Fourier Analysis (Graduate Studies in Mathematics)

00,

if and only if

(1.7)

Proof. The necessity of (1.7) follows from the uniform boundedness principle.

To see that it is sufficient, first note that if 9 is a trigonometric polynomial, then SN9 = 9 for N > degg. Therefore, since the trigonometric polynomials are dense in LP (see Corollary 1.11), if f E LP we can find a trigonometric polynomial g such that Ilf - gllp < E, and so for N sufficiently large

o If 1 < p < 00, then inequality (1.7) holds, as we will show in Chapter 3. When p = 1, the L1 operator norm of S N is again L N, and so by Lemma 1.7 the answer to the first question is no. When p = 2, the functions {e27fikx} form an orthonormal system (by (1.3)) which is complete (i.e. an orthonormal basis) by the density of the trigonometric polynomials in L2. Therefore, we can apply the theory of Hilbert spaces to get the following.

5. Summability methods

9

Theorem 1.9. The mapping f ~ {j(k)} is an isometry from L2 to £2, that ~s,

= 2::

Ilfll§ =

Ij(k)12.

k=-= Convergence in norm in L2 follows from this immediately. The second question is much more difficult. A. Kolmogorov (1926) gave an example of an integrable function whose Fourier series diverges at every point, so the answer is no if p = 1. If f E LP, 1 < p < 00, then the Fourier series of f converges almost everywhere. This was shown by L. Carleson (1965, p = 2) and R. Hunt (1967, p > 1). Until the result by Carleson, the answer was unknown even for f continuous.

5. Summability methods In order to recover a function f from its Fourier coefficients it would be convenient to find some other method than taking the limit of the partial sums of its Fourier series since, as we have seen, this approach does not always work well. One such method, Cesaro summability, consists in taking the limit of the arithmetic means of the partial sums. As is well known, if lim ak exists then .

a1

hm k----= also exists and has the same value.

+ ... + ak k

Define 1 aN f(x)

=

N

N

+

=1

1

2:: Skf(x) k=O

1

1

f(t) N

o

=

l

+

N

1

2:: Dk(X - t) dt k=O

!(t)FN(X - t) dt,

where FN is the Fejer kernel,

FN(t) =

1 N

tDk(t) =

+ 1 k=O

1 N

+

FN has the following properties: FN(t) > 0,

(sin(1f.(N + 1)t))2 1 sm(1ft)

10

1. Fourier Series and Integrals

(1.8) lim

IIFNIlI =

[FN(t) dt

f

FN (t) dt = 0 if 0 > O.

N-?= J8
= 1,

Because FN is positive, its L1 norm coincides with its integral and is 1. This is not the case for the Dirichlet kernel: its integral equals 1 because of cancellation between its positive and negative parts while its L1 norm tends to infinity with N.

Theorem 1.10. If f E V, 1 then

Proof. Since


00,

or if f is continuous and p

=

00,

J FN = 1, by Minkowski's inequality we have that

IlaN f - flip = 11/2 IIfC - t) - f(·) IIpFN(t) dt -1/2

<

f

J tl<8

IIfC - t) - f(')lIpFN(t) dt + 211fllp

1

Since for 1
f

FN(t) dt.

J8
< 00,

ETa IIf(' -

t) - f(·) lip = 0,

and the same limit holds if p = 00 and f is continuous, the first term can be made as small as desired by choosing a suitable O. And for fixed 0, by (1.8) the second term tends to O. 0

Corollary 1.11.

(1) The trigonometric polynomials are dense in LP, 1

00.

(2) If f is integrable and j(k) = 0 for all k, then f is identically zero. A second summability method is gotten by treating a Fourier series as the formal limit on the unit circle (in the complex plane) of

(1.9)

u(z) =

=

L

-1

L

j(k)zk +

k=O

j(k)zlkl,

z = re 27riO .

k=-=

Since {j(k)} is a bounded sequence, this function is well defined on Izl It can be rewritten as

u(re 27riO ) =

=

L

k=-=

j(k) r k e27rikO = 1

1

1

1/2

-1/2

f(t)Pr (()

-

t) dt,

< 1.

6. The Fourier transform of L1 functions

11

where

is the Poisson kernel. The Poisson kernel has properties analogous to those of the Fejer kernel:

l

(1.10)

Pr(t) > 0, Pr(t) dt = 1,

lim ( Pr (t) dt r-.1- J8
a result

Theorem 1.12. If f E LP, 1 then

= 0 if 6" > O.

analogous to Theorem 1.10.


lim IIPr r-.1-

*f

00/\

or if f is continuous and p

- flip

Since the function u is harmonic on Iz I Dirichlet problem: ~u

=0 u=f

=

00,

= O. <

1, it is the solution to the

if Iz I < 1, if Iz I = 1,

where the boundary condition is interpreted in terms of Theorem 1.12. In Chapter 2 we will study the almost everywhere convergence of (J N f (x) and Pr * f(x).

6. The Fourier transform of L1 functions Given a function

f

E L1(JRn ), define its Fourier transform by

j(~) = I f(x)e-27rix.~ dx, . )jRn

(1.11)

where x . ~ = x16 + x26 of the Fourier transform: (1.12) (1.13) (1.14) (1.15)

+ ... + xn~n.

The following is a list of properties

+ /3g)"'= oJ + /3fJ (linearity); Iljlloo < Ilflh and j is continuous;

(af

lim j(~)

I~I-'oo

= 0 (Riemann-Lebesgue);

(f * g)"'= jfJ;

1. Fourier Series and Integrals

12

(1.16)

(Thfr(~)

=

1(~)e21rih.f;" where Thf(x)

(fe21rih.xr(~)

(1.17)

=

I(~

= f(x + h);

- h);

if pEOn (an orthogonal transformation), then

(f(p'))l~) = I(p~);

= A-nf(>..-lx), then g(~) =

(1.18)

if g(x)

(1.19)

.( 8 f ) ~ 8xj (~) = 27ri~jf(~);

(1.20)

(-2"ixjfn/J =

I(A~);

;~ (~)

The continuity of 1 follows from the dominated convergence theorem; (1.14) can be proved like Lemma 1.4; the rest follow from a change of variables, Fubini's theorem and integration by parts. In (1.19) we assume that 8f j8xj E LI and in (1.20) that xjf ELI. Unlike on the torus, LI(JRn ) does not contain LP(JRn ) , P > 1, so (1.11) does not define the Fourier transform of functions in those spaces. For the same reason, the formula which should allow us to recover f from

I,

I,

may not make sense since (1.13) and (1.14) are all that we know about and they do not imply that is integrable. (In fact, is generally not integrable. )

1

1

7. The Schwartz class and tempered distributions A function f is in the Schwartz class, S(JR n ), if it is infinitely differentiable and if all of its derivatives decrease rapidly at infinity; that is, if for all a.,/3 E Nn , sup IxC! D/J f(x)1

= Pa,/J(f) < 00.

x

Functions in ergo are in S, but so are functions like e- 1x12 which do not have compact support. The collection {Pa,/J} is a countable family of seminorms on S, and we can use it to define a topology on S: a sequence {
7. The Schwartz class and tempered distributions

13

With this topology S is a Frechet space (complete and metrizable) and is dense in LP(IRn ), 1 < p < 00. In particular, SeLl and (1.11) defines the Fourier transform of a function in S. The space of bounded linear functionals on S, S', is called the space of tempered distributions. A linear map T from S to C is in S' if lim T(¢k)

k --'> 00

=0

whenever

lim ¢k

k --'> 00

=0

in S.

Theorem 1.13. The Fourier transform is a continuous map from S to S such that

r fff = JF.nr jg.J'Rn

(1.21 )

i

and (1.22)

Equality (1.22) is referred to as the inversion formula. To prove Theorem 1.13 we need to compute the Fourier transform of a particular function.

Lemma 1.14. If f(x)

= e-rrlxl2 then

j(~) = e-rrl~12.

Proof. We could prove this result directly by integrating in C, but we will give a different proof here. It is enough to prove this in one dimension, since in IRn j is the product of n identical integrals. The function f(x) = e- rrx2 is the solution of the differential equation

u'

+ 27rxU = 0, u(O) = 1.

By (1.19) and (1.20) we see that with the initial value

U(O)

=

Therefore, by uniqueness,

J.

u satisfies the same differential equation

u(x) dx

=

fa

e- rrx2 dx

=

1.

j = f.

Proof of Theorem 1.13. By (1.19) and (1.20) we have ~Cl D(3 j(~) = C(D Cl x(3 fr(~),

so

o

1. Fourier Series and Integrals

14

The Ll norm can be bounded by a finite linear combination of semi norms of f, which implies that the Fourier transform is a continuous map from S to itself. Equality (1.21) is an immediate consequence of Fubini's theorem since f(x)g(y) is integrable on IR n x IRn. From (1.18) and (1.21) we get

J f(x)g(Ax) dx = J j(x)A-ng(A-1x) dx. If we make the change of variables Ax

= y in the first integral, this becomes

J f(A-1x)g(x) dx = J j(x)g(A-lx) dx; if we then take the limit as A ~

00,

we get

f(O) J g(x) dx = g(O) J j(x) dx. Let g(x)

= e-?r 1x I2 ; then by Lemma 1.14, f(O) =

which is (1.22) for x

f(x)

=

= O.

Jj(~) d~,

If we replace f by Txf, then by (1.16),

(Txf)(O) =

J(Txfr(~) d~ = J j(~)e2?riX{, d~.

o If we let

j (x) = f (-x), we get the following corollary.

Corollary 1.15. For f E S, (jr- = j, and so the Fourier transform has period 4 (i. e. if we apply it four times, we get the identity operator). Definition 1.16. The Fourier transform of T E S' is the tempered distribution T given by A

T(f)

= T(j),

f

ES.

By Theorem 1.13, T is a tempered distribution, and in particular, ifT is an integrable function, then T coincides with the Fourier transform defined by equation (1.11). Likewise, if p., is a finite Borel measure (i.e. a bounded linear functional on Co (IRn), the space of continuous functions which vanish at infinity), then fl is the bounded continuous function given by

fl(~) = ( e- 2?rix.c; dp.,(x).

JRn

For 8, the Dirac measure at the origin, this gives us

J=

1.

8. The Fourier transform on LP, 1 < p < 2

15

Theorem 1.17. The Fourier transform is a bounded linear bijection from

S' to S' whose inverse is also bounded. Proof. If Tn

T in S', then for any f

---t

E

S,

Furthermore, the Fourier transform has period 4, so its inverse is equivalent to applying it 3 times; therefore, its inverse is also continuous. 0 _ If we define

('rr = T.

T by T(f)

T(j), then it follows from Corollary 1.15 that

=

And if TELl then by the inversion formula we get that

T(x)

= ( T(e)e 27rix .t;, de; JJRn

in particular, T is a bounded, continuous function.

