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:

X ∂αu Lu(x) = a . (1.1) α ∂xα α

By taking the Fourier transform, X d α (Lu)(ξ) = aα(−2πiξ) uˆ(ξ). (1.2) α

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

(dT f)(ξ) = m(ξ)fˆ(ξ). (1.4)

(1.2) shows that any classical diﬀerential operator is a Fourier multiplier. 2 2 Example 1: Suppose f ∈ L (R) and F is an analytic extension of f on R+ given by Z Z i 1 i (x − t) − iy F (x + iy) = − f(t)dt = − f(t)dt π x + iy − t π (x − t)2 + y2 Z Z 1 y i (x − t) = f(t)dt − f(t)dt. (1.5) π (x − t)2 + y2 π (x − t)2 + y2 R Letting y → 0, then 1 y f(t)dt → f(x) for a. e. x, and, in general, the second π (x−t)2+y2 R 1 term above has no limit. However, one can show p.v x−t f(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 Z 1 f(t) H(f)(x) = − dt. (1.7) π x − t

1 Example 2: Consider the Laplacian

Xn ∂2u 4u = . (1.8) ∂x2 j=1 j

By taking the Fourier transform,

Xn d 2 2 (− 4 u)(ξ) = (4π|ξj|) uˆ(ξ) = 4π|ξ| uˆ(ξ). j=1

Deﬁne the Riesz transforms Rj, 1 ≤ j ≤ n, by

ξ (Rdf)(ξ) = j fˆ(ξ). (1.9) j |ξ|

Then d2 ∂ u d ( )(ξ) = −4πξiξjuˆ(ξ) = (RiRj 4 u)(ξ). (1.10) ∂xi∂xj Thus, ∂2u = RiRj 4 u. (1.11) ∂xi∂xj 2 Since, it is easy to see that H and Rj, 1 ≤ j ≤ n, are bounded on L , so we obtain

klimy→0F (x + iy)k2 ≤ Ckfk2 (1.12) and ∂2u k k2 = kRiRj 4 uk2 ≤ Ck 4 uk2. (1.13) ∂xi∂xj The Hilbert transform, by taking the Fourier transform, can be written as

Hd(f)(ξ) = −isign(ξ)fˆ(ξ) (1.14) and the Riesz transform, by taking the inverse Fourier transform, can be written as Z y R (f)(x) = c p.v. j f(x − y)dy, 1 ≤ j ≤ n. (1.15) j n |y|1+n

In 1952, Calderon and Zygmund introduced the following ﬁrst generation of singular integral operators: Deﬁnition 1.16: Z Ω(y) T (f)(x) = p.v. f(x − y)dy, |y|n

2 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 Z Z Ω(y) Ω(y) p.v. f(y)dy = lim f(y)dy = |y|n ²→0 |y|n |y|>² Z Z Ω(y) Ω(y) [f(y) − f(0)]dy + f(y)dy (1.20) |y|n |y|n |y|<1 |y|≥1 for all f ∈ S since Ω has zero average. It is then easy to see that both integrals above converge. R Ω(y) Remark 1.21: A necessary condition for p.v. |y|n f(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, Z Z Ω(y) Ω(y) T (f)(x) = lim dy + f(x − y)dy. (1.21) ²→0 |y|n |y|n ²<|y|<1 |y|≥1 R 1 The second integral is convergent but the first equals lim²→0 Ω(y)dσ(y)log( ² ). Thus, Sn−1 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 ≤ Ckfkp. (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, Z 1 π m(ξ) = Kˆ (ξ) = Ω(t)[log( ) − i sign(t · ξ)]dσ(t). (1.24) |t · ξ| 2 Sn−1

3 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, Z Ω(y) m(ξ) = lim e−2πiy·ξdy ²→0 |y|n 1 ²<|y|< ²

1 Z Z1 Z² dt dt = lim Ω(y)[ (e−2πity·ξ − 1) + e−2πity·ξ ]dσ(y) ²→0 t t Sn−1 ² 1

1 Z Z1 Z² dt dt = lim Ω(y)[ (cos(2πty · ξ) − 1) + cos(2πty · ξ ]dσ(y) ²→0 t t Sn−1 ² 1

1 Z Z² dt −ilim Ω(y) sin(2πty · ξ) dσ(y) ²→0 t Sn−1 ² Making the change of variables s = 2πty · ξ and assume y · ξ 6= 0, the second term above will be

1 Z Z² Z Z∞ dt sin(s) lim Ω(y) sign(2πty · ξ) dσ(y) = Ω(y) sign(y · ξ) dsdσ(y) ²→0 t s Sn−1 ² Sn−1 0 Z π = Ω(t) sign(t · ξ)dσ(t). 2 Sn−1 The ﬁrst, after the change of variables, will be

1 Z Z1 Z² dt dt lim Ω(y)[ (cos(2πy · ξ) − 1) + cos(2πy · ξ ]dσ(y) = ²→0 t t Sn−1 ² 1

2π|y·ξ| 1 Z Z1 Z ² 2πZ|y·ξ| ds ds ds = Ω(y)[ (cos(s) − 1) + cos(s) − ]dσ(y) s s s Sn−1 2π|y·ξ|² 1 1 Z 1 = Ω(t)[log( )dσ(t) |t · ξ| Sn−1 since Ω has zero average. Now applying the Plancheral yields

d ˆ ˆ kT (f)k2 = k(T (f)k2 = kKˆ fk2 ≤ Ckfk2 = Ckfk2.

4 Step 2: We show T is of week type (1, 1): There is a constant C such that

kfk |{x ∈ Rn : |T (f)(x) > λ}| ≤ C 1 (1.25) λ 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; 1 | ∪ Q | ≤ kfk ; (1.28) j λ 1 Z 1 λ < f(x)dx ≤ 2nλ. (1.29) |Qj| Qj

The proof of this decompositionR is to use the so-calledR stopping time argument. First, 1 1 n choose a large cube Q so that |Q| f(x)dx ≤ |Q| f(x)dx ≤ λ. Then divide Q to 2 equal Q R 0 1 subcubes Q . Now we use the stopping time argument as follows: if |Q0| f(x)dx > λ, we Q0 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 thatR there is a sequence {Qn} 1 with|Qn| → 0 as n → ∞, so that x∈ Qn for all n and f(x)dx ≤ λ which shows |Qn| R Qn R 1 1 (1.27). To see (1.29), notice that if λ < f(x)dx then λ ≥ n f(x)dx which |Qj | |2 Qj | n R Qj R 2 Qj 1 1 yields (1.29). Finally, from λ < f(x)dx we obtain |Qj| < f(x)dx. Summing up |Qj | λ Qj Qj shows Z 1 X 1 | ∪ Q | ≤ f(x)dx ≤ kfk . j λ λ 1 j Qj

Now we apply the Calderon-Zygmund decomposition toR show T is of the week type (1, 1 1). Deﬁne g(x) = f(x) for x∈ / ∪Qj and g(x) = f(x)dx for x ∈ Qj, and b(x) = |Qj | Qj f(x) − g(x). Since T is bounded on L2, so Z λ kT (g)k2 kgk2 1 |{x ∈ Rn : T (g)(x) > }| ≤ C 2 ≤ C 2 ≤ C [ |f(x)|2dx 2 λ2 λ2 λ2 c (∪Qj ) X Z Z Z 1 1 n 1 + f(y)dydx] ≤ C [ |f(x)|dx + 2 | ∪ Qj|] ≤ C kfk1. |Qj| λ λ j c Qj Qj (∪Qj )

5 To estimate T (b)(x), it is easy to see

λ λ |{x ∈ Rn : |T (b)(x)| > }| ≤ | ∪ 2Q | + |{x ∈ (∪2Q )c : |T (b)(x)| > }|. 2 j j 2

c λ 1 By (1.28), it suﬃces to show |{x ∈ (R∪2Qj) : |T (b)(x)| > 2 }| ≤ C λ kfk1. Rewrite b(x) = P 1 2 bj(x), where bj(x) = [f(x) − f(x)dx]χQ (x) and the sum converges in L , as well |Qj | j j Qj c as pointwise. Ifx ∈ (∪2Qj) , then, by the fact that bj has zero average, Z Z

T (bj)(x) = K(x − y)bj(y)dy = [K(x − y) − K(x − xQj )]bj(y)dy

c where xQj is the center of Qj. Thus, by the fact thatx ∈ (∪2Qj) and the smoothness of K, Z (Qj) |T (bj)(x)| ≤ C n+1 |bj(y)|dy |x − xQj | Qj We now estimate the integral

We apply Fubini’sP theorem to change the order of integration in each of the double integrals and it gives C kbjk1. This, in turn, can be estimated by using (1.28) and (1.29). So we j get Z

|T (b)(x)|dx ≤ Ckfk1

c (∪2Qj )

c λ kfk1 which shows |{x ∈ (∪2Qj) : |T (b)(x)| > 2 }| ≤ C λ . This fact together with (1.28) shows λ kfk |{x ∈ Rn : |T (b)(x)| > }| ≤ C 1 . 2 λ n n λ n Noting |{x ∈ R : |T (f)(x)| > λ}| ≤ |{x ∈ R : |T (g)(x)| > 2 }| + |{x ∈ R : |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 ≤ Ckfkp.

