Harmonic Analysis and Its Applications

Transcription

1 Harmonic Analysis and Its Applications In these lectures, we concentrate on the motivations, development and applications of the Calderon-Zygmund operator theory. Lecture. The differential operators with constant coefficients and the first generation of Calderon-Zygmund operators Consider the following differential operator with constant coefficients: Lu(x) = α a α α u x α. (.) By taking the Fourier transform, (Lu)(ξ) = α a α ( 2πiξ) α û(ξ). (.2) This suggests one to consider the following more general Fourier multiplier: Definition.3: An operator T is said to be the Fourier multiplier if (T f)(ξ) = m(ξ) ˆf(ξ). (.4) (.2) shows that any classical differential operator is a Fourier multiplier. Example : Suppose f L 2 (R) and F is an analytic extension of f on R 2 + given by F (x + iy) = i π x + iy t f(t)dt = i π y (x t) 2 + y 2 f(t)dt i π (x t) iy (x t) 2 + y 2 f(t)dt (x t) (x t) 2 f(t)dt. (.5) + y2 = π Letting y 0, then y π (x t) 2 +y f(t)dt f(x) for a. e. x, and, in general, the second 2 term above has no limit. However, one can show p.v x tf(t)dt exists for a. e. x. Thus where H is called the Hilbert transform defined by lim y 0 F (x + iy) = f(x) + ih(f)(x) (.6) H(f)(x) = π f(t) dt. (.7) x t

2 Example 2: Consider the Laplacian By taking the Fourier transform, ( u)(ξ) = u = n j= 2 u x 2. (.8) j n (4π ξ j ) 2 û(ξ) = 4π ξ 2 û(ξ). j= Define the Riesz transforms R j, j n, by (R j f)(ξ) = ξ j ξ ˆf(ξ). (.9) Then Thus, ( 2 u )(ξ) = 4πξ i ξ j û(ξ) = (R i R j u)(ξ). (.0) x i x j 2 u x i x j = R i R j u. (.) Since, it is easy to see that H and R j, j n, are bounded on L 2, so we obtain lim y 0 F (x + iy) 2 C f 2 (.2) and 2 u 2 = R i R j u 2 C u 2. (.3) x i x j The Hilbert transform, by taking the Fourier transform, can be written as H(f)(ξ) = isign(ξ) ˆf(ξ) (.4) and the Riesz transform, by taking the inverse Fourier transform, can be written as R j (f)(x) = c n p.v. y j f(x y)dy, j n. (.5) y +n In 952, Calderon and Zygmund introduced the following first generation of singular integral operators: Definition.6: Ω(y) T (f)(x) = p.v. f(x y)dy, y n 2

3 where Ω satisfies the following conditions: Ω(λy) = Ω(y) (.7) for all λ > 0; Ω C (S n ); (.8) Ω(y)dσ(y) = 0. (.9) First we point out that the first generation of Calderon-Zygmund operators are well defined on S(R n ), the Schwartz test function space. To see this, one can define Ω(y) p.v. y n f(y)dy = lim ɛ 0 y < Ω(y) [f(y) f(0)]dy + y n y >ɛ y Ω(y) y n f(y)dy = Ω(y) f(y)dy (.20) y n for all f S since Ω has zero average. It is then easy to see that both integrals above converge. Remark.2: A necessary condition for p.v. Ω(y) y n f(x y)dy exists is that Ω has zero average on S n. In fact, let f S be such that f(x) = for x 2. Then for x <, T (f)(x) = lim ɛ 0 ɛ< y < Ω(y) y n dy + y Ω(y) f(x y)dy. (.2) y n The second integral is convergent but the first equals lim ɛ 0 Ω(y)dσ(y)log( ɛ ). Thus, S n if this limit is finite, then the integral of Ω on S n is zero. Theorem.22(Calderon and Zygmund): If T is an operator of the first generation of Calderon-Zygmund operator, then T is bounded on L p, < p <. Moreover, there is a constant C such that T (f) p C f p. (.23) The method of the proof of theorem.22 is called the real variable method of Calderon and Zygmund. This method includes the following steps. Step : T is bounded on L 2. 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 S n, m(ξ) = ˆK(ξ) = S n Ω(t)[log( t ξ ) iπ sign(t ξ)]dσ(t). (.24) 2 3

4 Indeed, since K is homogeneous of degree n, so m(ξ) is homogeneous of degree 0. Therefore, we may assume that ξ S n. Since Ω has zero average, m(ξ) = lim ɛ 0 ɛ< y < ɛ Ω(y) y n e 2πiy ξ dy = lim ɛ 0 S n Ω(y)[ ɛ (e 2πity ξ ) dt t + ɛ 2πity ξ dt e t ]dσ(y) = lim ɛ 0 S n Ω(y)[ ɛ (cos(2πty ξ) ) dt t + ɛ cos(2πty ξ dt t ]dσ(y) ilim ɛ 0 S n Ω(y) ɛ ɛ sin(2πty ξ) dt t dσ(y) Making the change of variables s = 2πty ξ and assume y ξ 0, the second term above will be lim ɛ 0 S n Ω(y) ɛ ɛ sign(2πty ξ) dt t dσ(y) = = S n The first, after the change of variables, will be S n Ω(y) π Ω(t) sign(t ξ)dσ(t). 2 0 sign(y ξ) sin(s) dsdσ(y) s lim ɛ 0 S n Ω(y)[ ɛ (cos(2πy ξ) ) dt t + ɛ cos(2πy ξ dt t ]dσ(y) = = S n Ω(y)[ 2π y ξ ɛ (cos(s) ) ds s + = S n 2π y ξ ɛ Ω(t)[log( t ξ )dσ(t) cos(s) ds s since Ω has zero average. Now applying the Plancheral yields 2π y ξ T (f) 2 = (T (f) 2 = ˆK ˆf 2 C ˆf 2 = C f 2. 4 ds s ]dσ(y)

