways to set up the Littlewood-Paley theory on Rn. One standard
way is as follows. Let φ(ξ) be a real radial bump function supported on {ξ ∈ Rn : |ξ| ≤ 2}
which equals 1 on {ξ ∈ Rn : |ξ| ≤ 1} Let ψ be the function: ψ(ξ) = φ(ξ) − φ(2ξ). Thus ψ
is a bump function supported on {ξ ∈ Rn: 1
2 ≤ |ξ| ≤ 2}. By construction we have
X
k
ψ(ξ/2k) = 1
for all ξ 6= 0. Thus we can partition unity into the function ψ(ξ/2k) for integers k, each of
which is supported on an annuls of the form |ξ| ∼ 2k. We now define the Littlewood-Paley
projection operators Qk and Pk by
d
(Qkf )(ξ) = ψ(ξ/2k) ˆf (ξ) (3.1)
d
(Pkf )(ξ) = φ(ξ/2k) ˆf (ξ). (3.2)
Informally, Qk is a frequency projection to time annuls {|ξ| ∼ 2k}, while Pk is a frequency
projection to the ball {|ξ| ≤ 2k}. Observe that Q
k = Pk − Pk−1. Also, if f ∈ L2, then
Pk(f ) → 0 in L2 as k → −∞, and Pk(f ) → f in L2 as k → ∞ , which follows from
the Plancherel theorem. By telescoping the series, we thus have the Littlewood-Paley decomposition
f =X
k
Qk(f ) (3.3)
for all f ∈ L2, where the series converges in the L2 norm.
We are now interested in how the Lp behavior of the Littlewood-Paley pieces Q kf
relate to the behavior of f . First, we write Pkf (x) = f ∗ (2nkφ(2k·)) =
Z
f (x + 2−ky)φ(y)dy. (3.4) Note that φ is a Schwartz function with the total mass R φ(y)dy = φ(0) = 1. Thus the function Pkf is an average of f localized to physical scales≤ 2−k. In particular, we expect
Pkf to be essentially constant at scales < 2−k. What does a function Qkf look like? Since
Qkf = Pk+2Qkf, we see from (3.4) that
Qkf (x) =
Z
Qkf (x + 2−2−ky)φ(y)dy. (3.5)
Thus, Qkf is essentially constant at physical scales< 2−k. On the other hand, we have
Pk−2Qk1f = 0, so from (3.3),
Z
for all x ∈ Rn.
This roughly assert that Qkf has mean zero at scales ≤ 2−k.
From (3.3) and the Minkowski’s inequality, for 1 ≤ p ≤ ∞, we have
kPkf kp ≤
Z
kf (x + 2−ky)kp|φ(y)|dy ≤ Ckf kp. (3.6)
Thus, Pkf does not get any bigger that f itself as measured in Lp, or in any translation
invariant Banach space. Similarly, kQkf kp ≤ Ckf kp. On the other hand, we have f =
P
k
Qkf, so by the triangle inequality, we obtain the cheap Littlewood-Paley inequality
sup
k kQkf kp ≤ Ckf kp ≤ C
X
k
kQkf kp. (3.7)
As the name suggests, the cheap Littlewood-Paley inequality is not the sharpest statement one can make connecting the Lp norms of Q
kf with those of f . By using the Fourier
transform, when p = 2, we get
kf k2 ∼ ( X k kQkf k22) 1 2. (3.8)
In fact, to see this, square both sizes and take Plancherel to obtain k ˆf k22 ∼X
k
k[ψ(·/2k) ˆf (·)k22.
Observe that for each ξ 6= 0, there are only two values of ψ(ξ/2k) which do not vanish,
and these two add up to 1. We can rewrite (3.8) as kf k2 ∼ k( X k |Qkf |2) 1 2k2. (3.9) The quantity (P k |Qkf |2) 1
2 is known as the Littlewood-Paley square function.
Now define Sf for the vector-valued function by Sf (x) = {Qkf }k, and |Sf | =
(P
k
|Qkf |2)
1
2 is the `2 norm of Sf .
Theorem 3.10: For 1 < p < ∞, then kSf kp ∼ kf kp with the implicit constant depending
on p.