Step 4: We observe that the adjoint T ∗ of T satisﬁes the same conditions as T , so T ∗ is bounded on Lp, 1 < p ≤ 2. The duality argument shows T is bounded on Lp, 2 ≤ p < ∞.

6 Example: Let f(x, y) be the density of a mass distribution in the plane. Then its New- 3 tonian potential in the half-space R+ is Z f(u, v) g(x, y, z) = 1 dudv. [(x − u)2 + (y − v)2 + z2] 2 R2 Formally, we have Z ∂g f(u, v)(x − u) limz→0 (x, y, z) = − 3 dudv. ∂x [(x − u)2 + (y − v)2] 2 R2 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 x considering Ω(x, y) = − x2+y2 . We can consider the first generation of Calderon-Zygmund operators as the Fourier multipliers. ∞ n Theorem 1.30: If m ∈ C (R \{0}) is a homogeneous function of degree 0, and Tm is d ˆ the Fourier multiplier defined by (Tmf) = mf, then there exist a, a complex number and Ω ∈ C∞(Sn−1) with zero average such that for any f ∈ S, Z Ω(y) T f = af + p.v. f(x − y)dy. (1.31) m |y|n

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. ∞ n Lemma 1.32: Let m ∈ C (R \{0}) be a homogeneous function of degree 0, and Tm is d ˆ the Fourier multiplier deﬁned by (Tmf) = mf, then there exist a, a complex number and ∞ n−1 Ω(y) Ω ∈ C (S ) with zero average such thatm ˆ (y) = p.v. |y|n . Proof: Since m is a tempered distribution,m ˆ exists. Thus,

dn ∂ m n ( n )(ξ) = Cξi mˆ (ξ), ∂xi

∂nm ∞ n where C is a constant. The function n is homogeneous of degree -n, in C (R \{0}) ∂xi and has zero average on Sn−1. Moreover, ∂nm ∂nm X = p.v. + C Dαδ, (1.33) ∂xn ∂xn α i i |α|≤k

∂nm where δ is the Dirac measure at the origin, since the diﬀerence between n and p.v. ∂xi ∂nm n is a distribution supported at the origin. By taking the Fourier transform on the ∂xi

7 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, mˆ 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 Z Z Ω(x) mˆ (φ) = φ(x)dx = c Ω(x)dσ(x), (1.34) |x|n

where c > 0. On the other hand, since φb is radial and m is homogeneous, Z mˆ (φ) = m(φb) = c0 m(x)dσ(x) = 0 R which together with (1.34) shows Ω(x)dσ(x) = 0. Finally, to see that mb is identical to Ω(x) Ω(x) p.v. |x|n , consider their difference mb −p.v. |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 dΩ(x) because both m and(p.v. |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 Tm1 Tm2 = Tm1m2 .Tm is invertible if and only if ∞ n n−1 T 1 ∈ A, and hence, 1/m ∈ C (R \{0}) which shows m(ξ) 6= 0 for any ξ ∈ S . We m P ∂αu now return to study Lu(x) = aα ∂xα . Deﬁne |α|=m

(Λdf)(ξ) = 2π|ξ|fˆ(ξ). (1.36)

Then Lu = T Λmu, (1.37) where the operator T is deﬁned by P (ξ) (dT u)(ξ) = im uˆ(ξ) (1.38) |ξ|m

P α P (ξ) where P (ξ) = aα(2πiξ) . The multiplier |ξ|m is homogeneous of degree zero and it |α|=m is a C∞ function on Sn−1, and hence, T is an operator in A. To solve X ∂αu Lu(x) = a = f, α ∂xα |α|=m

m −1 1 it suﬃces to solve (-4) 2 u = T f because Λ = (−4) 2 .

8 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 coefficients: X ∂αu Lu(x) = a (x) (2.1) α ∂xα |α|=m ∞ n where aα(x) ∈ C (R ). Formally, by taking the Fourier transform and then the inverse Fourier transform, we get X Z α 2πiξ·x Lu(x) = aα(x) (−2πiξ) uˆ(ξ)e dξ |α|=m Z X α 2πiξ·x = [ aα(x)(−2πiξ) ]û(ξ)e dξ. (2.2) |α|=m (2.2) can be written by more general form: Z σ(x, D)f(x) = σ(x, ξ)fˆ(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 Z T f(x) = L(x, x − y)f(y)dy (2.4)

where Z p.v. 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 Z ZZ T f(x) = σ(x, ξ)fˆ(ξ)e2πiξ·xdξ = σ(x, ξ)e2πiξ·xf(y)e−2πiy·ξdydξ Z Z Z = [ σ(x, ξ)e2πiξ·(x−y)dξ]f(y)dy = L(x, x − y)f(y)dy.

1 Using the operator Λ again, Calderon and Zygmund rewrote (2.2) as Z X 1 Lu(x) = ([ a (x)(−2πiξ)α ]|ξ|muˆ(ξ)e2πiξ·xdξ α |ξ|m |α|=m

= T (Λmu) (2.6) where T is deﬁned by Z T f(x) = σ(x, ξ)fˆ(ξ)e2πiξ·xdξ (2.7) P α aα(x)(−2πiξ) |α|=m with σ(x, ξ) = |ξ|m . Thus, σ is homogeneous of degree 0 in the variable ξ. By (2.5), Z T f(x) = K(x, x − y)f(y)dy (2.8)

R where K(x, y) = 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 Ω(x, y) K(x, y) = a(x)δ(y) + p.v. . |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 Ω(x, y)dσ(y) = 0 (2.12)

Sn−1 for all x ∈ Rn. The second generation of Calderon-Zygmund operators is the set of T deﬁned by Z Ω(x, y) T f(x) = p.v. f(x − y)dy. (2.13) |y|n

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

2 performing a spherical harmonic expansion of σ(x, ξ) on the Sn−1 for fixed x ∈ Rn, where Ω(x,y) σ(x, ξ) is the Fourier transform of p.v. |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

X∞ σ(x, ξ) = mk(x)hk(ξ) 0

P∞ with kmkk∞khkk∞ < ∞. 0 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 X∞ T = MkHk 0 and the series of operators is convergent in the L2 − norm. It is easy to see that the second generation of Calderon-Zygmund operators includes the first generation of Calderon- Zygmund operators. Now we consider formally a pseudo-differential operator defined by Z ˆ 2πix·ξ Tσf(x) = σ(x, ξ)f(ξ)e dξ.