8. The Fourier transform on LP, 1 < p If f E £P, 1 < P < for ¢ E S define

00,

<2

f can be identified with a tempered distribution:

then

Clearly this integral is finite. To see that Tj is continuous, suppose that ¢k ---t 0 in S as k ---t 00. Then by Holder's inequality, ITj(¢k)1

< IlfllplI¢kllp'·

Then II¢kllp' is dominated by the L= norm of functions of the form Xa¢k, and so by a finite linear combination of seminorms of ¢k; hence, the left-hand side tends to 0 as k ---t 00. Moreover, when 1 < P < 2 we have that

f is a function.

Theorem 1.18. The Fourier transform is an isometry on L2; that is, i E L2 and IIil12 = Ilf112. Furthermore,

i(e)

= lim (

f(x)e- 27rix .t;, dx

R-+= J1xl
and f(x)

=

lim

{

i(e)e 27riX .t;, de,

R-+= J1t;,I
where the limits are in L2. The identity

IIil12 = IIfl12 is referred to as the Plancherel theorem.

1. Fourier Series and Integrals

16 A

Proof. Given that

f, h

_

E 5, let g = h, so that fj = h. Then by (1.21) we have

(1.23) If we let h = f then we get IIfl12 = IIil12 for f E 5. Since 5 is dense in L2, the Fourier transform extends to all f in L2 with equality of norms.

Finally, the continuity of the Fourier transform implies the given formulas for f and i as limits in L2, since fXB(O,R) and iXB(O,R) converge to f and in L2. 0

i

If f E LP, 1 < P < 2, then it can be decomposed as f = h + 12, where hELl and 12 E L2. (For example, let h = fX{x:lf(x)l>l} and 12 = f - h.) Therefore, i = i1 + i2 E L oo + L2. However, by applying an interpolation theorem we can get a sharper result. Theorem 1.19 (Riesz-Thorin Interpolation). Let 1 < PO,P1, go, q1 < and for 0 < () < 1 define P and q by

1 - =

1-()

()

1

--+-, -

=

1-()

00,

()

--+-.

P Po PI q qo q1 If T is a linear operator from LPO + LPI to LqO + Lql such that

and

then

The proof of this result uses the so-called "three-lines" theorem for analytic functions; it can be found, for example, in Stein and Weiss [18, Chapter 5] or Katznelson [10, Chapter 4]. Corollary 1. 20 (Hausdorff-Young Inequality). If f E LP, 1 :S P < 2, then I f E LP and A

Proof. Apply Theorem 1.19 using inequality (1.13), Plancherel theorem, IIil12 = Ilf112.

Ililioo < Ilf111' and the 0

9. The convergence and summability of Fourier integrals

17

We digress to give another corollary of Riesz-Thorin interpolation which is not directly related to the Fourier transform but which will be useful in later chapters. Corollary 1.21 (Young's Inequality). If f E LP and g E Lq, then f L T , where 11r + 1 = lip + 1/q, and

*g E

Ilf * gilT < Ilfllpllgllq· Proof. If we fix

f

E LP we immediately get the inequalities

and

The desired result follows by Riesz-Thorin interpolation.

o

9. The convergence and summability of Fourier integrals The problem of recovering a function from its Fourier transform is similar to the same problem for Fourier series. We need to determine if and when lim R--+oo

(

iBR

j(e)e 21riX .f, de = f(x),

where BR = {Rx : x E B}, B is an open convex neighborhood of the origin, and the limit is understood either as in LP or as pointwise almost everywhere. If we define the partial sum operator S R by

(SRff =

XBRj,

then this problem is equivalent to determining if lim SRf

R--+oo

= j.

Analogous to Lemma 1.8, a necessary and sufficient condition for convergence in norm is that

where Cp is independent of R. When n = 1 this is the case; we will prove this in Chapter 3. We will also prove several partial results when n > 1, but in general there is no convergence in norm when p -I 2. We will discuss this in Chapter 8. In the case n

= 1, if B = (-1,1) then SRf(x) = DR * f(x),

18

1. Fourier Series and Integrals

where DR is the Dirichlet kernel,

rR e27rix~ d~ = sin(27fRx).

DR(X) =

J- R

7fX

This is clearly not integrable, but it is in well defined if f E LP, 1 < p < 00.

Lq(~)

for any q > 1, so DR

*f

is

Almost everywhere convergence depends on the bound

II sup ISRflll p< Cpllfllp· R

This holds if 1 < p < it here.

00

(the Carleson-Hunt theorem) but we cannot prove

For the Fourier transform, the method of Cesaro summability consists in taking integral averages of the partial sum operators,

oRf(x) =

~ foR Sd(x) dt,

and determining if limO"Rf(x) = f(x). When n = 1 and B = (-1,1),

* f(x),

O"Rf(x) = FR where FR is the Fejer kernel,

(1.24)

R r FR(X) = R J Dt(x)

sin2(7fRx)

1

o

dt

= R(7fx)2 .

Unlike the Dirichlet kernel, the Fejer kernel is integrable. Since it has properties analogous to (1.8), one can prove that in LP, 1 < p < 00, lim O"Rf

R-+oo

= f.

The proof is similar to that of Theorem 1.10. In: Chapter 2 we will prove two general results from which we can deduce convergence in LP and pointwise almost everywhere for this and the following summability methods. The method of Abel-Poisson summability consists in introducing the factor e-27rtl~1 into the inversion formula. Then for any t > 0 the integral converges, and we take the limit as t tends to o. If we instead introduce the factor e-7rt21~12, we get the method of Gauss-Weierstrass summability. More precisely, we define the functions

(1.25) (1.26)

r e-27rtl~lj(~)e27riX.~ d~, w(x, t) = r e-7rt21~12 j(Oe27riX.~ d~, JRn u(x, t) =

JRn

and then try to determine if

(1.27)

lim u(x, t)

t-+O+

=

f(x),

10. Notes and further results

19

lim w(x, t) = f(x)

(1.28)

t ........ O+

in LP or pointwise almost everywhere. One can show that u(x, t) is harmonic in the half-space ]R~+1 = ]Rn x (0, (0). When n = 1 we have an equivalent formula analogous to (1.9):

(1.29)

u(z)

=

f=

io

j(~)e27rizf. d~ +

fO

i-=

j(~)e27rizf. d~,

z

=

x + it,

which immediately implies that u is harmonic. The limit (1.27) can be interpreted as the boundary condition of the Dirichlet problem, on ]Rn+l + , x E ]Rn.

~u=O

u(x,O) = f(x), It follows from (1.25) that'

u(x, t) = Pt

* f(x),

where Pt(~) = e- 27rt1 f.1. One can prove by a simple calclilationif n = 1, and a more difficult one when n > 1 (see Stein and Weiss [18, p. 6]), that

(130) .

.

Rt(x) =

r

(ntl )

t (2 1 12) n+l t+x 2

-----'--~---~ n+l 7r2

.

This is called the Poisson kernel. In the case of Gauss-Weierstrass summability, one can show that the function w(x, t) = w(x, J47rt) is the solution of the heat equation

ow

-

ot

-

_

~w =

0

w(x,O) = f(x)

on ]Rn+l + '

xE

]Rn,

and (1.28) can be interpreted as the initial condition for the problem. We also have the formula

w(x, t)

= Wt * f(x),

where Wt is the Gauss-Weierstrass kernel,

(1.31)

Wt(x) = t-ne-7rlxI2 jt 2 •

This formula can be proved using Lemma 1.14 and (1.18).

10. Notes and further results 10.1. References.

The classic reference on trigonometric series is the book by Zygmund [21], which will also be a useful reference for results in the next few chapters. However, this work can be difficult, to consult at times. Another comprehensive reference on trigonometric series is the book by Bary [1].

1. Fourier Series and Integrals

20

There are excellent discussions of Fourier series and integrals in Katznelson [10] and Dym and McKean [4]. The book by R.E. Edwards [5] is an exhaustive study of Fourier series from a more modern perspective. The article by Weiss [20] and the book by Korner [12] are also recommended. An excellent historical account by J. P. Kahane on Fourier series and their influence on the development of mathematical concepts is found in the first half of [9]. The book Fourier Analysis and Boundary Value Problems, by E. Gonzalez-Velasco (Academic Press, New York, 1995), contains many applications of Fourier's method of separation of variables to partial differential equations and also contains historical information. (Also see by the same author, Connections in mathematical analysis: the case of Fourier series, Amer. Math. Monthly 99 (1992),427-441.) The book by O. G. J0rsboe and L. Melbro (The Carleson-Hunt Theorem on Fourier Series, Lecture Notes in Math. 911, Springer-Verlag, Berlin, 1982) is devoted to the proof of this theorem. The original references for this are the articles by L. Carleson (On convergence and growth of partial sums of Fourier series, Acta Math. 116 (1966), 135-157) and R. Hunt (On the convergence of Fourier series, Orthogonal Expansions and their Continuous Analogues (Proc. Conf., Edwardsville, Ill., 1967), pp. 235-255, Southern Illinois Univ. Press, Carbondale, 1968). Kolmogorov's example of an Ll function whose Fourier series diverges everywhere appeared in Une serie de Fourier-Lebesgue divergente partout (C. R. Acad. Sci. Paris 183 (1926), 1327-1328). 10.2. Multiple Fourier series. Let ']fnbe the n-dimensional torus (which we can identify with the quotient group ~n /'ZP). A function defined on ']fn is equivalent to a function on ~n which has period 1 in each variable. If fELl (~n) then we can define its Fourier coefficients by

j(v) =

J

f(x)e-27rix.v dx,

and construct the Fourier series of

L

v E Zn ,

f with these coefficients, j(v)e27rix.v.

vE1P

One can prove several results similar to those for Fourier series in one variable, but one needs increasingly restrictive regularity conditions as n increases. See Stein and Weiss [18, Chapter 7]. 10.3. The Poisson summation formula. Let

f be a function such that for some <5 > 0, If(x)1 < A(l + Ix!)-n-o

and

Ij(e)1 < A(l + lel)-n-o.

21

10. Notes and further results

(In particular,

f

and

j

are both continuous.) Then

This equality (or more precisely, the case when x = 0) is known as the Poisson summation formula and is nothing more than the inversion formula. The left-hand side defines a function on ']fn whose Fourier coefficients are precisely j (v). 10.4. Gibbs phenomenon.

Let f(x) = sgn(x) on (-1/2,1/2). By Dirichlet's criterion, for example, we know that S N f (x) converges to f (x) for all x. To the right of 0 the partial sums oscillate around 1 but, contrary to what one might expect, the amount by which they overstep 1 does not tend to 0 as N increases. One can show that

217r sin(y) dy

lim supSNf(x) = N-HXJ

7r

X

0

Y

~

. 1.17898 .... .... _

This phenomenon occurs whenever a function has a jump discontinuity. It is named after J. Gibbs, who announced it in Nature 59 (1899), although it had already been discovered by H. Wilbraham in 1848. See Dym and McKean [4, Chapter 1] and the paper by E. Hewitt and R. E. Hewitt (The Gibbs- Wilbraham phenomenon: an episode in Fourier analysis, Arch. Hist. Exact Sci. 21 (1979/80), 129-160). Gibbs phenomenon is eliminated by replacing pointwise convergence by Cesaro summability. For if m < f (x) < M, then by the first two properties of Fejer kernels in (1.8), m < ON f(x) ::; M. In fact, it can be shown that if m < f(x) < M on an interval (a, b), then for any E > 0, m - E < ON f(x) < M + E on (a + E, b - E) for N sufficiently large. 10.5. The Hausdorff'-Young inequality.