The proof of theorem 3.10 follows from the Calderon-Zygmund real variable method. In fact, we have the L2 result, and it suffices to see that S is a vector-valued
Calderon-Zygmund operator with vector-valued kernel K(x) = (2nkψ(2kx))
k. Since ψ is a Schwartz
first generation Calderon-Zygmund operators. Now the L2 result implies the Lp, 1 < p <
∞, results. By duality we also have kS∗(f
k)kp ≤ Ck(fk)kp. Thus, kX k Qkfkkp ≤ Ck( X k |fk|2) 1 2kp. Similarly, kX k e Qkfkkp ≤ Ck( X k |fk|2) 1 2kp, where eQk = Pk+2− Pk−2.
We apply this with fk= Qkf, since eQkQk = Qk, we obtain
kf kp ≤ k( X k |Qkf |2) 1 2kp.
As an application, we give a proof of the Hormander-Mikhlin multiplier theorem. Theorem 3.11: Let m(ξ) be a multiplier such that
|∂αm(ξ)| ≤ C|ξ||α| (3.12)
for all |α| ≥ 0, where the constant C depends on α. Let Tm be the Fourier multiplier with
symbol m : d(Tmf )(ξ) = m(ξ) ˆf (ξ). Then Tm is bounded on Lp, 1 < p < ∞.
Proof: We have Tm = X k,k0 QkTmQk0 = X k e QkQ¯k where eQk = QkTm and ¯Qk = P k−2<k0<k+2
Qk. From (3.12) we see that eQk and ¯Qk are
smooth frequency localization operators to the annuls {|ξ| ∼ 2k}. Then the theorem 3.11
follows from the Littlewood-Paley Lp inequality by composing eQ
k and ¯Qk.
A typical consequence of the Hormander-Mikhlin multiplier theorem is the estimate k∂xi∂xjf kp ≤ Ck 4 f kp
for all 1 < p < ∞, where 4 = Pn
i=1
∂2
xi is the Laplacian on R
n. This follows because
d
(∂xi∂xjf )(ξ) =
ξiξj
|ξ|2(4f )(ξ), and the symbol m(ξ) =d
ξiξj
|ξ|2 satisfies the condition (3.12).
The Littlewood-Paley Lp, 1 < p < ∞, inequality suggest that one can consider the
similar inequality for 0 < p ≤ 1. However, this fails even when p = 1. Notice that ψ defined in the Littlewood-Paley S function, is in S, the Schwartz test function space, so for any f ∈ S0, the temperate distribution space, Q
kf is well defined. This means that for any
f ∈ S0, Sf is well defined. Now we introduce the hardy space Hp as follows.
Definition 3.13: For 0 < p < ∞, Hp = {f ∈ S0 : Sf ∈ Lp} and if f ∈ Hp, the norm of f
It is easy to see that when 1 < p < ∞, Hp = Lp, by the Littlewood-Paley Lpinequality.
For 0 < p ≤ 1, Hp is a new space which is different from Lp. The best way to see this is
to get the so-called atomic decomposition of Hp. Here we only consider p = 1. To do this,
let S∞ = {f ∈ S :
R
f (x)xαdx = 0, for all |α| ≥ 0}. We then have
Theorem 3.14: Suppose f ∈ S∞ and
P
k
| bψ(2kξ)|2 = 1, where ψ is a bump function
supported on the annuls {|x| ∼ 2−k}. Then
f =X
k
QkQkf,
where the series converges in the topology of S. By a duality argument, for f ∈ S0 and g ∈ S
∞ we have <X k QkQkf, g >=< f, X k QkQkg >=< f, g > .
Hence, for any f ∈ S0, f =P k
QkQkf in (S∞)0 ≈ S0/P, S0 modulo polynomials.
Definition 3.15: A function a(x) is said to be an atom if a(x) satisfies (i) Supp a ⊆ Q, a cube in Rn;
(ii) kak2 ≤ |Q|−
1 2;
(iii) R a(x)dx = 0.
Theorem 3.15: f ∈ H1 if and only if f =P k
λkak, where ak are atoms and
P k |λk| < ∞. Moreover, kf kH1 ≈ inf{ X k |λk| : for allf = X k λkak}.
As mentioned before, Hp is well defined for 0 < p < ∞, but not for p = ∞. Next, we
consider the replacement of Hp when p = ∞.
Definition 3.16: For f ∈ L1
loc(Rn) we let kf k∗ = supQmQ|f − mQf |, where mQf is
the average of f over Q, and we define the space BMO(Rn)( functions of bounded mean
oscillation) to consist of those functions f such that kf k∗ < ∞.