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 operator: (dT f)(ξ) = m(ξ)fˆ(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

β α m−|α| |∂x ∂ξ σ(x, ξ)| ≤ Cα,β(1 + |ξ|) , (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. m Given a symbol in S , the operator Tσ 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

2πix·ξ 2 2 2πix·ξ (I − 4ξ)e = (1 + 4π |x| )e ,

3 and deﬁne the operator 2 2 −1 Lξ = (1 + 4π |x| ) (I − 4ξ),

N 2πix·ξ 2πix·ξ then (Lξ) e = e . Inserting this in (2.15) and carrying out the repeated integra- tions by parts gives Z N ˆ 2πix·ξ Tσf(x) = (Lξ) [σ(x, ξ)f(ξ)]e dξ

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 m to pointing out that if {σk} is a pointwise convergent sequence of symbols in S 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 ZZ 2πi(x−y)·ξ Tσf(x) = σ(x, ξ)e f(y)dydξ. (2.18)

However, the integral in (2.18) does not necessarily converge absolutely, even when f ∈ S. ∞ n n To with with this integral, ﬁx a function γ ∈ C0 (R ×R ) with γ(0, 0) = 1. Set σ²(x, ξ) = m m σ(x, ξ)γ(²x, ²ξ). Notice that if σ ∈ S , then σ² ∈ S 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 ZZ 2πi(x−y)·ξ Tσf(x) = lim²→0 σ²(x, ξ)e f(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. 0 Theorem 2.20: If σ ∈ S , then the operator Tσ initially deﬁned on S, extends to a bounded operator on L2. Proof: It suﬃces to show that kTσfk2 ≤ Ckfk2, (2.21) whenever f ∈ S, with C independent of f. 2 2 Indeed, suppose f ∈ L and let {fn} ∈ S so that fn → f in L . Then, by (2.21), Tσ(fn) 2 converges in L norm, and hence, Tσ(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 Z σ(x, ξ) = σb(µ, ξ)e2πiµ·xdµ

since σ has compact support in x variable. An integration by parts shows for each multi- index α, Z α α −2πiµ·x (2πiµ) σb(µ, ξ) = [∂x σ(x, ξ)]e dx

4 and α |(2πiµ) σb(µ, ξ)| ≤ cα, uniformly in ξ. As a result, we obtain

−N supξ|σb(µ, ξ)| ≤ CN (1 + |µ|) for arbitrary N ≥ 0. Now Z ZZ ˆ 2πix·ξ 2πiµ·x ˆ 2πix·ξ Tσf(x) = σ(x, ξ)f(ξ)e dξ = σb(µ, ξ)e f(ξ)e dξdµ Z = (T µf)(x)dµ, where µ 2πix·µ (T f)(x) = e (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 ˆ kTbσ(µ,ξ)fk2 ≤ sup |σb(µ, ξ)|kfk2 = sup |σb(µ, ξ)|kfk2 (2.22). ξ ξ

µ −N By (2.22), kT k ≤ CN (1 + |µ|) , which yields Z −N kTσk ≤ CN (1 + |µ|) dµ < ∞ 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 Z

Tσf(x) = 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, Z σ(x, ξ) = 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).

−N |k(x, z)| ≤ CN |z| 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 distribution whose Fourier transform is the function σ(x, ·) as we have written in (2.23). Next,

5 (−2πiz)αk(x, ·), with the distribution k(x, ·) thought of as acting on functions of z, equals α α the inverse Fourier transform of ∂ξ σ(x, ξ); by (2.17), ∂ξ σ(x, ξ) is integrable in ξ whenever |α| ≥ n + 1. This shows that k(x, ·) equals a function away from the origin, and that N |z| |k(x, z)| ≤ CN 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 n support in x. To begin with, we will show that, for each x0 ∈ R ,

Z Z 2 2 |f(x)| |Tσf(x)| dx ≤ CN N dx (2.25) (1 + |x − x0|) n |x−x0|≤1 R

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 ∈ C0 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 Z Z Z 2 2 2 2 |Tσf1| ≤ |Tησf1| ≤ C |f1| ≤ C |f| (2.26).

B(0,1) Rn Rn Rn

If x ∈ B(0, 1), since f2 is supported away from B(0, 2), the representation (2.23) holds: Z

Tσf2(x) = k(x, x − z)f2(z)dz. (2.27)

(B(0,2))c

Since |x − z| ≥ 1 when x ∈ B(0, 1) and z∈ / B(0, 2), using the estimate (2.24) yields Z −N |Tσf2(x)| ≤ CN |z| |f(z)|dz.

(B(0,2))c

By Schwartz inequality, we obtain, for N > n, Z Z 2 −N 2 |Tσf2(x)| dx ≤ CN (1 + |x|) |f(x)| dx. (2.28)

B(0,1)

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, n is. To see this, let τh, h ∈ R , 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 N > n, and interchanging order of integration. Theorem is proved.

6 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 β α −n−m−|α|−N |∂x ∂z k(x, z)| ≤ Cα,β,N |z| , 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 ∈ C0 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 : X∞ 1 = η(ξ) + δ(2−jξ), (2.31) j=1 for all ξ, and X∞ 1 = δ(2−jξ), (2.32) j=−∞ 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). R Let φ be the inverse Fourier transform of η, i.e., φb(ξ) = η(ξ). Then φ ∈ Sand φdx = 1. R b −n x Define ψ by ψ(ξ) = δ(ξ). Then ψdx = 0. Writing φt(x) = t φ( t ) and ψj = ψ2−j , we d −j d −j then have ψj = φj − φj−1, while (φj)(ξ) = η(2 ξ) and (ψj)(ξ) = δ(2 ξ). 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 X∞ I = S0 + 4j (2.33) j=1 and X∞ I = 4j. (2.34) j=−∞

Note that if f is a tempered distribution, Sj(f) is well deﬁned, and

XN S0(f) + 4j(f) = SN (f) → f, j=1

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.

7 We now return to the operator Tσ. Using (2.33), we write

X∞ X∞

Tσ = TS0 + T 4j = Tσj , j=1 j=0

−j where σ0(x, ξ) = σ(x, ξ)η(ξ), and σj(x, ξ) = σ(x, ξ)δ(2 ξ) for j ≥ 1.

Each of the pseudo-diﬀerential operators Tσj will be written in its singular integral form by Z

Tσj f(x) = 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 Z 2πiξ·z kj(x, z) = σj(x, ξ)e dz.

We claim the following estimate: If σ ∈ Sm, then

β α −N j[n+m−N+|α|] |∂x ∂z kj(x, z)| ≤ Cα,β,N |z| 2 , (2.35)

for all multi-indices α, β, and N ≥ 0, where Cα,β,N is independent of j ≥ 0. In fact, observe that Z γ β α γ α β 2πiξ·z (−2πiz) ∂x ∂z kj(x, z) ≤ ∂ξ [(2πiξ) ∂x σj(x, ξ)]e dξ.

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,

γ β α j[n+m−M+|α|] |z ∂x ∂z kj(x, z)| ≤ Cα,β,N 2 .

P∞ Taking the supremum over all γ with |γ| = M, gives (2.35). Since k(x, z) = kj(x, z), it j=0 ∞ P β α suffices to show |∂x ∂z kj(x, z)| satisfies the estimate given by the right side of (2.29). j=0 Consider the case when 0 < |z| ≤ 1, first. We break the above sum into two parts: j j the first where 2 ≤ |z|, the second where 2 > |z|. For the first sumP we use the estimate (2.35) with M = 0, which is then majorized by a multiple of 2j(n+m+|α|). This in 2j ≤|z| 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, forP the second sum, we choose M > n+m+|α|. By (2.35), we get the estimate O(|z|−M ) 2j[n+m+|α|−M] = O(|z|−n−m−|α|). The last term is 2j >|z|−1 O(|z|−n−m−|α|−N ) if N ≥ 0, since |z| ≤ 1.

8 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. 2 Now the L boundeness and the estimates of kernel of Tσ 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 α −(|α|−|β|) |∂x ∂ξ σ(x, ξ)| ≤ Cα,β|ξ| . (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:

β α −(|β|−|α|) |∂x ∂ξ σ(x, ξ)| ≤ Cα,β(1 + |ξ|) . (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 P α coefficients aα(x) of the differential operators aα(x)∂ . 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 = ( )T1 + ··· + ( )Tn, the Tj, 1 ≤ j ≤ n, being first ∂x1 ∂xn generation Calderon-Zygmund operators. d In dimension 1, T is replaced by DH, where D = −i 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 Z 1 T f(x) = p.v. K(x, y)f(y)dy ωn where a(x) − a(y) − (x, y) · 5a(y) K(x, y) = , (|x − y|2 + (a(x) − a(y))2)(n+1)/2

9 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 does not work anymore. This leads to third generation Calderon-Zygmund operators.