Corollary 1.20 was gotten by an immediate application of Riesz-Thorin interpolation. But in fact a stronger inequality is true: if 1 < P < 2 then ~

1IJllpi <

(

p 1/P ) (p')I/pl

n/2

IIfll p·

This inequality is sharp since equality holds for f(x) = e-7rlxI2. This result was proved by W. Beckner (Inequalities in Fourier analysis, Ann. of Math. 102 (1975), 159-182); the special case when p is even was proved earlier by K. 1. Babenko (An inequality in the theory of Fourier integrals (Russian), Izv. Akad. Nauk SSSR Ser. Mat. 25 (1961), 531-542).

1. Fourier Series and Integrals

22

In the same article, Beckner also proved a sharp version of Young's inequality (Corollary 1.21).

10.6. Eigenfunctions for the Fourier transform in L2(JR). Since the Fourier transform has period 4, if j is a function such that j = Aj, we must have that A4 = 1. Hence, A = ±1, ±i are the only possible eigenvalues of the Fourier transform. Lemma 1.14 shows that exp( _TrX2) is an eigenfunction associated with the eigenvalue 1. The Hermite functions give the remaining eigenfunctions: for n > 0, ( l)n dn hn(x) = - , eXP(Trx 2)-d exp( _TrX2) n. xn satisfies hn = (-i)nh n . If we normalize these functions,

en

n )-n. 1n 2 ,]1/2h = IIhhn l1 2 -[(4 Tr V ~n. n,

we get an orthonormal basis of L2(JR) such that

= n=O

Thus L2(JR) decomposes into the direct sum Ho E9 HI E9 H2 $ H 3 , where on the subspace H j , 0 < j < 3, the Fourier transform acts by multiplying functions by i j . This approach to defining the Fourier transform in L2(JR) is due to N. Wiener a.nd can be found in his book (The Fourier Integral and Certain of its Applications, original edition, 1933; Cambridge Univ. Press, Cambridge, 1988). Also see Dym and McKean [4, Chapter 2]. In higher dimensions, the eigenfunctions Qf the Fourier transform are products of Hermite functions, one in each coordinate variable. Also see Chapter 4, Section 7.2.

10.7. Interpolation of analytic families of operators. The Riesz-Thorin interpolation theorem has a powerful generalization due to E. M. Stein. (See Stein and Weiss [18, Chapter 5].) Let S = {z E C : 0 < Re z < I} and let {Tz } zES be a family of operators. This family is said to be admissible if given two functions f, 9 E LI(JR n ), the mapping Zf----t

r Tz(j)gdx

JlRn

is analytic on the interior of S and continuous on the boundary, and if there exists a constant a < Tr such that e-allmzllog is uniformly bounded for all

Z E

S.

r

JlR

n

Tz(f)g dx

10. Notes and further results

23

Theorem 1.22. Let {Tz } be an admissible family of operators, and suppose that for 1
IITiyfllqo < Mo(y)llfllpo where for some b <

and

II T l+iyfll q1 < Ml(y)llfll pl '

7f

supe-blyllogMj(Y) <

00,

j

= 1,2.

yElR

e

e

Then for 0 < < 1, Re z = and p, q defined as in Theorem 1.19, there exists a constant Me such that

10.8. Fourier transforms of finite measures.

As we noted above, if P is a finite Borel measure then p, is a- bounded, continuous function. The collection of all such functions obtained in this way is characterized by the following result. Theorem 1.23. If h is a bounded, continuous function, then the following are equivalent:

(1) h = P, for some positive, finite Borel measure Pi (2) h is positive definite: given any fELl (JR n ) ,

r r

JlRn JlRn

h(x - y)f(x)J(y) dxdy > O.

This theorem is due to S. Bochner (Lectures on Fourier Integrals, Princeton Univ. Press, Princeton, 1959; translated from Vorlesungen iiber Fouriersche Integrale, Akad. Verlag, Leipzig, 1932). Also see Katznelson [10, Chapter 6].

I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I

Chapter 2

The Hardy-Littlewood Maxilllal Function

1. Approximations of the identity Let ¢ be an integrable function on JRn such that J cP = 1, and for t > 0 define ¢t(x) = t-n¢(t-Ix). As t ~ 0, ¢t converges in S' to 6, the Dirac measure at the origin: if 9 E S then

¢t(g)

=

r

J~n

t-n¢(t-Ix)g(x) dx

r

=

J~n

cP(x)g(tx) dx,

and so by the dominated convergence theorem, lim cPt (g) = 9 (0) = 6 (g) .

t-tO

Since 6 * 9 = g, for 9 E S we have the pointwise limit lim ¢t

t-tO

* 9 (x) = 9 (x ) .

Because of this we say that {cPt: t

> O} is. an approximation of the identity.

The summability methods in the previous chapter can be thought of as approximations of the identity. For Cesaro summability, ¢ = FI and FR = ¢I/R- (See (1.24).) For Abel-Poisson summability, cP = PI (see (1.30)) and for Gauss-Weierstrass summability, ¢ = WI (see (1.31)). We see from the following result that these summability methods converge in £P norm. Theorem 2.1. Let {cPt: t

> O} be an approximation of the identity. Then lim II ¢t

t-tO

if f E LP, 1
<

00,

*f

- flip

=

0

and uniformly (i.e. when p

= (0)

if f E Co(JR n ).

-

25

2. The Hardy-Littlewood Maximal Function

26

Proof. Because ¢ has integral 1, ¢t Given

E

* f(x)

- f(x) =

r ¢(y)[j(x - ty) - f(x)] dy.

JlR

n

> 0, choose 6 > 0 such that if Ihl < 6,

Ilf(' + h) -

E

f(-)llp <

211¢III'

(Note that 6 depends on f.) For fixed 6, if t is sufficiently small then

r

J1yl ?8/t 1¢(y)1 dy < 411;11 p . Therefore, by Minkowski's inequality

II¢t * f

- flip <

r

J1yl <8/t

1¢(y)lllf(' + ty) -

+ 211fllp

r

f(')llp dy

1¢(y)1 dy

J1yl ?8/t

< E. D

As a consequence of this theorem, we know that there exists a sequence { t k}, depending on f, such that t k - t 0 and lim ¢tk

k-->oo

* f(x)

=

f(x)

a.e.

Hence, if lim.¢t * f(x) exists then it must equal f(x) almost everywhere. In Section 4 we will study the existence of this limit.

2. Weak-type inequalities and almost everywhere convergence Let (X,I-£) and (Y, v) be measure spaces, and let T be an operator from LP(X,I-£) into the space of measurable functions from Y to C. We say that T is weak (p, q), q < 00, if

v({y E Y: ITf(y)1 > A}) <

(GIl! lip )

q,

and we say that it is weak (p, (0) if it is a bounded operator from LP(X, 1-£) to LOO(Y, v). We say that T is strong (p, q) if it is bounded from LP(X, 1-£) to Lq(y, v). If T is strong (p, q) then it is weak (p, q): if we let E).. ITf(y)1 > A}, then

v(E ) =

).

r

JE)..

dv < -

r

JE)..

Tf(x) q dv < A -

IITfllg < Aq

-

= {y

(Cllfllp)q. A

E Y

2. Weak-type inequalities and almost everywhere convergence

27

When (X, J-L) = (Y, v) and T is the identity, the weak (p,p) inequality is the classical Chebyshev inequality. The relationship between weak (p, q) inequalities and almost everywhere convergence is given by the following result. In it we assume that (X, J-L) =

(Y, v). Theorem 2.2. Let {Ttl be a family of linear operators on LP(X, J-L) and

define T* f(x) = sup ITtf(x)l· t

If T* is weak (p, q) then the set {f E LP(X, J-L) : lim Ttf(x) = f(x) a.e.} t~to

is closed in LP(X, J-L). T* is called the maxhnal operator associated with the family {Ttl. Proof. Let {fn} be a sequence of functions which converges to J in £P (X, J-L) norm and such that Ttfn(x) converges to fn(x) almost everywhere. Then

J-L({x EX: lim sup ITtf(x) - f(x)1 > -\}) t~to

< J-L({x EX: limsuplTt(f - fn)(x) - (f - fn)(x) I > -\}) t~to

< J-L( {x EX: T*(f - fn)(x) > -\/2}) + J-L({x EX: I(f - fn)(x) I > -\/2}) 20 < ( Tllf - fnllp

)q + (2-\ II! - fnllp )P ,

and the last term tends to 0 as n

---7

00.

Therefore,

J-L({x EX: lim sup ITtf(x) - f(x)1 > O}) t~to

00

< LJ-L({x EX: limsupITtf(x) - f(x)1 > l/k}) t~to

k=l

=0.

o By the same technique we can also prove that the set

{f E LP(X, J-L) : lim Ttf(x) exists a.e.} t~to

is closed in LP(X, J-L). It suffices to show that

J-L ({ x EX: lim sup T t f (x) - lim inf T t f (x) > -\}) = 0, t~to

t~to

2. The Hardy-Littlewood Maximal Function

28

and this follows as in the above argument since lim sup Ttf(x) -liminfTtf(x) < 2T* f(x).

t--7 to

t--7 t o

(If Ttf(x) is complex, we apply this argument to its real and imaginary parts separately. ) Since for f E S approximations of the identity converge pointwise to f, we can apply this theorem to show pointwise convergence almost everywhere for f E LP, 1 < p < 00, or for f E Co if we can show that the maximal operator SUPt>o I(Pt * f (x) I is weakly bounded.

3. The Marcinkiewicz interpolation theorem Let (X, M) be a measure space and let f : X --+ C be a measurable function. We call the function aj : (0,00) --+ [0,00], given by

aj(>\) = fL({x EX: If(x)1 > A}), the distribution function of

f

(associated with fL).

Proposition 2.3. Let ¢ : [0,00) such that ¢(O) = 0. Then

Ix

--+

¢(If(x)l) dfL =

[0,00) be differentiable, increasing and

10

00

¢'(A)aj(A) dA.

To prove this it is enough to observe that the left-hand side is equivalent to

Ix lI(X)'

¢i(A) dAdIJ.

and then change the order of integration. If, in particular, ¢(A)

= AP then

(2.1) Since weak inequalities measure the size of the distribution function, this representation of the LP norm is ideal for proving the following interpolation theorem, which will let us deduce LP. boundedness from weak inequalities. It applies to a larger class of operators than linear ones (note that maximal operators are not linear): an operator T from a vector space of measurable functions to measurable functions is sublinear if

IT(fo

+ fl)(x)1 <

ITfo(x) I + ITh(x)l,

IT(Af)1 = IAIITf!'

A E C.