(BM O(Rn) is a semi-normed vector space, with the seminorm vanishing on the constant
functions. If we let C denote the vector space of constant functions, then the quotient of BMO by C is a Banach space, which we also denote by BMO. This space BMO was originally introduced by John and Nirenberg. They proved the following John-Nirenberg inequality.
Theorem 3.17: There exist two positive constants λ > 0 and C > 0 such that for any f ∈ BM O,
sup
Q mQ(exp(
λ
Proof: We assume that f is bounded, so that the above supremum makes sense for all λ, and we shall prove the theorem by finding a bound independent of kf k∞.
Let Q0 be a fixed cube and Q some dyadic cube. Recall that eQ is the unique dyadic
cube which contains Q and lies in the previous generation. It is easy to see that
|mQf − meQf | ≤ 2nkf k∗. (3.19)
Consider now the Calderon-Zygmund decomposition of the function (f − mQ0f )χQ0 for
λ = 2kf k∗. This yields a collection of dyadic cubes Qi, maximal with respect to inclusion,
satisfying
mQi|(f − mQ0f )χQ0| > 2kf k∗ (3.20)
and
|(f − mQ0f )χQ0| ≤ 2kf k∗ (3.21)
on (∪Qi)c.
Clearly, Qi ⊆ Q0 for each i, and | ∪ Qi| ≤ k(f −m2kf kQ0f )χ∗ Q0k1 ≤ |Q20|.
Since the Qi’s are maximal. meQ
i|(f − mQ0f | ≤ 2kf k∗ , and (3.19) gives
|mQif − mQ0f | ≤ (2
n+ 2)kf k ∗.
Let X(λ) = supQmQexp(kf kλ∗|f − mQf |, which is finite since we re assuming that f is
bounded. We obtain mQ0(exp( λ kf k∗ |f − mQf |) ≤ 1 |Q0| Z Q0\∪Qi e2λdx + 1 |Q0| X i |Qi| |Qi| [ Z Qi exp( λ kf k∗ |f − mQif |)dxe (2n+2)λ ] ≤ e2λ+ 1 2[exp(2 n+ 2)]X(λ).
From taking the supremum over all cube Q0 it follows that X(λ)[1 −12exp(2n+ 2)] ≤ e2λ,
which implies that X(λ) ≤ C, if λ is small enough, which proves the theorem.
A consequence of the theorem, which in fact is equivalent to it, is the following: There exist positive constant λ and C such that for every cube Q and every t > 0,
|{x ∈ Q : |f (x) − mQf | > tkf k∗} ≤ Ce−λt|Q|. (3.22)
For 1 ≤ p < ∞, then kf kp,∗ = supQ[mQ|f − mQf |p]
1
p are equivalent.
Now we are ready to prove the duality of H1 and BMO. We shall see that each
continuous linear functional on H1 can be realized as a mapping
`(g) = Z
when suitably defined, where f is a function in BMO.
For general f ∈ BM O and g ∈ H1, the integral in (3.23) does not converge absolutely.
For this reason, we take g ∈ H1 has finite atomic decomposition. We denote this subspace
by H1
a which is dense in H1.
Theorem 3.24: (a) Suppose f ∈ BM O. Then the linear functional given by (3.23), initially defined on H1
a, has a unique bounded extension to H1 and satisfies
k`k ≤ ckf k∗.
(b) Conversely, every continuous linear functional on H1 can be realized as above, with
f ∈ BM O, and with
kf k∗ ≤ ck`k.
Proof: To see (a), note that if a is an atom supported on Q, then
| Z f (x)a(x)dx| = | Z [f (x) − mQf ]a(x)dx| ≤ |Q| 1 2kf k2,∗kak2 ≤ ckf k∗. Thus, if g ∈ H1 a, then g = P k
λkak with the sum having a finete terms and ak are atoms.
So | Z f (x)g(x)dx| = |X k λk Z [f (x) − mQkf ]ak(x)dx| ≤X k |λkkQk| 1 2kf k2,∗kak2 ≤ ckf k∗ X k |λk|.