10 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 n 1 is a bump function supported on {ξ ∈ R : 2 ≤ |ξ| ≤ 2}. By construction we have X ψ(ξ/2k) = 1 k

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 k ˆ (Qkf)(ξ) = ψ(ξ/2 )f(ξ) (3.1)

d k ˆ (Pkf)(ξ) = φ(ξ/2 )f(ξ). (3.2) k Informally, Qk is a frequency projection to time annuls {|ξ| ∼ 2 }, while Pk is a frequency k 2 projection to the ball {|ξ| ≤ 2 }. Observe that Qk = Pk − Pk−1. Also, if f ∈ L , then 2 2 Pk(f) → 0 in L as k → −∞, and Pk(f) → f in L as k → ∞ , which follows from the Plancherel theorem. By telescoping the series, we thus have the Littlewood-Paley decomposition X f = Qk(f) (3.3) k for all f ∈ L2, where the series converges in the L2 norm. p We are now interested in how the L behavior of the Littlewood-Paley pieces Qkf relate to the behavior of f. First, we write Z nk k −k Pkf(x) = f ∗ (2 φ(2 ·)) = f(x + 2 y)φ(y)dy. (3.4) R Note that φ is a Schwartz function with the total mass φ(y)dy = φ(0) = 1. Thus the −k function Pkf is an average of f localized to physical scales≤ 2 . In particular, we expect −k Pkf to be essentially constant at scales < 2 . What does a function Qkf look like? Since Qkf = Pk+2Qkf, we see from (3.4) that Z −2−k Qkf(x) = Qkf(x + 2 y)φ(y)dy. (3.5)

−k Thus, Qkf is essentially constant at physical scales< 2 . On the other hand, we have Pk−2Qk1f = 0, so from (3.3), Z 2−k Qkf(x + 2 y)φ(y)dy = 0

1 for all x ∈ Rn. −k This roughly assert that Qkf has mean zero at scales ≤ 2 . From (3.3) and the Minkowski’s inequality, for 1 ≤ p ≤ ∞, we have Z −k kPkfkp ≤ kf(x + 2 y)kp|φ(y)|dy ≤ Ckfkp. (3.6)

p Thus, Pkf does not get any bigger that f itself as measured in L , or in any translation Pinvariant Banach space. Similarly, kQkfkp ≤ Ckfkp. On the other hand, we have f = Qkf, so by the triangle inequality, we obtain the cheap Littlewood-Paley inequality k X sup kQkfkp ≤ Ckfkp ≤ C kQkfkp. (3.7) k k

As the name suggests, the cheap Littlewood-Paley inequality is not the sharpest statement p one can make connecting the L norms of Qkf with those of f. By using the Fourier transform, when p = 2, we get

X 1 2 2 kfk2 ∼ ( kQkfk2) . (3.8) k

In fact, to see this, square both sizes and take Plancherel to obtain X ˆ 2 k ˆ 2 kfk2 ∼ k[ψ(·/2 )f(·)k2. k

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 X 2 1 kfk2 ∼ k( |Qkf| ) 2 k2. (3.9) k

P 2 1 The quantity ( |Qkf| ) 2 is known as the Littlewood-Paley square function. k Now deﬁne Sf for the vector-valued function by Sf(x) = {Qkf}k, and |Sf| = P 2 1 2 ( |Qkf| ) 2 is the ` norm of Sf. k

Theorem 3.10: For 1 < p < ∞, then kSfkp ∼ kfkp 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- nk k Zygmund operator with vector-valued kernel K(x) = (2 ψ(2 x))k. Since ψ is a Schwartz function, it is easy to check that K(x) satisﬁes the size and smoothness conditions for the

2 ﬁrst generation Calderon-Zygmund operators. Now the L2 result implies the Lp, 1 < p < ∗ ∞, results. By duality we also have kS (fk)kp ≤ Ck(fk)kp. Thus, X X 2 1 k Qkfkkp ≤ Ck( |fk| ) 2 kp. k k

Similarly, X X 2 1 k Qekfkkp ≤ Ck( |fk| ) 2 kp, k k

where Qek = Pk+2 − Pk−2. We apply this with fk = Qkf, since QekQk = Qk, we obtain X 2 1 kfkp ≤ k( |Qkf| ) 2 kp. k 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 d ˆ p symbol m : (Tmf)(ξ) = m(ξ)f(ξ). Then Tm is bounded on L , 1 < p < ∞. Proof: We have X X Tm = QkTmQk0 = QekQ¯k k,k0 k P where Qek = QkTm and Q¯k = Qk. From (3.12) we see that Qek and Q¯k are k−2

k∂xi ∂xj fkp ≤ Ck 4 fkp

n P 2 n for all 1 < p < ∞, where 4 = ∂xi is the Laplacian on R . This follows because i=1 d ξiξj d ξiξj (∂xi ∂xj f)(ξ) = |ξ|2 (4f)(ξ), and the symbol m(ξ) = |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 0 f ∈ S , the temperate distribution space, Qkf 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 is defined by kfkHp = kSfkp.

3 It is easy to see that when 1 < p < ∞,Hp = Lp, by the Littlewood-Paley Lp inequality. For 0 < p ≤ 1,Hp is a new space which is diﬀerent from Lp. The best way to see this is p to get the so-calledR atomic decomposition of H . Here we only consider p = 1. To do this, α let S∞ = {f ∈ S : f(x)x dx = 0, for all |α| ≥ 0}. We then have P b k 2 Theorem 3.14: Suppose f ∈ S∞ and |ψ(2 ξ)| = 1, where ψ is a bump function k supported on the annuls {|x| ∼ 2−k}. Then X f = QkQkf, k where the series converges in the topology of S. 0 By a duality argument, for f ∈ S and g ∈ S∞ we have X X < QkQkf, g >=< f, QkQkg >=< f, g > . k k

0 P 0 0 0 Hence, for any f ∈ S , f = QkQkf in (S∞) ≈ S /P, S modulo polynomials. k 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; − 1 (ii) kRak2 ≤ |Q| 2 ; (iii) a(x)dx = 0. 1 P P Theorem 3.15: f ∈ H if and only if f = λkak, where ak are atoms and |λk| < ∞. k k Moreover, X X kfkH1 ≈ inf{ |λk| : for allf = λkak}. k k As mentioned before, Hp is well defined for 0 < p < ∞, but not for p = ∞. Next, we consider the replacement of Hp when p = ∞. 1 n Definition 3.16: For f ∈ Lloc(R ) we let kfk∗ = supQ mQ|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

kfk∗ < ∞. (BMO(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 ∈ BMO, λ sup mQ(exp( |f − mQf| ≤ C. (3.18) Q kfk∗

4 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 kfk∞. Let Q0 be a ﬁxed cube and Q some dyadic cube. Recall that Qe is the unique dyadic cube which contains Q and lies in the previous generation. It is easy to see that

|m f − m f| ≤ 2nkfk . (3.19) Q Qe ∗

Consider now the Calderon-Zygmund decomposition of the function (f − mQ0 f)χQ0 for λ = 2kfk∗. This yields a collection of dyadic cubes Qi, maximal with respect to inclusion, satisfying

mQi |(f − mQ0 f)χQ0 | > 2kfk∗ (3.20) and

|(f − mQ0 f)χQ0 | ≤ 2kfk∗ (3.21) c on (∪Qi) . k(f−mQ0 f)χQ0 k1 |Q0| Clearly, Qi ⊆ Q0 for each i, and | ∪ Qi| ≤ ≤ . 2kfk∗ 2

Since the Qi’s are maximal. m |(f − mQ0 f| ≤ 2kfk∗ , and (3.19) gives Qei

n |mQi f − mQ0 f| ≤ (2 + 2)kfk∗.

λ Let X(λ) = sup mQexp( |f − mQf|, which is ﬁnite since we re assuming that f is Q kfk∗ bounded. We obtain Z λ 1 2λ mQ0 (exp( |f − mQf|) ≤ e dx kfk∗ |Q0| Q0\∪Qi Z 1 X |Q | λ n + i [ exp( |f − m f|)dxe(2 +2)λ] |Q | |Q | kfk Qi 0 i i ∗ Qi 1 ≤ e2λ + [exp(2n + 2)]X(λ). 2 1 n 2λ From taking the supremum over all cube Q0 it follows that X(λ)[1 − 2 exp(2 + 2)] ≤ e , 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,

−λt |{x ∈ Q : |f(x) − mQf| > tkfk∗} ≤ Ce |Q|. (3.22)

1 p p For 1 ≤ p < ∞, then kfkp,∗ = supQ[mQ|f − mQf| ] 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 Z `(g) = f(x)g(x)dx, g ∈ H1,

5 when suitably defined, where f is a function in BMO. For general f ∈ BMO 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 1 1 by Ha which is dense in H . Theorem 3.24: (a) Suppose f ∈ BMO. Then the linear functional given by (3.23), 1 1 initially defined on Ha , has a unique bounded extension to H and satisfies

k`k ≤ ckfk∗.