3. The Marcinkiewicz interpolation theorem

29

Theorem 2.4 (Marcinkiewicz Interpolation). Let (X, p,) and (Y, v) be measure spaces, 1 < Po < PI < 00, and let T be a sublinear operator from LPO (X, p,) + LPI (X, p,) to the measurable functions on Y that is weak (Po, po) and weak (PI,PI). Then T is strong (p,p) for Po < P < Pl. Proof. Given f E LP, for each)"

= JI = fo

> 0 decompose f as fo + JI, where fX{x:lf(x)l>c).}, fX{x:lf(x)l:Sc).};

the constant c will be fixed below. Then fo E LPo (p,) and Furthermore,

JI

E LPI (p,) .

ITf(x)1 < ITfo(x) I + ITJI(x) I, so

We consider two cases.

Case 1: PI = 00. Choose c = 1/(2A I ), where Al is such that Alllgiloo. Then aTh ()../2) = o. By the weak (po,po) inequality, aT!o (>,/2)

<

C~o IlioIIPo )

IITglloo <

pO ;

hence,

r

IITfll~ < P roo )..p-I-PO(2Ao)p° io r

r1f(x)l/c

= p(2Ao)PO i x If(x) IPO io =

Case 2: PI <

If(x)IPO dp,d)"

i{x:lf(x)l>cA}

)..p-I-po d)" dp,

P (2A o)PO(2A I )P-PO IlfiI P. P- Po P 00.

We now have the pair of inequalities i = 0,1.

From these we get (arguing as above) that

liTfll~ < P roo )..p-I-po (2Ao)p° io

r

If(x) IPO dp, d)"

i{x:lf(x)l>c).}

+ P roo )..p-I-PI (2Ad PI io

r

If(x) IPI dp, d)"

i{x:lf(x)l:Sc).}

p2PI API) P2pO APO . = ( P - Po cP-0po + PI - pCP-PI I IlfiIP P

o

2. The Hardy-Littlewood Maximal Function

30

We can write the strong (p, p) norm inequality in this theorem more precisely as (2.2) where 1

()

1-()

P

PI

Po

-=-+--,

O<()
When PI = 00 this is the constant which appears in the proof; when PI < 00 it is enough to take c such that (2Aoc)PO = (2A I c)Pl and then simplify.

4. The Hardy-Littlewood maximal function Let Br = B(O, r) be the Euclidean ball of radius r centered at the origin. The Hardy-Littlewood maximal function of a locally integrable function f on IR. n is defined by (2.3)

M f(x) = sup IBI r>O

I

r

r

JBr

If(x - y) I dy.

(This can equal +00.) If we let ¢ = IBII-IXBl' then (2.3) coincides for nonnegative f with the maximal operator associated with the approximation of the identity {¢t} as in Theorem 2.2. Sometimes we will define the maximal function with cubes in place of balls. If Qr is the cube [-r, r]n, define (2.4)

M' f(x) = sup ( l)n r>O

2r

r If(x - y) I dy.

JQr

When n = 1, M and M' coincide; if n > 1 then there exist constants en, depending only on n, such that

(2.5)

Cn

and

cnM'f(x) < Mf(x) < CnM'f(x).

Because of inequality (2.5), the two operators M and M' are essentially interchangeable, and we will use whichever is more appropriate, dep(;1nding on the circumstances. In fact, we can define a more general maximal function

(2.6)