5 Step 2: We show T is of week type (, ): There is a constant C such that {x R n : T (f)(x) > λ} C f λ (.25) for any λ > 0 and f L L 2. To do this, we need the following Calderon-Zygmund decomposition. Calderon-Zygmund decomposition.26: Given f L and non-negative, and given a positive λ, there exists a sequence {Q j } of disjoint cubes such that for x / Q j ; f(x) λ (.27) Q j λ f ; (.28) λ < f(x)dx 2 n λ. (.29) Q j Q j The proof of this decomposition is to use the so-called stopping time argument. First, choose a large cube Q so that Q f(x)dx Q f(x)dx λ. Then divide Q to 2 n equal Q subcubes Q. Now we use the stopping time argument as follows: if Q f(x)dx > λ, we Q keep this subcube Q. Otherwise, divide this subcube as above and keep this procedure. Now we get a sequence {Q j }. If x / Q j, this means that there is a sequence {Q n } with Q n 0 as n, so that x Q n for all n and Q n f(x)dx λ which shows Q n (.27). To see (.29), notice that if λ < Q j f(x)dx then λ 2 n Q j f(x)dx which Q j 2 n Q j yields (.29). Finally, from λ < Q j f(x)dx we obtain Q j < λ f(x)dx. Summing up Q j Q j shows Q j f(x)dx λ λ f. j Q j Now we apply the Calderon-Zygmund decomposition to show T is of the week type (, ). Define g(x) = f(x) for x / Q j and g(x) = Q j f(x)dx for x Q j, and b(x) = Q j f(x) g(x). Since T is bounded on L 2, so {x R n : T (g)(x) > λ 2 } C T (g) 2 2 λ 2 C g 2 2 λ 2 C λ 2 [ + j Q j Q j Q j f(y)dydx] C λ [ ( Q j ) c 5 ( Q j ) c f(x) 2 dx f(x) dx + 2 n Q j ] C λ f.

6 To estimate T (b)(x), it is easy to see {x R n : T (b)(x) > λ 2 } 2Q j + {x ( 2Q j ) c : T (b)(x) > λ 2 }. By (.28), it suffices to show {x ( 2Q j ) c : T (b)(x) > λ 2 } C λ f. Rewrite b(x) = b j (x), where b j (x) = [f(x) Q j f(x)dx]χ Qj (x) and the sum converges in L 2, as well j Q j as pointwise. Ifx ( 2Q j ) c, then, by the fact that b j has zero average, T (b j )(x) = K(x y)b j (y)dy = [K(x y) K(x x Qj )]b j (y)dy where x Qj K, is the center of Q j. Thus, by the fact thatx ( 2Q j ) c and the smoothness of T (b j )(x) C Q j (Q j ) x x Qj n+ b j(y) dy We now estimate the integral T (b)(x) dx C ( 2Q j ) c j ( 2Q j ) c Q j (Q j ) x x Qj n+ b j(y) dydx. We apply Fubini s theorem to change the order of integration in each of the double integrals and it gives C b j. This, in turn, can be estimated by using (.28) and (.29). So we j get ( 2Q j ) c T (b)(x) dx C f which shows {x ( 2Q j ) c : T (b)(x) > λ 2 } C f λ. This fact together with (.28) shows {x R n : T (b)(x) > λ 2 } C f λ. Noting {x R n : T (f)(x) > λ} {x R n : T (g)(x) > λ 2 } + {x Rn : T (b)(x) > } yields the proof of (.25) and the proof of step 2 is complete. λ 2 Step 3: By Marcinkiewicz s interpolation theorem, T is bounded on L p for < p 2, and T (f) p C f p. Step 4: We observe that the adjoint T of T satisfies the same conditions as T, so T is bounded on L p, < p 2. The duality argument shows T is bounded on L p, 2 p <. 6

7 Example: Let f(x, y) be the density of a mass distribution in the plane. Then its Newtonian potential in the half-space R 3 + is g(x, y, z) = R 2 f(u, v) dudv. [(x u) 2 + (y v) 2 + z 2 ] 2 Formally, we have g lim z 0 (x, y, z) = x R 2 f(u, v)(x u) dudv. [(x u) 2 + (y v) 2 ] 3 2 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 x 2 +y. 2 We can consider the first generation of Calderon-Zygmund operators as the Fourier multipliers. Theorem.30: If m C (R n \{0}) is a homogeneous function of degree 0, and T m is the Fourier multiplier defined by (T m f) = m ˆf, then there exist a, a complex number and Ω C (S n ) with zero average such that for any f S, Ω(y) T m f = af + p.v. f(x y)dy. (.3) 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 S n, theorem.30 is a consequence of the following lemma. Lemma.32: Let m C (R n \{0}) be a homogeneous function of degree 0, and T m is the Fourier multiplier defined by (T m f) = m ˆf, then there exist a, a complex number and Ω C (S n ) with zero average such that ˆm(y) = p.v. Ω(y) y. n Proof: Since m is a tempered distribution, ˆm exists. Thus, where C is a constant. The function n m x n i and has zero average on S n. Moreover, n m x n i ( n m x n )(ξ) = Cξi n ˆm(ξ), i = p.v. n m x n i is homogeneous of degree -n, in C (R n \{0}) + α k C α D α δ, (.33) where δ is the Dirac measure at the origin, since the difference between n m n m x n i x n i and p.v. is a distribution supported at the origin. By taking the Fourier transform on the 7