(b) Conversely, every continuous linear functional on H1 can be realized as above, with f ∈ BMO, and with kfk∗ ≤ ck`k.

Proof: To see (a), note that if a is an atom supported on Q, then Z Z 1 | f(x)a(x)dx| = | [f(x) − mQf]a(x)dx| ≤ |Q| 2 kfk2,∗kak2 ≤ ckfk∗.

1 P Thus, if g ∈ Ha , then g = λkak with the sum having a ﬁnete terms and ak are atoms. k So Z X Z

| f(x)g(x)dx| = | λk [f(x) − mQk f]ak(x)dx| k

X 1 X ≤ |λkkQk| 2 kfk2,∗kak2 ≤ ckfk∗ |λk|. k k To show (b), ﬁx a cube Q and let L2 be the space of all square integrable functions Q R 2 2 2 supported on Q. Let LQ,0 = {f ∈ LQ : f(x)dx = 0} . Note that every g ∈ LQ,0 is a 1 1 multiple of an atom and kgkH1 ≤ c|Q| 2 kgk2. Thus if is a given linear functional on H 2 1 with the norm≤ 1, then extends to a linear functional on LQ,0 with norm at most c|Q| 2 2 . By the Riesz representation theorem for the Hilbert space LQ,0, there exists an element F Q ∈ L2 so that Q,0 Z `(g) = F Q(x)g(x)dx, (3.25)

if g ∈ L2 , with Q,0 Z Q 1 1 ( |F (x)|dx) 2 ≤ c|Q| 2 . (3.26)

Hence for each Q, we get such a function F Q. We want to have a single function f so that, Q on each Q, f diﬀers from F by a constant. To construct this f, observe that if Q1 ⊆ Q2, Q1 Q2 Q1 Q2 then F − F is constant on Q1. Indeed, both F and F give the same functional on L2 , so they must diﬀer by a constant on Q . We can modify F Q, replacing it with Q1,0 1 Q Q Q f = F + cQ, where cQ is a constant chosen so that f has the average zero over the

6 Q1 Q2 unit cube centered at the origin. It follows that f = f on Q1, if Q1 ⊆ Q2. Finally, we deﬁne f on Rn by taking f(x) = f Q(x) for x ∈ Q. Observe that Z Z Z 1 1 2 1 1 Q 2 1 |f(x) − c |dx ≤ ( |f(x) − c | dx) 2 = ( |F | dx) 2 ≤ c, |Q| Q |Q| Q |Q| Q Q Q R Q 2 which shows f ∈ BMO with kfk∗ ≤ c. Also, by (3.25), `(g) = F (x)g(x)dx, if g ∈ LQ,0, 1 for some Q, in particular, this representation holds for all g ∈ Ha . The converse (b) of the theorem is proved. We now discuss the relationship between BMO and Carleson measures and Littlewood- Paley square functions. n+1 Deﬁnition 3.27: A Borel measure dµ on R+ is said to be a Carleson measure if Z 1 sup |dµ| ≤ C < ∞. (3.28) Q |Q| T (Q)

If dµ is a Carleson measure, we denote kdµkC , the Carleson norm of dµ, by the smallest constant C in (3.28). R Theroem 3.29: Suppose φ ∈ S with φ(x)dx = 1. Then dµ is a Carleson measure if and only if Z 2 2 |φt ∗ f(x)| dµ ≤ Ckfk2. (3.30)

n+1 R+

R∞ b 2 dt Theorem 3.31: Let ψ ∈ S with |ψ(tξ)| t = 1 for all ξ 6= 0. Then f ∈ BMO if and 0 2 dxdt only if |f ∗ ψt(x)| t 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 ˙ α,q ˙ α,q Besov spaces Bp and Triebel-Lizorkin spaces Fp as well as their inhomogeneous analogs. b 1 b Let us choose φ ∈ S so that Supp φ ⊆ {ξ : 2 ≤ |ξ| ≤ 2} and |φ(ξ)| ≥ c > 0 if 3 5 0 5 ≤ |ξ| ≤ 3 . For α ∈ R, p 6= ∞, 0 < p, q ≤ ∞ and f ∈ S we deﬁne X kα q 1 kfk ˙ α,q = k{ (2 |φ ∗ f|) } q k , (3.32) Fp k p k

and, for the same indices, and including p = ∞, X kα q 1 kfk ˙ α,q = { (2 kφ ∗ fk ) } q , (3.33) Bp k p k

kn k where φk(x) = 2 φ(2 x).

7 0 Note that φ ∗ f is a smooth function when φ ∈ S and f ∈ S . Also kfk ˙ α,q = 0 k Fp or kfk ˙ α,q = 0 if and only if φ ∗ f is the zero function for all k. But this is equivalent Bp k to having fˆ(ξ)φb(2kξ) be zero for all k. because of the conditions on φ,b this, in turn, is equivalent to Supp fˆ = {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 ˙ α,q ˙ α,q Fp and Bp 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 ∗ fkp to variants of the above expressions, where Φ ∈ S, b 5 Supp Φ ⊆ {ξ : |ξ| ≤ 2} and |Φ(ξ)| ≥ c > 0 if |ξ| ≤ 3 . The variants in question are as in P P α,q α,q (3.32) and (3.33) with replaced by . These spaces are denoted by Fp and Bp and k k≥1 they are spaces of tempered distributions; the necessity of considering such distributions modulo polynomials disappears since Φ(0)b 6= 0. By the results mentioned above by the Littlewood-Paley theory, we obtain the following identifications: p ˙ 0,2 L ∼ Fp when 1 < p < ∞; p ˙ 0,2 H ∼ Fp when 0 < p ≤ 1; ˙ 0,2 BMO ∼ F∞ ˙ 0,2 p α,2 ˙ p ˙ α,2 when F∞ is defined by the Carleson measure. We also can show Lα ∼ Fp and Lα ∼ Fp when α > 0 and 1 < p < ∞; α,∞ Λα ∼ F∞ and ˙ ˙ α,∞ Λα ∼ F∞ when α > 0. Suppose φe and Φe are two other functions satisfying the properties of φ and Φ announced above. One can show that replacing φ and Φ with φe and Φe in the definitions yields the same spaces with equivalent norms.

8 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 d the function a(x) and the operator DH, where D = −i 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 Z A(x) − A(y) [A, DH](f)(x) = p.v. f(y)dy. (4.1) (x − y)2 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 Xm djf Lf(x) = a (x) . (4.2) j dxj j−0 As we did in lecture 2, by Fourier transform and Fourier inversion, Z 1 Xm Lf(x) = a (x)(iξ)jfˆ(ξ)eixξdξ. (4.3) 2π j j−0

j The idea behind pseudo-differential operators is to replace the function aj(x)(iξ) 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 (Λdf)(ξ) = ψ(ξ)fˆ(ξ) where ψ is an infinitely differentiable function with ψ(ξ) = |ξ| if |ξ| ≥ 1, and let Z T f(x) = q(x, ξ)fˆ(ξ)eixξdξ + Rf(x), (4.4)

1 where Z Rf(x) = r(x, ξ)fˆ(ξ)eixξdξ. (4.5)

−m m Here q(x, ξ) = |ξ| am(x)(iξ) and r(x, ξ) is deﬁned by the relation

1 Xm a (x)(iξ)j = (q(x, ξ) + r(x, ξ))(ψ(ξ))m. 2π j j=0

Now we can write Lf = T Λmf.

d 2 It is easy to show that the operator R and R dx are bounded on L , 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 d 2 R and R dx are bounded on L . 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 Z Hf(x) = c sign(ξ)fˆ(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 Z a(x) − a(y) f(y) C f(x) = p.v. ( )k dy, k ≥ 1. (4.9) k x − y x − y