To show (b), fix a cube Q and let L2
Q be the space of all square integrable functions
supported on Q. Let L2
Q,0 = {f ∈ L2Q :
R
f (x)dx = 0} . Note that every g ∈ L2
Q,0 is a
multiple of an atom and kgkH1 ≤ c|Q| 1
2kgk2. Thus if is a given linear functional on H1
with the norm≤ 1, then extends to a linear functional on L2
Q,0 with norm at most c|Q|
1 2
. By the Riesz representation theorem for the Hilbert space L2
Q,0, there exists an element
FQ ∈ L2 Q,0 so that `(g) = Z FQ(x)g(x)dx, (3.25) if g ∈ L2 Q,0, with ( Z |FQ(x)|dx)1 2 ≤ c|Q|12. (3.26)
Hence for each Q, we get such a function FQ. We want to have a single function f so that,
on each Q, f differs from FQ by a constant. To construct this f , observe that if Q
1 ⊆ Q2,
then FQ1 − FQ2 is constant on Q
1. Indeed, both FQ1 and FQ2 give the same functional
on L2
Q1,0, so they must differ by a constant on Q1. We can modify F
Q, replacing it with
fQ = FQ + c
unit cube centered at the origin. It follows that fQ1 = fQ2 on Q
1, if Q1 ⊆ Q2. Finally, we
define f on Rn by taking f (x) = fQ(x) for x ∈ Q. Observe that
1 |Q| Z Q |f (x) − cQ|dx ≤ ( 1 |Q| Z Q |f (x) − cQ|2dx) 1 2 = ( 1 |Q| Z Q |FQ|2dx)12 ≤ c,
which shows f ∈ BM O with kf k∗ ≤ c. Also, by (3.25), `(g) =
R
FQ(x)g(x)dx, if g ∈ L2 Q,0,
for some Q, in particular, this representation holds for all g ∈ H1
a. The converse (b) of the
theorem is proved.
We now discuss the relationship between BMO and Carleson measures and Littlewood-Paley square functions.
Definition 3.27: A Borel measure dµ on Rn+1+ is said to be a Carleson measure if
sup Q 1 |Q| Z T (Q) |dµ| ≤ C < ∞. (3.28)
If dµ is a Carleson measure, we denote kdµkC, the Carleson norm of dµ, by the smallest
constant C in (3.28).
Theroem 3.29: Suppose φ ∈ S withR φ(x)dx = 1. Then dµ is a Carleson measure if and
only if Z
R+n+1
|φt ∗ f (x)|2dµ ≤ Ckf k22. (3.30)
Theorem 3.31: Let ψ ∈ S with
∞
R
0
| bψ(tξ)|2 dt
t = 1 for all ξ 6= 0. Then f ∈ BM O if and
only if |f ∗ ψt(x)|2 dxdtt is a Carleson measure.
Littlewood-Paley theory allows us to consider a large range of classical function spaces within a single framework. The general classes of spaces we will define are the homogeneous Besov spaces ˙Bα,q
p and Triebel-Lizorkin spaces ˙Fpα,q as well as their inhomogeneous analogs.
Let us choose φ ∈ S so that Supp bφ ⊆ {ξ : 12 ≤ |ξ| ≤ 2} and |bφ(ξ)| ≥ c > 0 if
3
5 ≤ |ξ| ≤ 53. For α ∈ R, p 6= ∞, 0 < p, q ≤ ∞ and f ∈ S0 we define
kf kF˙α,q p = k{ X k (2kα|φk∗ f |)q} 1 qkp, (3.32)
and, for the same indices, and including p = ∞, kf kB˙α,q p = { X k (2kαkφk∗ f kp)q} 1 q, (3.33) where φk(x) = 2knφ(2kx).
Note that φk ∗ f is a smooth function when φ ∈ S and f ∈ S0. Also kf kF˙α,q
p = 0
or kf kB˙α,q
p = 0 if and only if φk∗ f is the zero function for all k. But this is equivalent
to having ˆf (ξ)bφ(2kξ) be zero for all k. because of the conditions on bφ, this, in turn, is
equivalent to Supp ˆf = {0}. Finally, this means that the distribution f is a polynomial. Thus, we work modulo polynomials when considering (3.32) and (3.33); that is,. f ∈ S0/P
in these equalities, where P denote the class of polynomials on Rn. In particular, we define
˙ Fα,q
p and ˙Bpα,q to be the set of all such f for which the expression (3.32) and (3.33) is finite.
It is not difficult to see that these expressions are norms when 1 ≤ p, q ≤ ∞ and quasinorms in general. We are not include the case p = ∞ in the definition of the Triebel-Lizorkin spaces. In this case, the L∞ norm should be replaced by a Carleson measure condition.