8 both sizes of (.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 (.33) is a homogeneous function of degree 0 which is in C (R n \{0}). Since this valid for i n, ˆm coincides on R n \{0} with a homogeneous function of degree -n. We denote its restriction to S n by Ω. To see that Ω has zero average, fix a radial function φ S, which is supported on x 2 and positive for < x < 2. Then Ω(x) ˆm(φ) = x n φ(x)dx = c Ω(x)dσ(x), (.34) where c > 0. On the other hand, since φ is radial and m is homogeneous, ˆm(φ) = m( φ) = c m(x)dσ(x) = 0 which together with (.34) shows Ω(x)dσ(x) = 0. Finally, to see that m is identical to p.v. Ω(x) x, consider their difference m p.v. Ω(x) n x, which is supported at the origin. By taking n the Fourier transform of this difference, we get a polynomial which must be a constant because both m and(p.v. Ω(x) x ) are bounded. Furthermore, this constant must be zero since n both m and Ω have zero average on S n. Theorem.35: The set A of operators defined by theorem.30 is a commutive algebra. An element of A is invertible if and only if m is never zero on S n. Proof: If T m and T m2 are in A, then T m T m2 = T m m 2.T m is invertible if and only if T A, and hence, /m C (R n \{0}) which shows m(ξ) 0 for any ξ S n. We m now return to study Lu(x) = a α u α x. Define α Then α =m where the operator T is defined by where P (ξ) = α =m (Λf)(ξ) = 2π ξ ˆf(ξ). (.36) Lu = T Λ m u, (.37) (T u)(ξ) = i m P (ξ) û(ξ) (.38) ξ m a α (2πiξ) α. The multiplier P (ξ) ξ m is a C function on S n, and hence, T is an operator in A. To solve Lu(x) = α =m is homogeneous of degree zero and it a α α u x α = f, it suffices to solve (- ) m 2 u = T f because Λ = ( ) 2. 8

9 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: Lu(x) = a α (x) α u x α (2.) α =m where a α (x) C (R n ). Formally, by taking the Fourier transform and then the inverse Fourier transform, we get Lu(x) = a α (x) ( 2πiξ) α û(ξ)e 2πiξ x dξ α =m = [ α =m a α (x)( 2πiξ) α ]û(ξ)e 2πiξ x dξ. (2.2) (2.2) can be written by more general form: σ(x, D)f(x) = σ(x, ξ) ˆf(ξ)e 2πiξ x dξ. (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) = L(x, x y)f(y)dy (2.4) where p.v. L(x, y)e 2πiξ y dy = σ(x, ξ). (2.5) The relationship (2.5), discovered by Calderon and Zygmund in the 950s, 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) = σ(x, ξ) ˆf(ξ)e 2πiξ x dξ = σ(x, ξ)e 2πiξ x f(y)e 2πiy ξ dydξ = [ σ(x, ξ)e 2πiξ (x y) dξ]f(y)dy = L(x, x y)f(y)dy.

10 Using the operator Λ again, Calderon and Zygmund rewrote (2.2) as Lu(x) = ([ a α (x)( 2πiξ) α ξ m ] ξ m û(ξ)e 2πiξ x dξ where T is defined by a α (x)( 2πiξ) α α =m T f(x) = = T (Λ m u) (2.6) σ(x, ξ) ˆf(ξ)e 2πiξ x dξ (2.7) α =m with σ(x, ξ) = ξ. m Thus, σ is homogeneous of degree 0 in the variable ξ. By (2.5), T f(x) = K(x, x y)f(y)dy (2.8) where K(x, y) = σ(x, ξ)e 2πiy ξ dξ. For fixed x, K(x, ) is the inverse Fourier transform of σ(x, ). By theorem.30, for each fixed x there is a constant a(x) and a function Ω(x, ) C (S n ) with zero average on S n 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: for all x R n and all λ > 0; Ω(x, y) = Ω(x, λy) (2.0) Ω(x, y) C (R n S n ); (2.) S n Ω(x, y)dσ(y) = 0 (2.2) for all x R n. The second generation of Calderon-Zygmund operators is the set of T defined by Ω(x, y) T f(x) = p.v. f(x y)dy. (2.3) y n Then the operator T given by (2.3) is bounded on L p, < p <. To show this theorem, by the Calderon-Zygmund real variable method, it suffices to prove the L 2 boundedness of T. To see this, we follow Calderon and Zygmund, by 2

11 performing a spherical harmonic expansion of σ(x, ξ) on the S n for fixed x R n, where σ(x, ξ) is the Fourier transform of p.v. Ω(x,y) y with the variable y whenever x is fixed. The n proof of lemma.32 shows that σ satisfies (2.0), (2.) and (2.3) with Ω replaced by σ. By the regularity with respect to ξ, this gives a norm-convergent sequence σ(x, ξ) = m k (x)h k (ξ) 0 with m k h k <. 0 We extend h k (ξ) as a homogeneous function of degree 0, which is the symbol of an operator we denote by H k, and we write M k for the operator of pointwise multiplication by m k. This yields T = M k H k 0 and the series of operators is convergent in the L 2 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 T σ f(x) = σ(x, ξ) ˆf(ξ)e 2πix ξ 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: (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 S m, which is most common and useful of the general symbol classes. Definition 2.6: A function σ belongs to S m (and is said to be of order m) if σ(x, ξ) is a C function of (x, ξ) R n R n and satisfies the differential inequalities β x α ξ σ(x, ξ) C α,β ( + ξ ) m α, (2.7) 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 (2.7) is satisfied. Roughly speaking, the conditions (2.7) mean that the behavior of σ(x, ξ) looks like a polynomial of order m. Given a symbol in S m, the operator T σ will initially be defined on the Schwartz class of testing functions S. In fact, the integral (2.5) converges absolutely and is infinitely differentiable. An integration by parts argument shows that T σ (f) is a rapidly decreasing function. Indeed, note that (I ξ )e 2πix ξ = ( + 4π 2 x 2 )e 2πix ξ, 3