2 These operators are closely related to the boundary behavior of analytic functions given by Cauchy integrals Z 1 1 f(z(x)) = f(z(y)z0(y)dy (4.10) 2πi z(y) − z(x)

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 Z 1 a(x) − a(y) − (x − y) · 5a(y) T f(x) = p.v. n+1 f(y)dy, (4.11) ωn (|x − y|2 + (a(x) − a(y))2) 2 Rn n where ωn is the area of the unit sphere in R . 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 operators. 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 1 |K(x, y)| ≤ C , (4.13) |x − y|n

|x − x0|² |K(x, y) − K(x0, y)| ≤ C , (4.14) |x − y|n+² 0 1 if |x − x | ≤ 2 |x − y|, |y − y0|² |K(x, y) − K(x, y0)| ≤ C , (4.15) |x − y|n+² 0 1 if |y − y | ≤ 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 ZZ (T f, g) = K(x, y)g(x)f(y)dydx (4.16)

3 whenever f, g ∈ S and suppf ∩ suppg = φ. T is said to be a third generation Calderon-Zygmund operator if T ∈ CZSIO and it can be extended to a bounded operator on L2. We write T ∈ CZO. By the Calderon-Zygmund real variable method, we can easily get the following theorem. Theorem 4.17: If T ∈ CZO, then T is bounded on Lp for 1 < p < ∞, from H1 to L1, and from L∞ to BMO. A basic, important problem is how to understand when a Calderon-Zygmund singular integral operator is bounded on L2. As Meyer wrote ” To go beyond the context of convolution operators, it becomes independensable to have a criterion for L2 continuity, without which the theory collapses like a house of cards. This problem was solved by the celebrated T 1 theorem of David-Journe in 1983. This theorem represents the culmination of the theory started by Calderon and Zygmund some thirty years earlier. Let’s go back to the Calderon commutator. In 1965, Calderon proved the L2 boundedness of Calderon commutator. After 9 years, in 1974, Coifman and Meyer proved the same result for the second order Calderon commutator. in 1977, Calderon proved the L2 boundedness for the Cauchy integral on a Lipschitz curve with the Lip norm < ², where ² is a small number. Coifman, McIntosh and Meyer, in 1981, proved the L2 boundedness for the Cauchy integral on all Lipschitz curves. Before we can state the David-Journe T 1 theorem, we require some preliminary definition and concepts. We begin by assigning a meaning to T 1 and T ∗1 for T ∈ CZSIO. Even in the classical case of T ∈ CASIO, this requiresR care. Denote D the set of all f ∈ S with the compact support and D0 = {f ∈ D : f(x)dx = 0}. We now can define T 1 as a linear functional on D0 as follows. For any given f ∈ D0, choose χ1 ∈ D such that χ1(x) = 1 for x ∈ 2Q, where Q is a cube containing support of f, and 1 = χ1 + χ2. Since ∗ χ1 ∈ D, so < T χ1, f > is well defined. Note that, formally, < T χ2, f >=< χ2,T f > and ∗ this makes sense if < χ2,T f > is well defined. Indeed, by the fact thta f ∈ D0, we have ZZ Z Z ∗ < χ2,T f >= χ2(x)K(x, y)f(y)dydx = χ2(x) [K(x, y) − K(x, yQ)]f(y)dydx.

By the conditions on K, we obtain the absolute value of the last kfk1. This means < ∗ ∗ χ2,T f > is well defined. Now we define < T 1, f >=< T χ1, f > + < χ2,T f > . It is easy to seeR that the equality does not depend the choice of χ. We also define T 1 = 0 by the ∗ ∗ fact that T f(x)dx = 0 for all f ∈ D0, and T 1 = 0 is defined similarly. Note that D0 is a subset, so T 1 is a BMO function means that T 1, as a linear functional on D0, can be extended to be a linear functional on H1, and it is similar for T ∗1 being a BMO function. Next we consider a condition known as the weak boundedness property. Definition 4.18: We say that a linear continuous operator T : S → S0 satisfies the weak boundedness property, and we write T ∈ WBP, if for each bounded subset B of S there exists a constant C = C(B) such that for all f, g ∈ B,

z z −n n | < T (ft ), gt > | ≤ Ct , z ∈ R , t > 0, (4.19) z −n x−z where ft (x) = t f( t ).

4 Note that if T is bounded on Lp, for some fixed 1 < p < ∞, then T ∈ WBP. As an example, a rather large class of operators which satisfy the weak boundedness property is the class of CZSIO’s with antisymmetric kernel K, i.e., such that K(x, y) = −K(y, x). In this case, ZZ 1 < T f, g >= lim K(x, y)[f(y)g(x) − f(x)g(y)]dydx, ²→0 2 |x−y|>² and, using the definition of a Calderon-Zygmund singular integral kernel, it is easy to see that T ∈ WBP. This class of operators includes the all Calderon commutators. We are now ready to state the T 1 theorem of David-Journe. Theorem 4.20: Suppose T ∈ CZSIO. Then T ∈ CZO if and only if T 1 ∈ BMO,T ∗1 ∈ BMO, and T ∈ WBP. Proof: Step 1. Consider first the case when T 1 = T ∗1 = 0. BY the Littlewood-paley theory and the Calderon reproducing formula Z∞ Z∞ ds dt 1 kT fk ≤ CkS(T f)k = Ck{ |ψ ∗ (T ψ ∗ ψ ∗ f )| } 2 k . 2 2 t s s s t 2 0 0 A basic estimate under the condition of T is the following inequality: s t (t ∨ s)² |ψ ∗ (T ψ )(x, y)| ≤ C( ∧ )² , (4.21) t s t s (t ∨ s + |x − y|)n+²

where ψt ∗ (T ψs)(x, y) is the kernel of ψt ∗ (T ψs), a ∧ b = min(a, b), and a ∨ b = max(a, b). Using the estimate in (4.21) yields Z∞ Z∞ Z∞ ds dt ds |ψ ∗ (T ψ ∗ ψ ∗ f )|2 ≤ C M 2(ψ ∗ f) t s s s t s s 0 0 0 and then applying the Feﬀerman-Stein vector-valued maximal function inequality and, again, the Littlewood-paley theory, we obtain Z∞ 2 ds 1 kT fk ≤ Ck{ |ψ ∗ f| } 2 k ≤ Ckfk . 2 s s 2 2 0

Step 2. To reduce the general case to the case in step 1, we need the so-called para-product operator. Definition 4.22: Suppose b ∈ BMO and ψ and φ are functions as defined in the Littlewood-Paley theory. The para-product operator Πb, is defined by Z∞ dt Π f(x) = ψ ∗ [ψ ∗ b(·)φ ∗ f(·)] . (4.23) b t t t t 0

5 Theorem 4.24: If b ∈ BMO, then Πb ∈ CZO. Moreover,

kΠbk ≤ Ckbk∗.

∗ ∗ Now deﬁne Te = T − ΠT 1 − (ΠT ∗1) . Note that Πb(1) = b and (Πb) (1) = 0. It follows that Te ∈ CZSIO and Te(1) = (Te)∗(1) = 0. By step 1, Te is bounded on L2, and, hence, T is bounded on L2. As a beautiful application of the T 1 theorem, we prove all Calderon’s commutators are bounded on L2. Recall that the kth order Calderon commutator is deﬁned by Z a(x) − a(y) f(y) C f(x) = p.v. ( )k dy. (4.24) k x − y x − y

particularly, C0 = H, the Hilbert transform. So C0 is a CZO. To see C1 is bounded on 2 L , by the T 1 theorem it suﬃces to show C1(1) is a BMO function because all kernels of Ck for k ≥ 0, are antisymmetric, and, as mentioned above, Ck ∈ WBP for all k ≥ 0. 0 However, by the integration by parts, C1(1)(x) = H(a )(x) which shows C1(1) ∈ BMO. repeating this procedure, we prove Ck ∈ CZO. Moreover,