The spaces defined by the finiteness of the these norms are called the homogeneous Triebel-Lizorkin and Besov spaces, respectively. The inhomogeneous versions of these spaces are obtained by adding the term kΦ ∗ f kp to variants of the above expressions, where Φ ∈ S,
Supp Φ ⊆ {ξ : |ξ| ≤ 2} and |bΦ(ξ)| ≥ c > 0 if |ξ| ≤ 5
3. The variants in question are as in
(3.32) and (3.33) withP
k
replaced by P
k≥1
. These spaces are denoted by Fα,q
p and Bpα,q and
they are spaces of tempered distributions; the necessity of considering such distributions modulo polynomials disappears since bΦ(0) 6= 0.
By the results mentioned above by the Littlewood-Paley theory, we obtain the follow-ing identifications: Lp ∼ ˙Fp0,2 when 1 < p < ∞; Hp ∼ ˙Fp0,2 when 0 < p ≤ 1; BM O ∼ ˙F0,2 ∞ when ˙F0,2
∞ is defined by the Carleson measure. We also can show Lpα ∼ Fpα,2 and ˙Lpα ∼ ˙Fpα,2
when α > 0 and 1 < p < ∞;
Λα ∼ F∞α,∞
and
˙Λα ∼ ˙F∞α,∞
when α > 0. Suppose eφ and eΦ are two other functions satisfying the properties of φ and Φ announced above. One can show that replacing φ and Φ with eφ and eΦ in the definitions yields the same spaces with equivalent norms.
Lecture 4. Third generation Calderon-Zygmund operators and the T1 theorem
The pseudo-differential calculus is like that mythological bird. Its first birth was at the end of the 1930s, the founding fathers being Giraud and Marcinkiewicz. The second birth took place at the end of the 1950s, as we discussed in the Lecture 2, and it clearly benefited from the theory of distributions, developed by Schwartz during the 1940s. The third birth is the one to claim in this lecture. In order to deal with linear partial differential equations having coefficients which are only slightly regular and. in order to approach the problem of the regularity of solutions of non-linear partial differential equations, Calderon decided to make the pseudo-differential calculus include the operators A of pointwise multiplication by functions a(x) which are only slightly regular with respect to x. Of course, Calderon wanted to keep what had been gained during the previous decades: the classical pseudo-differential operators. An important step was taken in 1965 when Calderon proved that the commutator [A, DH] = ADH − DHA between the pointwise multiplication operator A by the function a(x) and the operator DH, where D = −i d
dx and H is the Hilber transform,
is bounded on L2 if a is a Lipschitz function, i.e., |a(x) − a(y)| ≤ c|x − y| for x, y ∈ R. We
note that the commutator [A, DH](f )(x) is given by [A, DH](f )(x) = p.v.
Z
A(x) − A(y)
(x − y)2 f (y)dy. (4.1)
This operator is called Calderon’s first commutator. To see why this operator plays an important role in the study of linear partial differential operators with variable coefficients, we follow Calderon’s 1978 International Congress lecture. Let L be an operator defined by
Lf (x) = m X j−0 aj(x) djf dxj. (4.2)
As we did in lecture 2, by Fourier transform and Fourier inversion, Lf (x) = 1
2π
Z Xm j−0
aj(x)(iξ)jf (ξ)eˆ ixξdξ. (4.3)
The idea behind pseudo-differential operators is to replace the function aj(x)(iξ)j by more
general functions σ(x, ξ) is such a way that the resulting class of operators is closed under composition, adjunction, and other basic operations. If we want this class of operators to be closed under composition, and, in particular, be able to freely compose linear differentila operators L, then it will only contain differential operators with infinitely differentiable coefficients, i.e., a ∈ C∞.
There is another algebra of operators, however, that can be used in the study of operators L as above with nonsmooth coefficients. Let Λ be the operator defined by
d
(Λf )(ξ) = ψ(ξ) ˆf (ξ) where ψ is an infinitely differentiable function with ψ(ξ) = |ξ| if |ξ| ≥ 1, and let
T f (x) = Z
where
Rf (x) = Z
r(x, ξ) ˆf (ξ)eixξdξ. (4.5)
Here q(x, ξ) = |ξ|−ma
m(x)(iξ)m and r(x, ξ) is defined by the relation
1 2π m X j=0 aj(x)(iξ)j = (q(x, ξ) + r(x, ξ))(ψ(ξ))m.