12 and define the operator L ξ = ( + 4π 2 x 2 ) (I ξ ), then (L ξ ) N e 2πix ξ = e 2πix ξ. Inserting this in (2.5) and carrying out the repeated integrations by parts gives T σ f(x) = (L ξ ) N [σ(x, ξ) ˆf(ξ)]e 2πix ξ dξ which shows T σ f(x) is rapidly decreasing. Since this argument works for any partial derivative of T σ, and, hence, T σ maps S to S, and this mapping is continuous. It is worth to pointing out that if {σ k } is a pointwise convergent sequence of symbols in S m that satisfy the conditions (2.7) uniformly in k, then T σk (f) T σ (f) in S for f S. An alternative way of writing T σ defined in (2.5) is as a repeated integral T σ f(x) = σ(x, ξ)e 2πi(x y) ξ f(y)dydξ. (2.8) However, the integral in (2.8) does not necessarily converge absolutely, even when f S. To with with this integral, fix a function γ C 0 (R n R n ) with γ(0, 0) =. Set σ ɛ (x, ξ) = σ(x, ξ)γ(ɛx, ɛξ). Notice that if σ S m, then σ ɛ S m and they satisfy the condition (2.7) uniformly in ɛ, for 0 < ɛ,. As mentioned above, T σɛ (f) T σ (f) in S when f S, as ɛ 0. Moreover, since the operator T σ defined in (2.8) converges when σ has compact support, we get that T σ f(x) = lim ɛ 0 σ ɛ (x, ξ)e 2πi(x y) ξ f(y)dydξ. (2.9) By the duality relation < T σ f, g >=< f, T σ g > when f, g S. The same proof shows T σ also maps S to S. Theorem 2.20: If σ S 0, then the operator T σ initially defined on S, extends to a bounded operator on L 2. Proof: It suffices to show that T σ f 2 C f 2, (2.2) whenever f S, with C independent of f. Indeed, suppose f L 2 and let {f n } S so that f n f in L 2. Then, by (2.2), T σ (f n ) converges in L 2 norm, and hence, T σ (f n ) converges to T σ (f) in the sense of distributions. We return to the proof of (2.2). First, we assume that σ(x, ξ) has compact support in x. We then write σ(x, ξ) = σ(µ, ξ)e 2πiµ x dµ since σ has compact support in x variable. An integration by parts shows for each multiindex α, (2πiµ) α σ(µ, ξ) = [ x α σ(x, ξ)]e 2πiµ x dx 4

13 and (2πiµ) α σ(µ, ξ) c α, uniformly in ξ. As a result, we obtain sup ξ σ(µ, ξ) C N ( + µ ) N for arbitrary N 0. Now T σ f(x) = σ(x, ξ) ˆf(ξ)e 2πix ξ dξ = σ(µ, ξ)e 2πiµ x ˆf(ξ)e 2πix ξ dξdµ = (T µ f)(x)dµ, where (T µ f)(x) = e 2πix µ (T σ(µ,ξ) f)(x). Since for each µ, T σ(µ,ξ) is a Fourier multiplier operator on the Fourier side, by Plancherel s theorem we have that T σ(µ,ξ) f 2 sup ξ By (2.22), T µ C N ( + µ ) N, which yields T σ C N ( + µ ) N dµ < σ(µ, ξ) ˆf 2 = sup σ(µ, ξ) f 2 (2.22). ξ if we choose N > n. Thus (2.2) is proved when σ has compact support in x. The proof for general symbols needs to use the singular integral realization of the operator T σ. That is, we shall write T σ f(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, σ(x, ξ) = k(x, z)e 2πiz ξdz. Thus, T σ 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) C N z N for all z and all N > 0, uniformly in x. To see this, note that T σ (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

14 ( 2πiz) α k(x, ), with the distribution k(x, ) thought of as acting on functions of z, equals the inverse Fourier transform of ξ ασ(x, ξ); by (2.7), α ξ σ(x, ξ) is integrable in ξ whenever α n +. 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 x 0 R n, x x 0 T σ f(x) 2 dx C N R n f(x) 2 dx (2.25) ( + x x 0 ) N for all N 0. We prove (2.25) first when x 0 = 0. To do this, we split f by f = f + f 2, with f supported in B(0, 3), f 2 supported outside B(0, 2), f and f 2 smooth, and with f, f 2 f. We fix η C0 so that η = in B(0, ). Then ηt σ (f ) = T ησ (f ), and the symbol η(x)σ(x, ξ) has compact support in x, so the previous result applies. Hence T σ f 2 T ησ f 2 C f 2 C f 2 (2.26). R n R n R n B(0,) If x B(0, ), since f 2 is supported away from B(0, 2), the representation (2.23) holds: T σ f 2 (x) = k(x, x z)f 2 (z)dz. (2.27) (B(0,2)) c Since x z when x B(0, ) and z / B(0, 2), using the estimate (2.24) yields T σ f 2 (x) C N z N f(z) dz. (B(0,2)) c By Schwartz inequality, we obtain, for N > n, T σ f 2 (x) 2 dx C N ( + x ) N f(x) 2 dx. (2.28) B(0,) Now T σ (f) = T σ (f ) + T σ (f 2 ), and combining (2.26) and (2.28) shows (2.25) when x 0 = 0. The passage to (2.25) for general x 0 can be achieved by noting that, while an individual pseudo-differential operator is not(in general) translation-invariant, the class S m, in fact, is. To see this, let τ h, h R n, denote the unitary translation operator given by (τ h f)(x) = f(x h). Then, τ h T σ τ h = T σh, where σ h (x, ξ) = σ(x h, ξ). Note that the symbol σ h satisfies the same estimates that σ does, uniformly in h. Hence, (2.25) holds for x 0 = 0, with σ replaced by σ h, with a bound independent of h. If we set h = x 0, we see that (2.25) is established. Finally, it is only a matter of integrating (2.25) with respect to x 0, choosing N > n, and interchanging order of integration. Theorem is proved. 6

15 By combining the L 2 result with the Calderon-Zygmund real variable method, we can also prove the L p boundedness of these operators. However, we need to realize pseudodifferential operators in the class S 0 as singular integrals. To be precise, we shall prove the following result. Proposition 2.29: Suppose σ S m. Then the kernel k(x, z) is in C (R n (R n \{0})), and satisfies β x α z k(x, z) C α,β,n z n m α N, z 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) = for ξ, 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 : = η(ξ) + δ(2 j ξ), (2.3) for all ξ, and = j= j= δ(2 j ξ), (2.32) for all ξ 0. It is worth to pointing out that for each ξ there are at most two nonzero terms in the sums (2.3) and (2.32). Let φ be the inverse Fourier transform of η, i.e., φ(ξ) = η(ξ). Then φ Sand φdx =. Define ψ by ψ(ξ) = δ(ξ). Then ψdx = 0. Writing φ t (x) = t n φ( x t ) and ψ j = ψ 2 j, we then have ψ j = φ j φ j, while (φ j )(ξ) = η(2 j ξ) and (ψ j )(ξ) = δ(2 j ξ). We now define the operator S j by S j (f) = f φ j and j (f) = S j (f) S j (f) = f ψ j. In parallel with (2.3) and (2.32), we have the operator identities and I = S 0 + I = j= Note that if f is a tempered distribution, S j (f) is well defined, and S 0 (f) + j (2.33) j= j. (2.34) N j (f) = S N (f) f, j= as N, in the sense of distribution. However, it is not true that S M (f) 0 as M, for arbitrary f. It fails when f =. Thus (2.34) holds only under some restriction on f. 7