M" f(x) =

~~~ I~I

k

If(y)1 dy,

where the supremum is taken over all cubes containing x. Again, Mil is pointwise equivalent to M. One sometimes distinguishes between M' and Mil by referring to the former as the centered and the latter as the noncentered maximal operator. Alternatively, we could define the non-centered maximal function with balls instead of cubes.

4. The Hardy-Littlewood maximal function

31

Theorem 2.5. The operator M is weak (1, 1) and strong (p,p), 1 < p

< 00.

We remark that by inequality (2.5), the same result is true for M' (and also for Mil). It is immediate from the definition that

(2.7) ,

11Mflloo < Ilflloo,

so by the Marcinkiewicz interpolation theorem, to prove Theorem 2.5 it will be enough to pr;ove that M is weak (1, 1). Here we will prove this when n = 1; we will prove the general case in Section 6. In the one-dimensional case we need the following covering lemma whose simple proof we leave to the reader. Lemma 2.6. Let {Iah~EA be a collection of intervals in JR and let K be a compact set contained in their union. Then there exists a finite subcollection { I j } such that

c

K

UI

L Xlj (x) < 2,

and

j ,

j

x E JR.

j

Proof of Theorem 2.5 for n = 1. Let E).. = {x E JR : Mf(x) x E E).. then there exists an interval Ix centered at x such that

(2.8)

ILl

LIf I

> A}. If

>A

Let K c E),. be compact. Then K C U Ix, so by Lemma 2.6 there exists a finite collection {Ij} of intervals such that K C Uj I j and Lj Xlj < 2. Hence,

IKI < ~ IIjl < ~ ~ hJ. J

J

lfl <

h

~ ~Xljlfl < ~ Ilf111; J

the second inequality follows from (2.8). Since this inequality holds for every compact K C E),., the weak (1,1) inequality for M follows immediately. D Lemma 2.6 is not valid in dimensions greater than 1, and though one could replace it with similar results, this is not the approach we are going to take here. (For such a proof, see Section 8.6.) The importance of the maximal function in the study of approximations of the identity comes from the following result. Proposition 2.7. Let ¢ be a function which is positive, radial, decreasing (as a function on (0,00)) and integrable. Then

sup I¢t t>O

* f(x)1 < 11¢lh M f(x).

2. The Hardy-Littlewood Maximal Function

32

Proof. If we assume in addition to the given hypotheses that ¢ is a simple function, that is, it can be written as

¢(x)

=

L ajXBrj (x) j

with aj > 0, then 1

¢

* f(x) = ~ajlBrjllBrjlXBrj * f(x) <

' //¢//lMf(x)

J

since //¢//l = L ajlBrj I· An arbitrary function ¢ satisfying the hypotheses can be approximated by a sequence of simple functions which increase to it monotonically. Any dilation ¢t is another positive, radial, decreasing function with the same integral, and it will satisfy the same inequality. The desired conclusion follows at once. 0 Corollary 2.8. If 1¢(x)1 < 'lj;(x) almost everywhere, where'lj; is positive, radial, decreasing and integrable, then the maximal function SUPt I¢t * f (x) I is weak (1,1) and strong (p,p), 1 < p < 00.

This is an immediate consequence of Proposition 2.7 and Theorem 2.5. If we combine this corollary with Theorem 2.2 we get the following result. Corollary 2.9. Under the hypotheses of the previous corollary, if f E V, 1 < p < 00, or if f E Co, then

l~ qJ, * f(x) =

(J qJ) . f(x) a.e.

In particular, the summability methods discussed in Chapter 1, Section 9 (Cesaro, Abel-Poisson and Gauss- Weierstrass summability) , each converge to f(x) almost everywhere if f is in one of the given spaces.

Proof. Since we have convergence for f E 5,. by Theorem 2.2 we have convergence for f E 5 = LP (or f E Co if p = (0). The Poisson kernel (1.30) and the Gauss-Weierstrass kernel (1.31) are decreasing; the Fejer kernel (1.24) is not but Fl(X) < min(l, (7TX)-2). 0

5. The dyadic maximal function In ]Rn we define the unit cube, open on the right, to be the set [0, 1)n, and we let Qo be the collection of cubes in]Rn which are congruent to [0, l)n and whose vertices lie on the lattice 7l n . If we dilate this family of cubes by a factor of 2- k we get the collection Qk, k E Z; that is, Qk is the family of

5. The dyadic maximal function

33

cubes, open on the right, whose vertices are adjacent points of the lattice (2- kz)n. The cubes in Uk Qk are called dyadic cubes. From this construction we immediately get the following properties:

(1) given x E ~n there is a unique cube in each family Qk which contains it;

(2) any two dyadic cubes are either disjoint or one is wholly contained in the other; (3) a dyadic cube in Qk is contained in a unique cube of each family Qj, j < k, and contains 2n dyadic cubes of Qk+1. Given a function

f

E Lfoc(~n),

Ekf(x)

=

define

L (,~,l f) XQ(x); QEQk

.

Q

Ekf is the conditional expectation of f with respect to the a-algebra generated by Qk. It satisfies the following fundamental identity: if 0 is the union of cubes in Qk, then

Ekf is a discrete analog of an approximation of the identity. The following theorem makes this precise; first, define the dyadic maximal function by (2.9)

Mdf(x)

=

sup IEkf(x)l· k

Theorem 2.10.

(1) The dyadic maximal function is weak (1, 1).

(2) If f E Lfoc! lim Ekf(x) k~=

= f(x)

a.e.

Proof. (1) Fix fELl; we may assume that f is non-negative: if f is real, it can be decomposed into its positive and negative parts, and if it is complex, into its real and imaginary parts.

Now let

{x E ~n: Mdf(x) > A}

= UOk, k

where

Ok

= {x

E ~n :

Ekf(x) > A and Ejf(x) < A if j < k};

that is, x E Ok if Ekf(x) is the first conditional expectation of f which is greater than A. (Since f E L1, Ekf(x) ~ 0 as k ~ -00, so such a k exists.)

2. The Hardy-Littlewood Maximal Function

34

The sets Ok are clearly disjoint, and each one can be written as the union of cubes in Qk. Hence,

(2) This limit is clearly true if f is continuous, and so by Theorem 2.2 it holds for fELl. To complete the proof, note that if f E Ltoc then fXQ E L1 for any Q E Qo. Hence, (2) holds for almost every x E Q, and so for almost every x E }Rn. 0 This proof uses a decomposition of }Rn which has proved to be extremely useful. It is called the Calder6n-Zygmund decomposition and we state it precisely as follows. Theorem 2.11. Given a function f which is integrable and non-negative, and given a positive number A, there exists a sequence {Qj} of disjoint dyadic cubes such that

(1)

f(x)

< A for almost every x

ret

UQj'j

(2) (3) Proof. As in the proof of Theorem 2.10, form the sets Ok and decompose each into disjoint dyadic cubes contained in Qk; together, all of these cubes form the family {Q j }.

Part (2) of the theorem is then just the weak (1,1) inequality of Theorem 2.10.

If x ret Uj Qj then for every k, Ekf(x) < A, and so by part (2) of Theorem 2.10, f(x) < A at almost every such point. Finally, by the definition of the sets Ok, the average of f over Q j is greater than A; this is the first inequality in (3). Furthermore, if Qj is the dyadic cube containing Qj whose sides are twice as long, then the average

6. The weak (1,1) inequality for the maximal function

of

35

f over Qj is at most .\.. Therefore, _1_

r

IQjl lQj

f < 10jl--2:-

r

f < 2n.\..

IQjllQjllQj

o 6. The weak (1,1) inequality for the maximal funct~n . We are now going to use Theorem 2.10 to prove Theorem 2.5. In fact, it is an immediate consequence of the following lemma and inequality (2.5). (Recall that M' is the maximal operator on cubes defined by (2.4).)

Lemma 2.12. If f is a non-negative function, then

I{x E}Rn:

M'f(x) > 4n.\.} I < 2nl{x

E

jRn: Mdf(x) >

.\.}I.

Given this lemma, by the weak (1,1) inequality for Md proved in Theorem 2.10,

I{x E jRn : M'f(x) > .\.}I < 2n l{x E jRn : J'1I[df(x) > 4- n .\.} I < (Since M' f

= M'(lfl), we

may a~sume that

gn

~llflh-

f is non-negative.)

Proof of Lemma 2.12. As before, we form the decomposition

{x

E}Rn :

Mdf(x) > .\.}

=

UQj. j

Let 2Qj be the cube with the same center as Qj and whose sides are twice as long. To complete the proof it will suffice to show that

{x

E

jRn: M'f(x) > 4n.\.}

C

U2Qj. j

Fix x tf. Uj 2Qj and let Q be any cube centered at x. Let l(Q) denote the side length of Q, and choose k E Z such that 2k - 1 < l(Q) < 2k. Then Q~ntersects m < 2n dy?-dic cubes in Qk; call them Rl, R2,' .. ,Rm. None of these cubes is contained in any of the Qj's, for otherwise we would have x E Uj 2Qj. Hence, the average of f on each Ri is at most .\., and so

o

36

2. The Hardy-Littlewood Maximal Function

As a consequence of the weak (1,1) inequality and Theorem 2.2 we get a continuous analog of the second half of Theorem 2.10. Corollary 2.13 (Lebesgue Differentiation Theorem). If f E Ltoc(JRn ) then

lim IBI

r---+O+

r

I

r

JBr f(x -

y) dy = f(x) a.e.

From this we see that If(x)1 < Mf(x) almost everywhere. The same is true if we replace M by M' or Mil. We can make the conclusion of Corollary 2.13 sharper: . hm

(2.10)

-I1-I

r---+O+ Br

1 Br

If(x - y) - f(x)1 dy = 0 a.e.

This follows immediately from the fact that

I~rl

L"

If(x - y) - f(x)1 dy < Mf(x)

+ If(x)l·

The points in JRn for which limit (2.10) equals 0 are called the-Lebesgue poillts-of f. If x is a Lebesgue point and if {B j } is a sequence of balls such that B1 :=> B2 :=> .•. and j Bj = {x} (note that the balls need not be centered at x), then

n

lim _1_

r 'F= f(x).

IB j I JBj

j---+oo

This follows immediately from the inclusion B j C B(x,2rj), where rj is the radius of B j . A similar argument shows that the set of Lebesgue points of f does not change if we take cubes instead of balls. The weak (1,1) inequality for M is a substitute for the strong (1,1) inequality, which is false. In fact, it never holds, as the following result shows. Proposition 2.14. If fELl and is not identically 0, then M f (j. L1.

The proof is simple: since that

Now if

Ixl > R,

BR

C

f

is not identically 0, there exists R

B(x, 2Ixl), so 1

Mf(x) >

r

(2Ixl)n JBR If I >

Nevertheless, we do have the following.

E

2n lxl n

'

> 0 such

7. A weighted norm inequality

37

ofrr~,n,

Theorem 2.15. If B is a bounded subset

then

where log + t = max (log t, 0). Proof.

Is

Mf <2

100

< 21BI Decompose

f

E

Mf(x) > 2-\}1 d-\

E B :

+ 2 1,00 I{x E B

h + 12, where h =

as

{x

I{x

B : Mf(x) > 2-\}

: Mf(x)

fX{x:lf(x)I>A} C

{x

E

> 2-\}1 d-\.

and

12 =

f -

h.

Then

B : Mh(x) > -\}.

Hence,

rOO I{x E B

: M f (x) > 2-\} I d-\ <

Jl

roo ~ ( Jl


1" lR.n

If (x) I dx d-\ -

J{x:lf(x)I>A} I,max(lf(x)l,l) d-\

If(x)1

\ 1

dx A

= C { If(x)llog+ If(x)1 dx. JlR.n

o 7. A weighted norm inequality Theorem 2.16. If w is a non-negative, measurable function and 1 00,


then there exists a constant Cp such that

Furthermore, (2.11)

( J{x:Mf(x»A}

w(x) dx <

~l

(

If(x)IMw(x) dx.

JlR.n

Proof. It will suffice to show that 11M fllux)(w) < Ilfllu"'(Mw) and that the weak (1,1) inequality holds; the strong (p,p) inequality then follows from the Marcinkiewicz interpolation theorem. If Mw(x) = 0 for any x, then

2. The Hardy-Littlewood Maximal Function

38

w(x) = 0 almost everywhere and there is nothing to prove. Therefore, we may assume that for every x, Mw(x) > O. If a > IUlluXl(Mw) then f

1{x:lf(x)l>a}

Mw(x) dx = 0,

and so I{x E lRn : If(x)1 > a}1 = 0; that is, If(x)1 < a almost everywhere. From this it follows that M f(x) < a a.e., so 11M fIILOO(w) < a. Hence,

IIMfIlLoo(w) < IlfIILOO(Mw)· To prove the weak (1,1) inequality we may assume that f is non-negative and f E LI(lRn). (If f E LI(Mw) then fj = fXB(O,j) is a sequence of integrable functions which increase pointwise to f.) If {Q j} is the Calder6nZygmund decomposition of f at height). > 0, then as we showed in the proof of Lemma 2.12,