0 k kCkk ≤ (Cka k∞)

for some constant C. Finally, the Cauchy integral on a Lipschitz curve can be decomposed by Z 1 X∞ f(y)dy = (−i)kC f(x) x − y + i(a(x) − a(y)) k k=0 0 2 which shows that if ka k∞ < ², then the Cauchy integral is bounded on L . Calderon’s research program was motivated by the study of elliptic partial differential equations in domains with irregular boundary. Calderon’s method consists of replacing the partial differential equation on the interior by a pseudo-differential equation on the boundary. But if the boundary is only Lipschitz, the nature of this equation changes: the operators which appear are no longer pseudo-differential but are of the third generation of Calderon-Zygmund operators. When Calderon inaugurated his program, there were two difficulties. The very existence of the operators, needed for the method, was problematic. And, supporsing that, in the fullness of time, such operators could be constructed, it would be necessary to solve the equations on the boundary that this process led to. The regrettable absence of such a symbolic calculus was signaled above, and is the second problem of Calderon’s program. We now illustrate these remarks by examining a classical problem, which goes back to Poincare, Neumann, and Hilbert. This is the solution, by the double-layer potential method, of the Dirichlet and Neumann problem for a domain Ω in Rn+1. When Ω is a bounded, regular, open set, the operator of Calderon’s method are classical pseudo-differential operators. They are also singular integral operators whose kernels are given by a double-layer potential. After the reduction given by Calderon’s method, resolving a Dirichlet or Neumann problem just amounts to inverting an operator

6 1 2 of the form 2 + K acting on the boundary. The ambient Banach space will be L (∂Ω, dσ), where dσ is the surface measure on the boundary ∂Ω of Ω. When Ω is a bounded, regular, open set(of class C1+α, for some α > 0), the operator 2 2 1 K : L (∂Ω, dσ) → L (∂Ω, dσ) is compact, so Fredholm theory allows one to invet 2 + K. When Ω is just of class C1, the principal difficulty is to prove that K is continuous on the space L2(∂Ω, dσ). Once this continuity had been established(Calderon, 1977), the operator K was still compact, so that the rest could be done just as in the regular case. However, K no longer was a classical pseudo-differential operator. Finally, for Ω a bounded, Lipschitz, open set, the continuity of K was established in 1981. But the operator K is 1 no longer compact. This essential difficulty was overcome by showing that 2 + K was invertible, using the Jerison and Kenig energy inequality.

7 Lecture 5. Spaces of Homogenenous Type and nonhomogeneous space

The Calderon-Zygmund operator theory may be substantially widened by replacing Rn with an abstract space having a few simple properties. This is space of homogeneous type introduced by Coifman and Weiss in 1970. A quasimetric ρ on a set X is a function ρ : X × X → [0, ∞] satisfying

ρ(x, y) = 0 (5.1)

if and only ifx = y, ρ(x, y) = ρ(y, x) (5.2) for all x, y, ρ(x, y) ≤ A[ρ(x, z) + ρ(z, y)] (5.3) for all x, y, z, where A < ∞ is a constant. Any quasimetric defines a topology, for which the balls B(x, r) = {y ∈ X : ρ(x, y) < r} form a base. However, the balls themselves need not to be open when A > 1. A general setting for the Calderon-Zygmund operator theory is as follows. Definition 5.4: A space of homogeneous type (X, ρ, dµ) is a set X together with a quasimetric ρ and a nonnegative measure dµ on X such that µ(B(x, r)) < ∞ for all x ∈ X and r > 0, and dµ satisfies the doubling condition, that is, there exists a constant C < ∞ such that for all x ∈ X and r > 0,

µ(B(x, 2r)) ≤ Cµ(B(x, r)). (5.5)

It is unsetting that the balls associated with a quasimetric need not to be open, in general. Fortunately, a result of Macias and Segovia asserts that the given quasimetric may always be replaced by an equivalent one with better properties. Theorem 5.6: Let ρ be a quasimetric on a set X. Then there exist C < ∞, ² > 0 and a quasimetric ρ0 on X, equivalent to ρ in the sense that C−1ρ0(x, y) ≤ ρ(x, y) ≤ Cρ0(x, y) for all x, y ∈ X, such that for all x, y, z ∈ X,

|ρ0(x, z) − ρ0(y, z)| ≤ Cρ0(x, y)²[ρ0(x, z) + ρ0(y, z)]1−². (5.7)

Then the balls associated with ρ0 also form a base for the original topology, but are easily seen to be open as a consequence of (5.7). It is also easy to verify, using (5.7), the existence 0 δ0 of δ0 > 0 such that (ρ ) is a metric, not merely a quasi-metric. Before we are going to set up the Calderon-Zygmund operator theory on space of homogeneous type, let us examine some examples. Example 1. Rn, with the Euclidean metric and Lebesgue measure. Example 2. Any C∞ compact Riemannian manifold, with the Riemannian metric and volume.

1 Example 3. The graph of a Lipschitz function F: Rn → R, with the induced Euclidean metric and with µ(F (E)) = |E|, the Lebesgue measure of E ⊆ Rn. Similarly, the boundary of any bounded Lipschitz domain in Rn, that is, any bounded open set whose boundary may be represented locally as the graph of a Lipschitz function after a suitable rotation of coordinates. This includes all polygons in R2. Pn aj n Example 4. Let n ≥ 1 and a1, ..., an > 0. Set ρ(x, y) = |xj − yj| for x, y ∈ R . Then j=1 ρ is a quasimetric, though not in general a metric, and Rn becomes a space of homogeneous type when equipped with ρ and Lebesgue measure. Example 5. Let n ≥ 1 be an integer and d ∈ (0, n] a real number. Let E ⊆ Rn be a closed subset whose d-dimensional Hausdorﬀ measure Λd(E) is ﬁnite and positive. Suppose it happens that there exists C < ∞ so that for every x ∈ E and r > 0,

−1 d d C r ≤ Λd(E ∩ B(x, r)) ≤ Cr .

Then with the Euclidean metric and µ = Λd,E becomes a space of homogeneous type. Example 6. Let G be a nilpotent group with a left-invariant Riemannian metric ρ. Let µ be the induced measure. Then (G, ρ, µ) is a space of homogeneous type. n ∞ Example 7. Let Ω be an open set in R and let X1, ..., Xk be C vector ﬁelds in Ω. Suppose that {Xj} satisfy the condition of Hormander, which is that they, together with all their commutators of all orders, span the tangent space to Rn at each x ∈ Ω. Say that a Lipschitz curve γ : [0, 1] → Ω is admissible if for almost every t,

dγ Xk = c (t)X (γ(t)) dt j j j=1

P 2 where |cj(t)| ≤ 1. Deﬁne ρ(x, y) to be the inﬁmum of the set of all r for which there exists an admissible curve with γ(0) = x and γ(r) = y. It is a theorem that such an admissible curve exists for any x, y ∈ Ω, provided Ω is connected. Then ρ is clearly a metric. If one stays away from the boundary of Ω, then with Lebesgue measure d, (Ω, ρ, d) becomes a space of homogeneous type. Example 8. This is a special case of Example 7. Let U be a bounded open subset of C2 with C∞ boundary. At any z ∈ ∂U the vector space of linear combinations

∂ ∂ a1 + a2 ∂z¯1 ∂z¯2

belonging to the complexﬁed tangent space to ∂U has dimension 1 over C. Fix a nonvanishing C∞ complex vector ﬁeld

∂ ∂ L¯(z) = a1(z) + a2(z) ∂z¯1 ∂z¯2

2 which is tangent to ∂U at every point z in some open set V ⊆ ∂U. Write L¯ = X + iY where X,Y are real vector fields. If U is strictly pseudoconvex, then the pair X,Y satisfies the Hormander condition and V becomes a space of homogeneous type. We set down the basic definitions and results concerning the Calderon-Zygmund operator theory on spaces of homogeneous type. The first ingredient of the general theory is of course the analogue of the maximal function of Hardy and Littlewood. 1 Definition 5.8: For f ∈ Lloc(X), the maximal function of f is defined by Z 1 Mf(x) = sup |f(y)|dµ(y). (5.9) r>0 µ(B(x, r)) B(x,r)

Theorem 5.10: M is bounded on Lp(X) for all 1 < p < ∞, and is of weak type (1, 1). We also have the Calderon-Zygmund decomposition. 1 −1 Theorem 5.11: Let f ∈ L (X) and λ > 0, and assume that µ(X) > λ kfk1. Then f may be decomposed as f = g + b where