16 We now return to the operator T σ. Using (2.33), we write T σ = T S 0 + T j = j= T σj, where σ 0 (x, ξ) = σ(x, ξ)η(ξ), and σ j (x, ξ) = σ(x, ξ)δ(2 j ξ) for j. Each of the pseudo-differential operators T σj will be written in its singular integral form by T σj f(x) = k j (x, z)f(x z)dz. Since σ j have compact ξ support and are smooth, the kernel k j will also be smooth, and the integrals above will converge for all x. The kernels k j are given by k j (x, z) = j=0 σ j (x, ξ)e 2πiξ z dz. We claim the following estimate: If σ S m, then β x α z k j (x, z) C α,β,n z N 2 j[n+m N+ α ], (2.35) for all multi-indices α, β, and N 0, where C α,β,n is independent of j 0. In fact, observe that ( 2πiz) γ x β z α k j (x, z) γ ξ [(2πiξ)α x β σ j (x, ξ)]e 2πiξ z dξ. Noting that the integrand is supported in 2 j ξ 2 j+ and estimates on σ(x, ξ) and δ(2 j ξ), we obtain that when γ = M, z γ β x α z k j (x, z) C α,β,n 2 j[n+m M+ α ]. Taking the supremum over all γ with γ = M, gives (2.35). Since k(x, z) = k j (x, z), it suffices to show x β z α k j (x, z) satisfies the estimate given by the right side of (2.29). j=0 Consider the case when 0 < z, first. We break the above sum into two parts: the first where 2 j z, the second where 2 j > z. For the first sum we use the estimate (2.35) with M = 0, which is then majorized by a multiple of 2 j(n+m+ α ). This in 2 j z turn is O( z n m α ) when n+m+ α > 0, or O(log( z )+) when n+m+ α 0. In either case we get the estimate O( z n m α M ), under restriction that z, 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 ) 2 j[n+m+ α M] = O( z n m α ). The last term is 2 j > z O( z n m α N ) if N 0, since z. 8 j=0

17 Finally, when z, 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. The proof of the proposition is therefore concluded. Now the L 2 boundeness and the estimates of kernel of T σ allow us to apply the Calderon-Zygmund real variable method, and hence, the L p, < 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 α ξ σ(x, ξ) C α,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: β x α ξ σ(x, ξ) C α,β ( + ξ ) ( β α ). (2.37) Several years later(965), 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 a α (x) of the differential operators a α (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 = ( x )T + + ( x n )T n, the T j, j n, being first generation Calderon-Zygmund operators. In dimension, T is replaced by DH, where D = i d dx and H is the Hilbert transform. In 965, Calderon showed that the commutator [A, DH] was bounded on L 2 (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 H 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) = p.v. K(x, y)f(y)dy ω n where K(x, y) = a(x) a(y) (x, y) a(y) ( x y 2 + (a(x) a(y)) 2 ) (n+)/2, 9

18 and f L 2 (R n ). The Lipschitz domain is defined (locally) by t > a(x), where x R n, t R, and the function a(x) is Lipschitz. If n =, 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 L 2 boundedness for first and second generation Calderon-Zygmund operators does not work anymore. This leads to third generation Calderon-Zygmund operators. 0

19 Lecture 3. Littlewood-Paley Theory and Function Spaces There is a number of ways to set up the Littlewood-Paley theory on R n. One standard way is as follows. Let φ(ξ) be a real radial bump function supported on {ξ R n : ξ 2} which equals on {ξ R n : ξ } Let ψ be the function: ψ(ξ) = φ(ξ) φ(2ξ). Thus ψ is a bump function supported on {ξ R n : 2 ξ 2}. By construction we have ψ(ξ/2 k ) = k for all ξ 0. Thus we can partition unity into the function ψ(ξ/2 k ) for integers k, each of which is supported on an annuls of the form ξ 2 k. We now define the Littlewood-Paley projection operators Q k and P k by (Q k f)(ξ) = ψ(ξ/2 k ) ˆf(ξ) (3.) (P k f)(ξ) = φ(ξ/2 k ) ˆf(ξ). (3.2) Informally, Q k is a frequency projection to time annuls { ξ 2 k }, while P k is a frequency projection to the ball { ξ 2 k }. Observe that Q k = P k P k. Also, if f L 2, then P k (f) 0 in L 2 as k, and P k (f) f in L 2 as k, which follows from the Plancherel theorem. By telescoping the series, we thus have the Littlewood-Paley decomposition f = Q k (f) (3.3) k for all f L 2, where the series converges in the L 2 norm. We are now interested in how the L p behavior of the Littlewood-Paley pieces Q k f relate to the behavior of f. First, we write P k f(x) = f (2 nk φ(2 k )) = f(x + 2 k y)φ(y)dy. (3.4) Note that φ is a Schwartz function with the total mass φ(y)dy = φ(0) =. Thus the function P k f is an average of f localized to physical scales 2 k. In particular, we expect P k f to be essentially constant at scales < 2 k. What does a function Q k f look like? Since Q k f = P k+2 Q k f, we see from (3.4) that Q k f(x) = Q k f(x k y)φ(y)dy. (3.5) Thus, Q k f is essentially constant at physical scales< 2 k. On the other hand, we have P k 2 Q k f = 0, so from (3.3), Q k f(x k y)φ(y)dy = 0