{x

E

lR n : M'f(x) > 4n).}

U2Qj;

C

j

hence,

f

w( x) dx <

l{x:MI f(x»4n)..}

L j

=

f

w (x) dx

12Qj

L 2nlQjll 2Ql .1 j

J

f

w(x) dx

12Qj

<2;~ l f.f(y)( 12Ql.,1. J

< 2:C

Q)

L

J

W

(X)dx) dy

2Q)

j(y)M"w(y) dy.

Since M"w(y) < CnMw(y), we get the desired inequality.

o

If w is such that Mw(x) < w(x) a.e., then these inequalities hold with the same weight w on both sides. Functions which satisfy this condition are called Al weights, and we will consider them in greater detail in Chapter 7.

8. Notes and further results 8.1. References. The maximal function for n = 1 was introduced by G. H. Hardy and J. E. Littlewood (A maximal theorem with function-theoretic applications, Acta Math. 54 (1930), 81-116), and for n > 1 by N. Wiener (The ergodic theorem, Duke Math. J. 5 (1939), 1-18). In their article, Hardy and Littlewood first consider the discrete case, about which they say: "The problem

8. Notes and further results

39

is most easily grasped when stated in the language of cricket, or any other game in which a player compiles a series of scores of which an average is recorded." Their proof, which uses decreasing rearrangements (see 8.2 below) can be found in Zygmund [21]. Our proof follows the ideas of Calderon and Zygmund ( On the existence of certain singular integrals, Acta Math. 88 (1952), 85-139). The decomposition which bears their name also first appeared in this paper. The method of rotations, discussed in Chapter 4, lets us deduce the strong (p, p) inequality for n > 1 from the one-dimensional result. For a discussion of questions related to Theorem 2.2, and in particular the necessity of the condition in that theorem, see de Guzman [7]. The Marcinkiewicz interpolation theorem was announced by J. Marcinkiewicz (Sur l'interpolation d'operations, C. R. Acad. Sci. Paris 208 (1939), 12721273). However, he died in World War II and a complete proofwas finally given by A. Zygmund (On a theorem of M arcinkiewicz concerning interpolation of operations, J. Math. Pures Appl. 34 (1956), 223-248). Both Marcinkiewicz interpolation and Riesz-Thorin interpolation in Chapter 1 have been generalized considerably; see, for example the book by C. Bennett and R. Sharpley (Interpolation of Operators, Academic Press, New York, 1988). The weighted norm inequality for the maximal operator is due to C. Fefferman and E. M. Stein (Some maximal inequalities, Amer. J. Math. 93 (1971), 107-115).

8.2. The Hardy operator, one-sided maximal functions, and decreasing rearrangements of functions.

Given a function g on 1R+ defined by

Tg(t)

= (0,00)

the Hardy operator acting on g is

lint g(s) ds, t

=-

t

E

1R+.

0

If 9 E L1 (1R+) is non-negative, then, since Tg is continuous, one can show that

(2.12)

E(A) =

~r

/\ JE()..)

g(t) dt,

where E(A) = {t E lR+ : Tg(t) > A}. From this and (2.1) we get IITgllp < p'llgllp, 1 < p < 00. Other proofs of this result and generalizations of the operator can be found in the classical book by G. H. Hardy, J. E. Littlewood and G. Polya (Inequalities, Cambridge Univ. Press, Cambridge, 1987, first edition in 1932), or in Chapter 2 of the book by Bennett and Sharpley cited above.

40

2. The Hardy-Littlewood Maximal Function

For a function f on IR, the one-sided Hardy-Littlewood maximal functions are defined by M+ f(t)

= sup -h1it+h If(s)1 ds and h>O

M- f(t)

= sup -h1 i t If(s)1 ds. h>O

t

t-h

The maximal function as defined by Hardy and Littlewood corresponds to M- for functions on IR+ (with 0 < h < t in the definition). When If I is decreasing this maximal function coincides with the Hardy operator acting on Ifl. If f is a measurable function on IR n , we can define a decreasing function f* on (0,00), called the decreasing rearrangement of f, that has the same distribution function as f:

f*(t) = inf{A : af(A) < t}. Because f and f* have the same distribution function, by (2.1) their LP norms are equal, as well as any other quantity which depends only on their distribution function. (Cf. Section 8.3 below.) The action of the Hardy operator on f* is usually denoted by f**. . Hardy and Littlewood showed that for functions on IR+, (2.13)

I{x : M- f(x) > A}I < I{x : f**(x) > A}I,

so the weak (1,1) and strong (p,p) inequalities for M- follow from the corresponding ones for T. A beautiful proof of the weak (1,1) inequality for M+ was given by F. Riesz as an application of his "rising sun lemma" (Sur un theoreme du maximum de MM. Hardy et Littlewood, J. London Math. Soc. 7 (1932), 10-13). Given a function F on IR, a point x is a shadow point of F if there exists y > x such that F(y) > F(x). The set of shadow points of F is denoted by S (F) . Lemma 2.17. Let F be a continuous function such that

lim F(x)

X~+=

= -00 and

lim F(x)

x~-=

=

+00.

Then S(F) is open and can be written as the disjoint union Uj(aj, bj ) of finite open intervals such that F (aj) = F (bj ) . Given f E Ll(IR) and A > 0, define F(x) = foX If(t)1 dt - AX. Then E(A) = {x : M+ f(x) > A} equals S(F). Using Lemma 2.17 one can deduce that

E(A)

=

~(

/\ JE(>..)

If(x) I dx,

8. Notes and further results

41

which is similar to (2.12) for the Hardy operator. The strong (p, p) inequality now follows as before with constant p' (which is sharp). We leave the details of the proof of this and of Lemma 2.17 to the reader. An inequality similar to (2.13) holds for the maximal operator acting on functions on ~n; in fact, we have the following pointwise inequality: there exist positive constants Cn and C n such that

cn(Mf)*(t) < f**(t) < Cn(Mf)*(t),

t

E ~+.

The left-hand inequality is due to F. Riesz; the right-hand inequality is due to C. Herz (n = 1) and C. Bennet and R. Sharpley (n > 1). See Chapter 3 of the above-cited book by these authors for a proof and further references.

8.3. The Lorentz spaces

!y,q.

Let (X, J-L) be a measure space. LP,q(X) denotes the space of measurable functions f which satisfy

Ilfllp,q = when 1 < p <

00,

1
(~1=[tl/Pf*(t)]q ~ty/q < 00,

00

and

Ilfllp,oo = sup t 1/ pf*(t) < 00 t>O

when 1
00.

When p

=

q,

Ilfllp,p = Ilf*llp = Ilfllp general, however, I . IIp,q

and we recover!Y. In is not a norm since the triangle inequality only holds when 1 < q < P < 00 or p = q = 00. But when 1 < p < 00 and 1 < q < 00, if we replace f* in the definition of Ilfllp,q with f**, we get a quantity which is equivalent to Ilfllp,q and which defines a norm. If ql < q2 then

11f11p,q2 < Ilfllp,qll so LP,ql

C

LP,q2.

An operator T is weak (p, p) precisely when

IITfllp,oo < Cllfll p· The Marcinkiewicz interpolation theorem can be generalized to these spaces; this lets us, for example, give a version of the Hausdorff-Young inequality which is stronger than Corollary l.20: if f E LP(~n), 1 < p < 2, then I f E LP ,p and there exists a constant Bp such that A

Ilillp/,p < Bpllfllp· For more information on decreasing rearrangements and the LP,q spaces, see Stein and Weiss [18, Chapter 5] and the book by Bennett and Sharpley cited above. Also see this latter book for more on interpolation theorems, particularly the so-called real method of interpolation which is well suited

2. The Hardy-Littlewood Maximal Function

42

to Lorentz spaces. These spaces were introduced by G. G. Lorentz in two papers (Some new functional spaces, Ann. of Math. 51,(1950), 37-55, and On the theory of spaces A, Pacific J. Math. 1 (1951),411-429). 8.4. LlogL.

In the proof of Proposition 2.14, the integrability of M f failed at infinity, and did not exclude local integrability. However, the example f(x) = x-I (log x )-2 X(O,1/2] shows that even local integrability can fail. A partial converse of Theorem 2.15 is true and characterizes when M f is locally integrable. Theorem 2.18. If f is an integrable function supported on a compact set B, then Mf E LI(B) if and only if flog+ f E LI(B).

This is due to E. M. Stein (Note on the class L log L, Studia Math. 32 (1969), 305-310); a proof can also be found in Garcia-Cuerva and Rubio de Francia [6, p. 146]. At the heart of the proof is a stronger version of the weak (1,1) inequality and a "reverse" weak (1,1) inequality for the maximal function:

(2.14) (2.15)

I{x

E

JRn : Mf(x)

> A}I <~ {

I{x E JRn : M f(x) > A}I >

J{x: If (x) 1>)../2}

~;;

If(x)1 dx,

If(x)1 dx.

{x:lf(x)I>)..}

Inequality (2.14) follows from the weak (1,1) inequality applied to h fX{x:lf(x)I>)"/2}; inequality (2.15) follows from the Calder6n-Zygmund decomposition if we replace M by Mil. In this and related problems it would be useful to consider functions f such that f log+ fELl as members of a Banach space. Unfortunately, the expression J f log+ f does not define a norm. One way around this is to introduce the Luxemburg norm: given a set 0 ~ JR n , define

(2.16)

IlfIiLlogL(l1) = inf

{A > 0: 10 If~)IIog+ Cf~)I) dx < I}.

By using this we can strengthen the conclusion of Theorem 2.18 to the following inequality: IIMfll£l(B) < CllfIiLlogL(B)' (See Zygmund [21, Chapter 4].) By replacing the function tlog+ t in (2.16) by any convex, increasing function <1>, we get a class of Banach function spaces, L
8. Notes and further results

43

Groningen, 1961), and M. M. Rao and Z. D. Ren, (Theory of Orlicz Spaces, Marcel Dekker, New York, 1991). 8.5. The size of constants.

In the proofs in this chapter, the constants which appear in the weak (1,1) and strong (p,p) inequalities, 1 < p < 00, for M and M' are of exponential type with exponent n. For the strong (p, p) inequality, the method of rotations (see Corollary 4.7) gives a better constant of the form Cpn for M. Furthermore, one can show that there exist constants Cp , independent of n, such that 1

00.

This result is due to E. M. Stein, and the proof appeared in an article by E. M. Stein and J.-O. Stromberg (Behavior of maximal functions in]Rn for large n, Ark. Mat. 21 (1983), 259-269); also see E. M. Stein, Three variations on the theme of maximal functions (Recent Progress in Fourier Analysis, 1. Peral and J. L. Rubio de Francia, eds., pp. 229-244, North-Holland, Amsterdam, 1985). A partial generalization of this result is possible: let B be a convex set which is symmetric about the origin and define MBf(x)

Then if p that

= sup I IBI1 r>O r

If(x - y)1 dy.

rB

> 3/2 there exists a constant Cp independent of Band

n such

For the weak (1,1) inequality, the best constant known for M is of order n. (See the article by Stein and Stromberg cited above.) For the non-centered maximal function defined by (2.6) (but with balls instead of cubes), the best constant in the strong (p, p) inequality must grow exponentially in n. See L. Grafakos and S. Montgomery-Smith (Best constants for uncentered maximal functions, Bull. London Math. Soc. 29 (1997), 60-64). We note that when n = 1 the best constants are known for the one-sided and for the non-centered maximal operator Mil but not for M. In the case of the weak (1,1) inequality for Mil, one can use Lemma 2.6 to get C = 2, and the function f(t) = X[o,l] shows this is the best possible. For the strong (p, p) inequality for Mil, see the article by Grafakos and Montgomery-Smith cited above. For lower and upper bounds on the weak (1,1) constants for M, see the article by J. M. Aldaz (Remarks on the Hardy-Littlewood maximal function, Proc. Roy. Soc. Edinburgh Sect. A 128 (1998), 1-9).

44

2. The Hardy-Littlewood Maximal Function

8.6. Covering lemmas.

A standard approach to proving that the maximal fmiction in JRn is weak (1,1) is to use covering lemmas. Here we give two; the first is a Vitali-type lemma due to N. Wiener (in the paper cited above). If B = B(x, r) and t > 0, then we let tB = B(x, tr). Theorem 2.19. Let {Bj hEJ be a collection of balls in JR n . Then there exists an at most countable subcollection of disjoint balls {B k } such that