Now we can write
Lf = T Λmf.
It is easy to show that the operator R and Rdxd are bounded on L2, say, provided the
coefficients aj are bounded. The function q(x, ξ) is regular, homogeneous of degree 0 in ξ,
and bounded. The corresponding operator T can be generalized by allowing q(x, ξ) to be a general function with these three properties, and allowing R to be any operator such that R and Rdxd are bounded on L2. To avoid some pathologies it turns out to be necessary to
restrict the class slightly and assume, in addition, that q(x, ξ) is Lipschitz in x. The class of operators L, given by (4.6), with T in this more general class, at least contains the linear differential operators whose coefficients are bounded, and, for the highest terms, bounded and Lipschitz. Let A be the operator corresponding to multiplication by the Lipschitz function a(x), and let H be the Hilbert transform. Obviously, A is one of the operators T , and if we recall that
Hf (x) = c Z
sign(ξ) ˆf (ξ)eixξdξ, (4.7) then it becomes clear that H is in this class as well. To prove that the class of operators T as above is closed under composition, it is necessary to show that AH and HA are also of the same general type. For AH this is trivial. For HA, if we write HA = AH +(HA−AH), then it becomes clear that HA is also of the right type if (AH − HA)D is bounded on L2.
Now
(AH − HA)D = [A, HD] + HDA − HAD
and DA − AD is just multiplication by a0(x), which of course is bounded on L2 since a0
is a bounded function. Hence, HA belongs to the class if and only if [A, HD] is bounded on L2, and this is Calderon’s result since HD = DH. Now to show that the composition
of two general operators T in the class is still in the class can be reduced to the special cases we just considered. The fact that the class is closed under composition can be used to prove existence and uniqueness results, a priori estimates, etc. for partial differential equations.
There are also many other operators which are not convolution operators that arise naturally in analysis. Calderon’s kth commutator, for example, is given by
Ckf (x) = p.v. Z (a(x) − a(y) x − y ) k f (y) x − ydy, k ≥ 1. (4.9)
These operators are closely related to the boundary behavior of analytic functions given by Cauchy integrals f (z(x)) = 1 2πi Z 1 z(y) − z(x)f (z(y)z 0(y)dy (4.10)
on Lipschitz curve z(x) = x + ia(x), a0 ∈ L∞. Another nonconvolution operator is the
double layer potential associated with a domain Ω. In local coordinates this operator takes the form
T f (x) = p.v. 1 ωn
Z
Rn
a(x) − a(y) − (x − y) · 5a(y)
(|x − y|2+ (a(x) − a(y))2)n+12 f (y)dy, (4.11)
where ωn is the area of the unit sphere in Rn. To solve the Dirichlet problem in a Lipschitz
domain by the method of layer potentials, one needs to prove the boundedness of L2 of
the above operator with a Lipschitz.
We emphasize that while for convolution operators, boundedness on L2 is a simple
application of Plancherel’s theorem, the L2 − boundedness for non-convolution operators
like the ones above is highly nontrivial.
In 1978 Coifman and Meyer introduced the third generation Calderon-Zygmund op-erators. Let T be a continuous linear operator from the Schwartz class S of test functions to its dual S0. By the Schwartz kernel theorem there is a distribution K in S0 such that
(T f, g) = (K, g ⊗ f )
for all f, g ∈ S and here ( , ) denotes the distribution pairing, linear in each coordinate, rather than the pairing < , >, which is conjugate linear in the second coordinate, and g ⊗ f (x, y) = g(x)f (y). The distribution K is called the kernel of T .
Definition 4.12: We say that K is a Calderon-Zygmund kernel if its restriction to the set Ω = {(x, y) ∈ Rn× Rn: x 6= y} is a continuous function K(x, y) which satisfies
|K(x, y)| ≤ C 1 |x − y|n, (4.13) |K(x, y) − K(x0, y)| ≤ C |x − x 0|² |x − y|n+², (4.14) if |x − x0| ≤ 1 2|x − y|, |K(x, y) − K(x, y0)| ≤ C |y − y 0|² |x − y|n+², (4.15) if |y − y0| ≤ 1 2|x − y|,
for some constant C and some ² in (0, 1]. We call T a Calderon-Zygmund singular integral operator, and write T ∈ CZSIO, or T ∈ CZSIO(²), if the kernel of T satisfies these conditions. In particular, if T ∈ CZSIO, then
(T f, g) = Z Z