20 for all x R n. This roughly assert that Q k f has mean zero at scales 2 k. From (3.3) and the Minkowski s inequality, for p, we have P k f p f(x + 2 k y) p φ(y) dy C f p. (3.6) Thus, P k f does not get any bigger that f itself as measured in L p, or in any translation invariant Banach space. Similarly, Q k f p C f p. On the other hand, we have f = Q k f, so by the triangle inequality, we obtain the cheap Littlewood-Paley inequality k sup Q k f p C f p C k k Q k f p. (3.7) As the name suggests, the cheap Littlewood-Paley inequality is not the sharpest statement one can make connecting the L p norms of Q k f with those of f. By using the Fourier transform, when p = 2, we get f 2 ( k Q k f 2 2) 2. (3.8) In fact, to see this, square both sizes and take Plancherel to obtain ˆf 2 2 k [ψ( /2 k ) ˆf( ) 2 2. Observe that for each ξ 0, there are only two values of ψ(ξ/2 k ) which do not vanish, and these two add up to. We can rewrite (3.8) as f 2 ( k Q k f 2 ) 2 2. (3.9) The quantity ( Q k f 2 ) 2 is known as the Littlewood-Paley square function. k Now define Sf for the vector-valued function by Sf(x) = {Q k f} k, and Sf = ( Q k f 2 ) 2 is the l 2 norm of Sf. k Theorem 3.0: For < p <, then Sf p f p with the implicit constant depending on p. The proof of theorem 3.0 follows from the Calderon-Zygmund real variable method. In fact, we have the L 2 result, and it suffices to see that S is a vector-valued Calderon- Zygmund operator with vector-valued kernel K(x) = (2 nk ψ(2 k 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

21 first generation Calderon-Zygmund operators. Now the L 2 result implies the L p, < p <, results. By duality we also have S (f k ) p C (f k ) p. Thus, k Q k f k p C ( k f k 2 ) 2 p. Similarly, k Q k f k p C ( k f k 2 ) 2 p, where Q k = P k+2 P k 2. We apply this with f k = Q k f, since Q k Q k = Q k, we obtain f p ( k Q k f 2 ) 2 p. As an application, we give a proof of the Hormander-Mikhlin multiplier theorem. Theorem 3.: Let m(ξ) be a multiplier such that α m(ξ) C ξ α (3.2) for all α 0, where the constant C depends on α. Let T m be the Fourier multiplier with symbol m : (Tm f)(ξ) = m(ξ) ˆf(ξ). Then T m is bounded on L p, < p <. Proof: We have where Q k = Q k T m and Q k = T m = k,k Q k T m Q k k 2<k <k+2 = k Q k Qk Q k. From (3.2) we see that Q k and Q k are smooth frequency localization operators to the annuls { ξ 2 k }. Then the theorem 3. follows from the Littlewood-Paley L p inequality by composing Q k and Q k. A typical consequence of the Hormander-Mikhlin multiplier theorem is the estimate for all < p <, where = xi xj f p C f p n x 2 i i= is the Laplacian on R n. This follows because ( xi xj f)(ξ) = ξ iξ j ξ ( f)(ξ), and the symbol m(ξ) = ξ iξ j 2 ξ satisfies the condition (3.2). 2 The Littlewood-Paley L p, < p <, inequality suggest that one can consider the similar inequality for 0 < p. However, this fails even when p =. Notice that ψ defined in the Littlewood-Paley S function, is in S, the Schwartz test function space, so for any f S, the temperate distribution space, Q k f is well defined. This means that for any f S, Sf is well defined. Now we introduce the hardy space H p as follows. Definition 3.3: For 0 < p <, H p = {f S : Sf L p } and if f H p, the norm of f is defined by f H p = Sf p. 3

22 It is easy to see that when < p <, H p = L p, by the Littlewood-Paley L p inequality. For 0 < p, H p is a new space which is different from L p. The best way to see this is to get the so-called atomic decomposition of H p. Here we only consider p =. To do this, let S = {f S : f(x)x α dx = 0, for all α 0}. We then have Theorem 3.4: Suppose f S and k ψ(2 k ξ) 2 =, where ψ is a bump function supported on the annuls { x 2 k }. Then f = k Q k Q k f, where the series converges in the topology of S. By a duality argument, for f S and g S we have < k Q k Q k f, g >=< f, k Q k Q k g >=< f, g >. Hence, for any f S, f = k Q k Q k f in (S ) S /P, S modulo polynomials. Definition 3.5: A function a(x) is said to be an atom if a(x) satisfies (i) Supp a Q, a cube in R n ; (ii) a 2 Q 2 ; (iii) a(x)dx = 0. Theorem 3.5: f H if and only if f = k Moreover, f H inf{ k λ k a k, where a k are atoms and k λ k : for allf = k λ k a k }. λ k <. As mentioned before, H p is well defined for 0 < p <, but not for p =. Next, we consider the replacement of H p when p =. Definition 3.6: For f L loc (Rn ) we let f = sup Q m Q f m Q f, where m Q f is the average of f over Q, and we define the space BMO(R n )( functions of bounded mean oscillation) to consist of those functions f such that f <. (BMO(R n ) 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 3.7: There exist two positive constants λ > 0 and C > 0 such that for any f BMO, λ sup m Q (exp( f m Q f C. (3.8) Q f 4