U B C U5B j

jEJ

k.

k

The second is due independently to A. Besicovitch and A. P. Morse; for a proof and further references, see the book by M. de Guzman (Differentiation of Integrals in ]Rn, Lecture Notes in Math. 481, Springer-Verlag, Berlin, 1985). Theorem 2.20. Let A be a bounded set in JRn , and suppose that {Bx}xEA is a collection of balls such that Bx = B (x, r x), r x > 0. Then there exists an at most countable subcollection of balls {B j } and a constant Cn, depending only on the dimension, such that

A

C

UBj

and

LXBj(X) < Cn· j

J

The same result holds if balls are replaced by cubes; more generally, the point x need not be the center of the ball or cube but must be uniformly close to the center. Using the Besicovitch-Morse lemma, we can extend our results for the maximal function to LP spaces with respect to other measures. Given a non-negative Borel measure p" define the maximal function Mp,f(x)

= sup (~) r>O P,

r

r

} Br

If(x - y)1 dp,(y).

(If P,(Br) = 0, define the p,-average of f on Br to be zero.) Then one can show that Mp, is weak (1,1) with respect to p,; hence, by interpolation it is bounded on LP(p,), 1 < p < 00. Note that if we define M~ as the maximal operator with respect to cubes, then Mp, and M~ need not be pointwise equivalent unless p, satisfies an additional doubling condition: there exists C such that for any ball B, p,(2B) < Cp,(B). Furthermore, if we define the non-centered maximal operator M~, then this need not be weak (1, 1) if n > 1 unless p, satisfies a doubling condition. In this case the weak (1,1) inequality follows from

8. Notes and further results

45

Wiener's lemma; when n = 1, Lemma 2.6 can be applied even to nondoubling measures. An example of J-L such that M~ is not weak (1,1) is due to P. Sjogren (A remark on the maximal function for measures on JR n , Amer. J. Math. 105 (1983), 1231-1233). For certain kinds of measures a weaker hypothesis than doubling implies that M~ is weak (1,1); see, for example, the paper by A. Vargas (On the maximal function for rotation invariant measures in JRn , Studia Math. 110 (1994),9-17).

8.7. The non-tangential Poisson maximal function. Given f E V(}Rn), u(x, t) = P t * f(x) defines the harmonic extension of f to the upper half-space }R~:+I = }Rn x (0,00), and limt~O Pt * f(x) is the limit of this function on the boundary as we approach x "vertically", that is, along the line perpendicular to }Rn. More generally, one can consider a non-tangential approach: fix a > 0 and let

ra(x) = {(y, t) : Iy -

xl < at}

be the cone with vertex x and aperture a. We can then ask whether lim

(y,t)~(x,O)

u(y, t) = f(x)

a.e. x E

}Rn.

(y,t)Er a(x)

Since this limit holds for all x if f is continuous and has compact support, it suffices to consider the maximal function u~(x)

=

sup

lu(y, t)l,

(y,t)Era(x)

so as to apply Theorem 2.2. But one can show that there exists a constant Ca , depending on a, such that u~(x)

and so

Ua

< CaM f(x), is weak (1,1) and strong (p,p), 1 < p < 00.

These results can be generalized to "tangential" approach regions, provided the associated maximal function is weakly bounded. For results in this direction, see, for example, the articles by A. Nagel and E. M. Stein (On certain maximal functions and approach regions, Adv. in Math. 54 (1984), 83-106) and D. Cruz-Uribe, C. J. Neugebauer and V. Olesen (Norm inequalities for the minimal and maximal operator, and differentiation of the integral, Publ. Mat. 41 (1997), 577-604).

8.8. The strong maximal function. Let R(h I , .. . ,hn ) = [-hI, hI] x ... x [-hn, h n ]. If f E Lfoc' define the strong maximal function of f by

M,J(x) =

h,,~~Lo IR(hl,.l . , hn ) Il(h',

,hn ) If(x - y)1 dy.

2. The Hardy-Littlewood Maximal Function

46

Then Ms is bounded on LP(ffi.n) , p > 1; this is a consequence of the onedimensional result. However, Ms is not weak (1,1): the best possible inequality is

I{x E IRn

:

M,j(x) >

A}I < C

JIf~)1

(1 +log+

If~)I) n~l dx

This result is due to B. Jessen, J. Marcinkiewicz and A. Zygmund (Note on the differentiability of multiple integrals, Fund. Math. 25 (1935), 217-234). A geometric proof was given by A. Cordoba and R. Fefferman (A geometric proof of the strong maximal theorem, Ann. of Math. 102 (1975), 95-100); also see the article by R. J. Bagby ( Maximal functions and rearrangements: some new proofs, Indiana Univ. Math. J. 32 (1983), 879-891). The analog of the Lebesgue differentiation theorem, lim max(hi)-'O

1 IR(h 1 , ... ,hn)1

r

f(x - y) dy = f(x) a.e.

JR(hl, ... ,hn )

is false for some fELl, but is true if f(l + log+ and in particular if f E Lfoc for some p > 1.

Ifl)n-1

is locally integrable,

If in the definition of Ms we allow rectangles (that is, parallelepipeds) with arbitrary orientation (and not just with edges parallel to the coordinate axes), then the resulting operator is not bounded on any LP, p < 00, and the associated differentiation theorem does not hold even for f bounded.

For these and other problems related to differentiation of the integral see de Guzman [.7], the book by the same author cited above, and the monograph by A. M. Bruckner (Differentiation of integrals, Amer. Math. Monthly 78 (1971), Slaught Memorial Papers, 12). 8.9. The Kakeya maximal function.

Given N > 1, let RN be the set of all rectangles in ffi.n with n - 1 sides of length h and one side of length N h, h > O. The Kakeya maximal function is defined by lCN f(x)

=

sup xERERN

IR11

rIf I·

JR

Since each rectangle in RN is contained in a ball of radius cnNh, the Kakeya maximal function is bounded pointwise by the Hardy-Littlewood maximal function:

(2.17) It follows immediately that lCN is bounded on LP, 1 < p <

00.

An important problem is to determine the size of the constants as functions of N for the LP estimates for the Kakeya maximal function. Inequality

8. Notes and further results

47

(2.17) gives the trivial estimate N n - 1 for the constant in the weak (1,1) in"equality, and it is clear that in L=, lCN has norm 1. Interpolating between these values we get

IllCN flip < CnN(n-l)/Pllfll p,

1

< p < 00.

It is conjectured that the constant can be improved to a power of log N when p = n, which, again by interpolation, would give the following bounds:

(1) if p > n then IllCNfllp < Cn(logN)apllfllp; (2) if 1 < p < n then IllCNfllp < CnNn/p-l(logNYl!pllfll p· This conjecture is closely related to two other problems: the Hausdorff dimension of the Kakeya set-a set with Lebesgue measure zero which contains a line segment in each direction (see Stein [17, p. 434]); and the boundedness properties of Bochner-Riesz multipliers (see Chapter 8, Sections 5 and 8.3). This conjecture has been proved completely only when n = 2: see the paper by A. Cordoba (The Kakeya maximal function and the spherical summation multipliers, Amer. J. Math. 99 (1977), 1-22). In this paper he also discussed the connection with Bochner-Riesz multipliers. Cordoba later proved the result in all dimensions when 1 < p < 2; see A note on Bochner-Riesz operators (Duke Math. J. 46 (1979), 505-511). M. Christ, J. Duoandikoetxea and J. L. Rubio de Francia (Maximal operators related to the Radon transform and the Calder6n-Zygmund method of rotations, Duke Math. J. 53 (1986), 189-209) extended the range to 1 < p < (n + 1)/2. A great deal of activity on this problem has been spurred by the work of J. Bourgain (B esicovitch type maximal operators and applications to Fourier analysis, Geom. Funct. Anal. 1 (1991), 147-187). In this paper he proved the conjecture for 1 < p < (n + 1)/2 + En, where En is given by an inductive formula (for instance, E3 = 1/3). T. Wolff (An improved bound for Kakeya type maximal functions, Rev. Mat. Iberoamericana 11 (1995), 651-674) improved this to 1 < p < (n+2)/2. Recently, J. Bourgain (On the dimension of Kakeya sets and related maximal inequalities, Geom. Funct. Anal. 9 (1999), 256-282) has shown that there exists c > 1/2 such that the conjecture is true for p < en if n is large enough. For a discussion of recent results on this problem and its connection with the Kakeya set, see the survey article by T. Wolff (Recent work connected with the Kakeya problem, Prospects in Mathematics, H. Rossi, ed., pp. 129162, Amer. Math. Soc., Providence, 1999).

Chapter 3

The Hilbert Transform

1. The conjugate Poisson kernel Given a function f in S(IR), its harmonic extension to the upper half-plane is given by u(x, t) = Pt * f(x), where Pt is the Poisson kernel. We can also write (see (1.29))

'00

, u(z) where z

= x + it.

=

(

Jo

j(~)e27riZ'f, d~ + (

0

J-oo

j(~)e27riz.f, d~,

If we now define

iv(z)

= (00 j(~)e27riz.f, d~ _ (0 j(~)e27riz.f, d~,

Jo

J-oo

IRt

then v is also harmonic in and both u and v are real if f is. Furthermore, u + iv is analytic, so v is the harmonic conjugate of u. Clearly, v can also be written as

v(z)

=

k_isgn(~)e-27rtlf,lj(~)e27rix.f, d~,

which is equivalent to

(3.1)

v(x, t) = Qt

* f(x),

where

(3.2) If we invert the Fourier transform we get

(3.3)

-49

3. The Hilbert Transform

50

the conjugate Poisson kernel. One can immediately verify that Q(x, t) = Qt(x) is a harmonic function in the upper half-plane and the conjugate of the Poisson kernel Pt (x). More precisely,

Pt(x) which is analytic in 1m Z

. 1 t + ix + 2Qt(X) = 1ft - 2 2 +x

i

= -, 1fZ

> o.

In Chapter 2 we studied the limit as t - t 0 of u( x, t) using the fact that {Pt } is an approximation of the identity. We would like to do the same for v(x, t), but we immediately run into an obstacle: {Qt} is not an approximation of the identity and, in fact, Qt is not integrable for any t > o. Formally, 1 lim Qt(x) = - ;

t-+O

1fX

this is not even locally integrable, so we cannot define its convolution with smooth functions.

2. The principal value of l/x We define a tempered distribution called the principal value of viated p. v. l/x, by

1

1 . ¢(x) p.v.-(¢)=hm dx, x E-+O jXj>E X

To see that

~his

p. v.

l/x,

abbre-

¢ES.

expression defines a tempered distribution, we rewrite it as

r

~(¢) = x

Jjxj
¢(x) - ¢(O) dx + x

Jjxj>l

this holds since the integral of l/x on E < that p. v.

r

Ixl < 1 is zero.

¢(x) dx; x It is now immediate

~(¢) < C(II¢'lloo + Ilx¢lloo). x

,. 1 1 Proposition 3.1. In S , hm Qt = - p. v. -. t-+O

X

1f

Proof. For each E > 0, the functions 'l/JE(X) = X-1X{jxj>E} are bounded and define tempered distributions. It follows at once from the definition that in

S' , 1 lim'l/JE = p. v. -. E-+O x Therefore, it will suffice to prove that in S'

lim (Qt -

t-+O

~'l/Jt)

1f

=

O.

51

3. The theorems of M. Riesz and Kolmogorov

Fix ¢ E 5; then (nQt - 'l/;t)(¢)

=

r ~¢(x)2 dx - r

+x = r ~¢(x)2 J1xl
¢(x) dx x

J1xl>t

dx

+

_r

r (2t + x 2- ~) ¢(x) dx x r ¢(tx) d X

J1xl>t

x¢(tx) d _ - J 1xl<1 1 + x 2 x J 1xl>1 x(l

+ x 2)

X.

If we take the limit as t --7 0 and apply the dominated convergence theorem, we get two integrals of odd functions on symmetric domains. Hence, the limit equals O. D

As a consequence of this proposition we get that lim Qt t~O

* f(x) = .!.lim n

E~O

1

f(x - y) dy,

IYI>E

y

and by the continuity of the Fourier transform on 5' and by (3.2) we get

Given a function f E 5, we define its Hilbert transform by anyone of the following equivalent expressions:

Hf = t---70 limQt * f, Hf

1

=-

1

p. v. -

* f,

x (Hfr(~) = -isgn(Oj(~)· n

The third expression also lets us define the Hilbert transform of functions in L2(~); it satisfies