2 kgk2 ≤ Cλkfk1, (5.12) X b = bj (5.13) j

where each bj is supported on some ball B(xj, rj), Z

bj(x)dµ(x) = 0, (5.14)

kbjk1 ≤ Cλµ(B(xj, rj)), (5.15) X −1 µ(B(xj, rj)) ≤ Cλ kfk1. (5.16) j

Deﬁnition 5.17: A kernel K : X ×X\{x = y} :→ C is said to be a singular integral kernel 1 if there exist ² > 0 and C < ∞ such that for all x 6= y ∈ X and z with ρ(x, z) ≤ 2A ρ(x, y),

|K(x, y)| ≤ Cµ(B(x, r))−1 (5.18)

where r = ρ(x, y),

ρ(x, z) |K(x, y) − K(z, y)| + |K(y, x) − K(y, z)| ≤ C( )²µ(B(x, r))−1 (5.19) ρ(x, y)

where r = ρ(x, y). 0 Deﬁnition 5.20: A continuous linear operator T :Λδ → (Λδ) , where Λδ is the space of all bounded function on X , Holder continuous of order δ > 0 with respect to ρ and having

3 0 bounded support, and (Λδ) is its dual space, is said to be a singular integral operator if it is associated with a singular integral kernel K such that ZZ < T f, g >= K(x, y)f(y)g(x)dµ(y)dµ(x) (5.21)

for all f, g ∈ Λδ whose supports are separated by a positive distance. Theorem 5.22: Any singular integral operator which is bounded on L2 is also bounded on Lp, 1 < p < ∞, and is of weak type (1, 1). 1 Deﬁnition 5.23: If f ∈ Lloc(X), then Z 1 kfkBMO = sup inf |f(y) − c|dµ(y) x,r c µ(B(x, r)) B(x,r)

and BMO is the set all all equivalence classes modulo constants of locally integrable functions with ﬁnite BMO norm. Theorem 5.24: A singular integral operator which is bounded on L2 maps L∞ boundedlly to BMO. To formulate the T 1 theorem, we need the notion of weak boundedness. For δ ∈ (0, 1], x ∈ X and r > 0, deﬁne A(δ, x, r) to be a set of all φ ∈ Λδ supported in B(x, r) −δ δ satisfying kφk∞ ≤ 1 and |φ(y) − φ(z)| ≤ r ρ(y, z) for all y, z ∈ X.

Deﬁnition 5.25: A singular integral operator T is weak bounded if there exist δ ∈ (0, δ0] and C < ∞ such that for all x ∈ X, r > 0, and φ, ψ ∈ A(δ, x, r)

| < T φ, ψ > | ≤ Cµ(B(x, r)). (5.26)

Of course, the inequality would be an immediate consequence of L2 boundedness. It hold automatically for singular integral operators with antisymmetric kernels. Theorem 5.27: A singular integral operator T on a space of homogeneous type is bounded on L2 if and only if it is weak bounded and T 1, T ∗1 ∈ BMO. We also have analogue of dyadic cubes for a space of homogeneous type. Theorem 5.28: let (X, ρ, dµ) be a space of homogeneous type. Then there exists a family k of subsets Qα ⊆ X, deﬁned for all integers k, and constants δ, ² > 0, C < ∞ such that

k µ(X\ ∪α Qα) = 0 (5.29) for all k,

` k ` k for anyα, β, k, with` ≥ k, either Qβ ⊆ Qα or Qβ ∩ Qα = φ, (5.30)

k each Qβ has exactly one parent for all k ≥ 1, (5.31)

4 k each Qβ has at least one child, (5.32)

k+1 k k+1 k Qα ⊆ Qβ then µ(Qα ) ≥ ²µ(Qβ), (5.33)

k k k for each (α, k) there exists xα,k ∈ X such that B(xα,k, δ ) ⊆ Qα ⊆ B(xα,k, Cδ ), (5.34) k k k ² k µ{y ∈ Qα : ρ(y, X\Qα) ≤ tδ } ≤ Ct µ(Qα) for 0 < t ≤ 1 and all α, k.(5.35) For most of the examples of spaces of homogeneous type we indicate natural examples of associated singular integral operators. Example 1. Compact manifolds. Any pseudo differential operators with symbol in the 0 ∞ 2 class S1,0, on a compact C manifold without boundary. L boundedness always holds. Example 2. Parabolic dilations in Rn. Let K be a tempered distribution on Rn with is ∞ −d n C away fromP the origin, and satisfies K(δtx) = t K(x) for all x ∈ R \{0} and t > 0, where d = aj is called the homogeneous dimension. Then convolution with K defines a j singular integral operator T . By translation-invariance, T maps constant to constant, so that T (1) = 0 in BMO. The same is true for the transpose. It is also easy to verify the weak boundedness. For a special example from differential equations, take aj = 1 for all j ≤ n − 1, and an = 2. Take K to be the tempered distribution whose Fourier transform ξiξj is the function (|ξ0|2+iτ) for some i, j ≤ n − 1, where the Fourier transform variable is written ξ = (ξ0, τ) ∈ Rn−1 × R. This multiplier arises from regularity estimates for the heat equation in n-1 space variables. Example 3. d-dimensional subset of Euclidean space. The Cauchy integral on an Ahlfors- regular one-dimensional subset of the complex plane.

Example 4. Nilpotent Lie groups. Let X1, ..., Xk be a basis for the linear space of left- P 2 invariant vector fields on a simply connected nilpotent Lie group. Let L = − Xj , a j −1 − 1 left-invariant second-order differential operator. Then L and L 2 may be defined by −1 − 1 means of spectral theory,and for any i and j, XjXiL and XjL 2 are singular integral operators.

In recent years it has been ascertained that central results of classical Calderon- Zygmund theory hold true in very general situations in which the standard doubling condition on the underlying measure is not satisﬁed. This has come as a great surprise to that homogeneous spaces were not only a convenient setting for developing Calderon- Zygmund theory, but that they were essentially the right context. The motivation of non-homogeneous space coma from the so-called problem of analytic capacity. A compact subset E of the plane is said to be removable for bounded analytic function if given any open set Ω and any bounded analytic function f on Ω\E, then f extends to an analytic function on Ω. This can be easily shown to be equivalent to any bounded analytic function on the complement of E being constant. For example, a set reduced to a point is removable and a closed disc is not. Painleve proved more than one hundred years ago that if E is a compact set of zero length, then E is removable. He also asked for a metric, or, even better,

5 geometric characterization of removable sets. This was called the Painleve problem. If one restricts the attention to subsets of the real line, then the converse of Painleve’s result is true, so that the removable subset of the real line are precisely those of zero length. Denjoy believe that the same would happen if the real axis were replaced by a general rectifiable curve, but the argument he found had a gap. A complete proof of the ” Denjoy conjecture did not arrive until Calderon obtained in 1977 his famous result on the L2 boundedness of the Cauchy integral on Lipschitz graphs with small Lipschitz constant. Notice that the solution of the Denjoy conjecture provides an answer to Painleve’s problem for a restricted class of compact sets. In 1967 Vitushkin conjectured that among sets of finite length removability is characterized by pure unrectifiability. That removability implies pure unrectifiability is again a consequence of Calderon’s theorem, by essentially the same argument that gives the Denjoy conjecture. The other implication is the difficult one. Roughly speaking, if one wants to proof that a pure unrectifiable set E is removable, by using at some point Calderon-Zygmund theory, you are in trouble because your set is irregular, that is, homogeneity is missing in this case. To be precise, let µ be a positive Borel measure in the plane. Given ² > 0 and a compact supported function in L1(µ), set Z f(ξ) C (fµ)(z) = dµ(ξ), z ∈ C. ² ξ − z |ξ−z|>²

One says that the above Cauchy integral maps boundedly Lp(µ) into Lp(µ) whenever

kC²(fµ)kp ≤ C(p)kfµkp, where C(p) is a positive constant independent of ² and of the compactly supported function f ∈ Lp(µ). The key facts that µ satisﬁes only the growth condition

µ(D) ≤ C radius (D), for each disc D. In recent years, the list of results that one can prove without resorting to the doubling condition is amazing: the Calderon-Zygmund decomposition and the derivation of weak type (1, 1), and Lp bounds, 1 < p < ∞, from L2 bounds, Cotlar’s inequality for maximal singular integral, the T 1 and T b theorems, and many others.