23 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 f. Let Q 0 be a fixed cube and Q some dyadic cube. Recall that Q is the unique dyadic cube which contains Q and lies in the previous generation. It is easy to see that m Q f m Q f 2 n f. (3.9) Consider now the Calderon-Zygmund decomposition of the function (f m Q0 f)χ Q0 for λ = 2 f. This yields a collection of dyadic cubes Q i, maximal with respect to inclusion, satisfying m Qi (f m Q0 f)χ Q0 > 2 f (3.20) and (f m Q0 f)χ Q0 2 f (3.2) on ( Q i ) c. Clearly, Q i Q 0 for each i, and Q i (f m Q 0 f)χ Q0 2 f Q 0 2. Since the Q i s are maximal. (f m Q0 f 2 f m Qi, and (3.9) gives m Qi f m Q0 f (2 n + 2) f. Let X(λ) = sup Q m Q exp( λ f f m Q f, which is finite since we re assuming that f is bounded. We obtain λ m Q0 (exp( f m Q f ) e 2λ dx f Q 0 + Q i Q 0 Q i i [ Q i Q 0 \ Q i λ exp( f m Qi f )dxe (2 f e 2λ + 2 [exp(2n + 2)]X(λ). n +2)λ ] From taking the supremum over all cube Q 0 it follows that X(λ)[ 2 exp(2n + 2)] e 2λ, 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) m Q f > t f } Ce λt Q. (3.22) For p <, then f p, = sup Q [m Q f m Q f p ] p are equivalent. Now we are ready to prove the duality of H and BMO. We shall see that each continuous linear functional on H can be realized as a mapping l(g) = f(x)g(x)dx, g H, 5

24 when suitably defined, where f is a function in BMO. For general f BMO and g H, the integral in (3.23) does not converge absolutely. For this reason, we take g H has finite atomic decomposition. We denote this subspace by H a which is dense in H. Theorem 3.24: (a) Suppose f BM O. Then the linear functional given by (3.23), initially defined on H a, has a unique bounded extension to H and satisfies l c f. (b) Conversely, every continuous linear functional on H can be realized as above, with f BMO, and with f c l. Proof: To see (a), note that if a is an atom supported on Q, then f(x)a(x)dx = [f(x) m Q f]a(x)dx Q 2 f 2, a 2 c f. Thus, if g Ha, then g = λ k a k with the sum having a finete terms and a k are atoms. k So f(x)g(x)dx = λ k [f(x) m Qk f]a k (x)dx k k λ k Q k 2 f 2, a 2 c f λ k. To show (b), fix a cube Q and let L 2 Q be the space of all square integrable functions supported on Q. Let L 2 Q,0 = {f L2 Q : f(x)dx = 0}. Note that every g L 2 Q,0 is a multiple of an atom and g H c Q 2 g 2. Thus if is a given linear functional on H with the norm, then extends to a linear functional on L 2 Q,0 with norm at most c Q 2. By the Riesz representation theorem for the Hilbert space L 2 Q,0, there exists an element F Q L 2 Q,0 so that l(g) = F Q (x)g(x)dx, (3.25) k if g L 2 Q,0, with ( F Q (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, on each Q, f differs from F Q by a constant. To construct this f, observe that if Q Q 2, then F Q F Q 2 is constant on Q. Indeed, both F Q and F Q 2 give the same functional on L 2 Q,0, so they must differ by a constant on Q. We can modify F Q, replacing it with f Q = F Q + c Q, where c Q is a constant chosen so that f Q has the average zero over the 6

25 unit cube centered at the origin. It follows that f Q = f Q 2 on Q, if Q Q 2. Finally, we define f on R n by taking f(x) = f Q (x) for x Q. Observe that Q Q f(x) c Q dx ( Q Q f(x) c Q 2 dx) 2 = ( Q Q F Q 2 dx) 2 c, which shows f BMO with f c. Also, by (3.25), l(g) = F Q (x)g(x)dx, if g L 2 Q,0, for some Q, in particular, this representation holds for all g H 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 R n+ + is said to be a Carleson measure if sup Q Q T (Q) dµ C <. (3.28) If dµ is a Carleson measure, we denote dµ C, the Carleson norm of dµ, by the smallest constant C in (3.28). Theroem 3.29: Suppose φ S with φ(x)dx =. Then dµ is a Carleson measure if and only if φ t f(x) 2 dµ C f 2 2. (3.30) R n+ + Theorem 3.3: Let ψ S with only if f ψ t (x) 2 dxdt t 0 2 dt ψ(tξ) t is a Carleson measure. = for all ξ 0. Then f BMO if and 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 Ḃα,q p and Triebel-Lizorkin spaces F p α,q as well as their inhomogeneous analogs. Let us choose φ S so that Supp φ {ξ : 2 ξ 2} and φ(ξ) c > 0 if 3 5 ξ 5 3. For α R, p, 0 < p, q and f S we define f F α,q p = { k (2 kα φ k f ) q } q p, (3.32) and, for the same indices, and including p =, f Ḃα,q p = { k (2 kα φ k f p ) q } q, (3.33) where φ k (x) = 2 kn φ(2 k x). 7

26 Note that φ k f is a smooth function when φ S and f S. Also f F p α,q = 0 = 0 if and only if φ k f is the zero function for all k. But this is equivalent or f Ḃα,q p to having ˆf(ξ) φ(2 k ξ) be zero for all k. because of the conditions on φ, 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 S /P in these equalities, where P denote the class of polynomials on R n. In particular, we define F p α,q and Ḃα,q p 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 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 Φ f p to variants of the above expressions, where Φ S, Supp Φ {ξ : ξ 2} and Φ(ξ) c > 0 if ξ 5 3. The variants in question are as in (3.32) and (3.33) with replaced by. These spaces are denoted by Fp α,q and Bp α,q and k k they are spaces of tempered distributions; the necessity of considering such distributions modulo polynomials disappears since Φ(0) 0. By the results mentioned above by the Littlewood-Paley theory, we obtain the following identifications: L p F p 0,2 when < p < ; when 0 < p ; H p F 0,2 p BMO F 0,2 when F 0,2 is defined by the Carleson measure. We also can show L p α Fp α,2 when α > 0 and < p < ; Λ α F α, and Λ α F α, and L p α F α,2 p when α > 0. Suppose φ and Φ are two other functions satisfying the properties of φ and Φ announced above. One can show that replacing φ and Φ with φ and Φ in the definitions yields the same spaces with equivalent norms. 8

27 Lecture 4. Third generation Calderon-Zygmund operators and the T theorem The pseudo-differential calculus is like that mythological bird. Its first birth was at the end of the 930s, the founding fathers being Giraud and Marcinkiewicz. The second birth took place at the end of the 950s, as we discussed in the Lecture 2, and it clearly benefited from the theory of distributions, developed by Schwartz during the 940s. 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 pseudodifferential operators. An important step was taken in 965 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 L 2 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. A(x) A(y) (x y) 2 f(y)dy. (4.) 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 978 International Congress lecture. Let L be an operator defined by Lf(x) = m j 0 a j (x) dj f dx j. (4.2) As we did in lecture 2, by Fourier transform and Fourier inversion, Lf(x) = m a j (x)(iξ) j ˆf(ξ)e ixξ dξ. (4.3) 2π j 0 The idea behind pseudo-differential operators is to replace the function a j (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 (Λf)(ξ) = ψ(ξ) ˆf(ξ) where ψ is an infinitely differentiable function with ψ(ξ) = ξ if ξ, and let T f(x) = q(x, ξ) ˆf(ξ)e ixξ dξ + Rf(x), (4.4)

28 where Rf(x) = r(x, ξ) ˆf(ξ)e ixξ dξ. (4.5) Here q(x, ξ) = ξ m a m (x)(iξ) m and r(x, ξ) is defined by the relation 2π m a j (x)(iξ) j = (q(x, ξ) + r(x, ξ))(ψ(ξ)) m. j=0 Now we can write Lf = T Λ m f. It is easy to show that the operator R and R d dx are bounded on L2, say, provided the coefficients a j 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 R d dx 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 sign(ξ) ˆf(ξ)e ixξ dξ, (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 L 2. Now (AH HA)D = [A, HD] + HDA HAD and DA AD is just multiplication by a (x), which of course is bounded on L 2 since a is a bounded function. Hence, HA belongs to the class if and only if [A, HD] is bounded on L 2, 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 C k f(x) = p.v. ( a(x) a(y) ) k f(y) dy, k. (4.9) x y x y 2

29 These operators are closely related to the boundary behavior of analytic functions given by Cauchy integrals f(z(x)) = 2πi z(y) z(x) f(z(y)z (y)dy (4.0) on Lipschitz curve z(x) = x + ia(x), a 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. a(x) a(y) (x y) a(y) f(y)dy, (4.) ω n ( x y 2 + (a(x) a(y)) 2 ) n+ 2 R n where ω n is the area of the unit sphere in R n. To solve the Dirichlet problem in a Lipschitz domain by the method of layer potentials, one needs to prove the boundedness of L 2 of the above operator with a Lipschitz. We emphasize that while for convolution operators, boundedness on L 2 is a simple application of Plancherel s theorem, the L 2 boundedness for non-convolution operators like the ones above is highly nontrivial. In 978 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 S. By the Schwartz kernel theorem there is a distribution K in S 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 4.2: We say that K is a Calderon-Zygmund kernel if its restriction to the set Ω = {(x, y) R n R n : x y} is a continuous function K(x, y) which satisfies if x x 2 x y, K(x, K(x, y) C x y n, (4.3) K(x, y) K(x, y) C x x ɛ y) K(x, y ) C y y ɛ, (4.4) x y n+ɛ, (4.5) x y n+ɛ if y y 2 x y, for some constant C and some ɛ in (0, ]. We call T a Calderon-Zygmund singular integral operator, and write T CZSIO, or T CZSIO(ɛ), if the kernel of T satisfies these conditions. In particular, if T CZSIO, then (T f, g) = K(x, y)g(x)f(y)dydx (4.6) 3

30 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 L 2. We write T CZO. By the Calderon-Zygmund real variable method, we can easily get the following theorem. Theorem 4.7: If T CZO, then T is bounded on L p for < p <, from H to L, and from L to BMO. A basic, important problem is how to understand when a Calderon-Zygmund singular integral operator is bounded on L 2. As Meyer wrote To go beyond the context of convolution operators, it becomes independensable to have a criterion for L 2 continuity, without which the theory collapses like a house of cards. This problem was solved by the celebrated T theorem of David-Journe in 983. 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 965, Calderon proved the L 2 boundedness of Calderon commutator. After 9 years, in 974, Coifman and Meyer proved the same result for the second order Calderon commutator. in 977, Calderon proved the L 2 boundedness for the Cauchy integral on a Lipschitz curve with the Lip norm < ɛ, where ɛ is a small number. Coifman, McIntosh and Meyer, in 98, proved the L 2 boundedness for the Cauchy integral on all Lipschitz curves. Before we can state the David-Journe T theorem, we require some preliminary definition and concepts. We begin by assigning a meaning to T and T for T CZSIO. Even in the classical case of T CASIO, this requires care. Denote D the set of all f S with the compact support and D 0 = {f D : f(x)dx = 0}. We now can define T as a linear functional on D 0 as follows. For any given f D 0, choose χ D such that χ (x) = for x 2Q, where Q is a cube containing support of f, and = χ + χ 2. Since χ D, so < T χ, f > is well defined. Note that, formally, < T χ 2, f >=< χ 2, T f > and this makes sense if < χ 2, T f > is well defined. Indeed, by the fact thta f D 0, we have < χ 2, T f >= χ 2 (x)k(x, y)f(y)dydx = χ 2 (x) [K(x, y) K(x, y Q )]f(y)dydx. By the conditions on K, we obtain the absolute value of the last f. This means < χ 2, T f > is well defined. Now we define < T, f >=< T χ, f > + < χ 2, T f >. It is easy to see that the equality does not depend the choice of χ. We also define T = 0 by the fact that T f(x)dx = 0 for all f D 0, and T = 0 is defined similarly. Note that D 0 is a subset, so T is a BMO function means that T, as a linear functional on D 0, can be extended to be a linear functional on H, and it is similar for T being a BMO function. Next we consider a condition known as the weak boundedness property. Definition 4.8: We say that a linear continuous operator T : S S satisfies the weak boundedness property, and we write T W BP, if for each bounded subset B of S there exists a constant C = C(B) such that for all f, g B, where f z t (x) = t n f( x z t ). < T (f z t ), g z t > Ct n, z R n, t > 0, (4.9) 4