(3.4) (3.5)

IIH fl12 = Ilflb,

(3.6)

!Hf'9=-!f.H9.

H(Hf)=-f,

3. The theorems ofM. Riesz and Kolmogorov The next theorem shows that the Hilbert transform, now defined for functions in 5 or L2, can be extended to functions in LP, 1 < P < 00. Theorem 3.2. For f E

S(~)7

the following assertions are true:

52

3. The Hilbert Transform

(1) (Kolmogorov) H is weak (1,1):

I{x

E

C

IR: IHf(x)1 > A}I < ~IIfI11.

(2) (M. Riesz) H is strong (p,p), 1 < p <

00:

IIH flip < Cpllfllp· Proof. (1) Fix A > 0 and f non-negative. Form the Calder6n-Zygmund decomposition of f at height A (see Theorem 2.11); this yields a sequence of disjoint intervals {Ij } such that

f(x) < A for a.e. x tj.

n = UI j , j

Inl < 1.1 A < _1 IJ

1

A"f"1'

J, f < 2A. Ij

Given this decomposition of IR, we now decompose functions, g and b, defined by g

(x) = { f 1(x)J IIj I

if x tj.

f

as the sum of two

n,

. Ij f If x E I j ,

and

b(x)

=

L

bj(x),

j

where

bj(x) = (f(X)

~ I~I J,; f) Xlj(X)

Then g(x) < 2A almost everywhere, and bj is supported on I j and has zero integral. Since H f = H g + Hb,

I{x E IR: IHf(x)1 > A}I

< I{x

E IR :

IHg(x)1 > A/2}1

+ I{x E IR : IHb(x)1 > A/2}1·

We estimate the first term using (3.4):

I{x E ffi.:

IHg(x)1 > ,\/2}1 <

Gr

l1H9(xWdX

= : 2 l g(x)2 dx <

~l

g(x) dx

53

3. The theorems of M. Riesz and Kolmogorov

=

8

rj(x) dx.

A JlR

Let 2Ij be the interval with the same center as I j and twice the length, and let 0* = Uj 2Ij . Then 10*1 < 2101 and

I{x E IR:

IHb(x)1 >

A/2}1 < 10*1 + I{x rf. 0* : IHb(x)1 > A/2}1 <

~lljlll + ~ A

r

A JlR\o*

IHb(x)1 dx.

Now IHb(x)1 < Lj IHbj(x) I almost everywhere: this is immediate if the sum is finite, and otherwise it follows from the fact that L bj and L Hbj converge to band Hb in L2. Hence, to complete the proof of the weak (1,1) inequality it will suffice to show that

Even though bj

rf. S,

when x

rf. 2Ij

Hbj(x) =

the formula

r bj(y) dy

JI- XJ

Y

is still valid. Denote the center of I j by ej; then since bj has zero integral,

The last inequality follows from the fact that Ix - ej 1/2. The inner integral equals 2, so

Iy - ejl < IIjl/2 and Ix - yl >

Our proof of the weak (1,1) inequality is for non-negative j, but this is sufficient since an arbitrary real function can be decomposed into its positive and negative parts, and a complex function into its real and imaginary parts.

3. The Hilbert Transform

54

(2) Since H is weak (1,1) and strong (2,2), by the Marcinkiewicz interpolation theorem we have the strong (p,p) inequality for 1 < p < 2. If p > 2 we apply (3.6) and the result for p < 2:

IIHIii p

= sup {

= sup {

L

1} LI· Hg : IIgll", < 1} HI· 9

:

iigiip' <

< IIIllpsup{IIHgllp' : Ilgllpl < I} < Cp/ilillp,

o The functions 9 and b in the proof of the first part of this theorem are traditionally referred to as the good and bad parts of I. It follows from these proofs that the constants in the strong (p,p) and (p', p') inequalities coincide and tend to infinity as p tends to 1 or infinity. More precisely,

Cp = O(p)

as

p

and

---* 00,

Cp= O((p - 1)-1)

as

p ---* 1.

By using the inequalities in Theorem 3.2 we can extend the Hilbert transform to functions in LP, 1 < p < 00. If I E Ll and {In} is a sequence of functions in S that converges to I in Ll (i.e. lim IIIn - Ilh = 0), then by the weak (1, 1) inequality the sequence {HIn} is a Cauchy sequence in measure: for any E > 0, lim I{x m,n->oo

E ffi. :

I(H In - HIm)( X ) I > E} I = 0,

.

i

Therefore, it converges in measure to a measurable function which we define to be the Hilbert transform of I, If I E LP, 1 < p < 00, and {In} is a sequence of functions in S that converges to I in LP, by the strong (p,p) inequality, {HIn} is a Cauchy sequence in LP, so it converges to a function in LP which we call the Hilbert transform of I.

In either case, a subsequence of {H In}, depending on I, converges pointwise almost everywhere to HI as defined. The strong (p,p) inequality is false if p = 1 or p = seen if we let I = X[O,lj' Then

HI(x)

=

1 x -log - - , 7r

x-I

and HI is neither integrable nor bounded.

00;

this can easily be

4. Truncated integrals and pointwise convergence

55

In the Schwartz class it is straightforward to characterize the functions whose Hilbert transforms are integrable: for ¢ E S, H ¢ E L1 if and only if J ¢ = O. We leave the proof of this as an exercise for the reader. For more information on the behavior of the Hilbert transform on L1, see Section 6.5. In Chapter 6 we will examine the space of integrable functions whose Hilbert transforms are again integrable, and the space (larger than LIX)) in which we can define Hf when I is bounded. (Also see Section 6.6.)

4. Truncated integrals and pointwise convergence For E > 0, the functions y-1 X{iyi>E} belong to Lq(JR}, 1 functions

HEI(x) = ~

r

<

q

<

00,

so the

f(x - y) dy

7r JiYi>E

Y

are well defined if I E LP, P > 1. Moreover, HE satisfies weak (1,1) and strong (p,p) estimates like those in Theorem 3.2 with constants that are uniformly bounded for all E. To see this, we first note that (~) =

! ·!

lim

N-,;IX)

=

E
l1m

N-,;IX)

=

e-27riye Y

.

-~

E
-2i sgn(~) lim N -';IX)

dy

sin(27ry~) dy Y

27rN'~' sin(t) dt. 127rEI~1 t

This is uniformly bounded, so the strong (2,2) inequality holds with constant independent of E. We can now prove the weak (1,1) inequality exactly as in Theorem 3.2, and the strong (p, p) inequalities follow by interpolation and duality.

If we fix I E LP, 1 < p < 00, then the sequence {HEI} converges to HI as defined above in LP norm if p > 1 and in measure if p = 1. To see this, fix a sequence {In} converging to I in LP. Then

HI = n-,;IX) lim H In = lim lim H(In = lim lim HEIn = lim HEI; n-,;IX) E-';O E-';O n-,;IX) E-';O the second and third equalities hold because of the corresponding (uniform) (p, p) inequality. We now want to show that the same equality holds pointwise almost everywhere.

3. The Hilbert Transform

56

Theorem 3.3. Given

f

E

LP, 1

00,

Hf(x) = lim Htf(x)

(3.7)

t---40

then a.e. xEIR.

Since we know that (3.7) holds for some subsequence {Htkf}, we only need to show that lim Htf(x) exists for almost every x. By Theorem 2.2 (and the remarks following it) it will suffice to show that the maximal operator

H* f(x) = sup IHtf(x)1 t>O

is weak (p, p). This however, follows from the next result. Theorem 3.4. H* is strong (p,p), 1 < p <

00,

and weak (1,1).

To prove this we need a lemma which is referred to as Cotlar's inequality.

< M(Hf)(x) + CMf(x).

Lemma 3.5. If f E S then H* f(x)

Proof. It will suffice to prove this inequality for each HE with a constant independent of E.

Fix a function ¢ E S(IR) which is non-negative, even, decreasing on (0,00), supported on {x E IR : Ixl < 1/2} and has integral 1. Let ¢t(x) = E- 1¢(X/E). Then

(cp, * p. v. ~) (Y) + [;X{lYI>'} - (cp, * p. v. ~) (Y)] ,

;X{lYI>'} =

and the convolution of the first term on the right-hand side with f is dominated by M(H f) (x) (d. Proposition 2.7). It will s,uffice to find the pointwise estimate for the second term when E = 1 since it follows for any other E by dilation. If

Iyl > 1 then 1

r

y- J 1

¢(x) d xl<1/2 Y - x x

=

1

Ixl<1/2

r

<

- Jxl<1/2 1

¢(x)

(1 1) - - -Y Y- x

¢(x)lxl Iylly - xl

dx

dx

C

<2; y

if

Iyl < 1 then · - 11m 8---40

1 Ixl>8

¢(y - x) d x < x

r

Jxl<2 1

¢(y - x) - ¢(y) dx < C. x

57

4. Truncated integrals and pointwise convergence

Hence,

(I>, * p. v.

;X{IYI>'l -

D

(y) < 1

~ y2'

and by Proposition 2.7 the convolution of the right-hand term with dominated by Mf(x).

f

is 0

Proof of Theorem 3.4. Since both the maximal function and the Hilbert transform are strong (p,p), 1 < p < 00, it follows at once from Lemma 3.5 that H* is strong (p, p). To show that H* is weak (1,1), we argue initially as in the proof of Theorem 3.2. We may assume that f > O. Now fix A > 0 and form the Calder6n-Zygmund decomposition of f at height A. Then we can write f as

f

= 9

+b= 9+ L

bj

.

j

The part of the argument involving 9 proceeds as in Theorem 3.2 using the fact that H* is strong (2,2). Therefore, the problem reduces to showing that

I{x rf. 0* Fix x holds:

rf.

0*,

E

: H*b(x) >

A}I < ~ Ilblh·

> 0 and bj with support I j . Then one of the following

(1) (x-E,x+E)nIj=Ij ,

(2) (x - E, X + E) n I j = 0, (3) x - E E I j or x + E E I j . In the first case, HEbj(x) = O. In the second, HEbj(x) = Hbj(x); hence, if we let Cj denote the center of Ij , since bj has zero average,

IIjl Ibj(y)1 dy < I 1211bjlll. lj X - Y x - Cj x - Cj In the third case, since x rf. 0*, I j C (x - 3E, X + 3E), and for all y Ix - yl > E/3. Therefore, 1

1

IHEbj(x) I < /, ~ -

r ibj{y)11 dy < ~ r x-y

lH,bj{x) I :s

X

If we sum over all j's we get

IHEb(x)1 < L. Ix J

+3,

Ibj{y)1 dy.

E }X-3E

}lj

31

11 .1 _J

.1211bjlll + -E c J

x

+3E

X-3E

IIjl - ~ Ix _ Cjl211bjlll + CMb(x).

<

J

Ib(y)1 dy

E Ij ,

3. The Hilbert Transform

58

. It follows from this that

I{x >-}I < {x Aj2} J

+ I{x E JR.: Mb(x) > '\/2G}1 <

~llbjlh" r Ix -IIjlCj 12 dx + ~'lIblh /\ 7 JlR\2Ij /\

G" < Tllbl1 1 .

o 5. Multipliers Given a function m E Loo(JR.n ), we define a bounded operator Tm on L2(JR.n ) by

(3.8)

(Tmff(~) = m(~)j(~).

By the Plancherel theorem, Tmf is well defined if f E L2 and

IITmfl12 < Ilmll oo llfl12. It is easy to see that the operator norm of Tm is IImll oo : fix E > 0 and let A be a measurable subset of {x E JR.n : Im(x)1 > Ilmll oo - E} whose measure is finite and positive, and let f be the L2 function such that j = XA. Then

We say that m is the multiplier of the operator Tm , though occasionally we will refer to the operator itself as a multiplier. When Tm can be extended to a bounded operator on LP we say that m is a multiplier on LP. For example, the multiplier of the Hilbert transform, m(~) = -i sgn(~), is a multiplier on LP. More generally, given a, bE JR., a < b, define ma,b(~) = X(a,b) (0· Let Sa,b be the operator associated with this multiplier:

(Sa,bff (~)

=

X(a,b) (~)j(~).

We have the equivalent expression (3.9) where Ma is the operator given by pointwise multiplication by e27fiax,

Maf(x)

= e27fiax f(x).

To see this, note that by (1.16) the multiplier of iMaHM-a is sgn(~ - a). Ma is clearly bounded on LP, 1 < P < 00, with norm 1. Therefore, from (3.9) and the strong (p,p) inequality for H, we get that Sa,b is bounded on

5. Multipliers

59

LP. By making the obvious changes in the argument we see that this is still true if a = -00 or b = 00. Hence, we have the following result. Proposition 3.6. There exists a constant C p, 1 < p < a and b, -00 < a < b < 00,

00,

such that for all

For an application of this result, let a = -R, b = R. Then Sa ,b is the partial sum operator SR introduced in Chapter 1: SRf = DR * f, where DR is the Dirichlet kernel. Hence,

IISRfll p < Cpllfllp with a constant independent of R. This yields the following corollary. Corollary 3.7. If f E LP(JR), 1


00,

then

lim IISRf - flip R-HX)

When p

= 1 we do

= O.

not have convergence in norm but only in measure:

lim I{x E JR :

R----'>oo

ISRf (x) -

f (x) I > E} I = O.

From this result and from Corollary 3.7 we see that there exists a sequence {SRkf(x)} that converges to f(x) for almost every x, but the sequence depends on f. Given a family of uniformly bounded operators on LP, any convex combination of them is also bounded. Hence, starting from Proposition 3.6 we can prove another corollary. Corollary 3.8. If m is a function of bounded variation on JR, then m is a multiplier on LP, 1 < p < 00. Proof. Since m is of bounded variation, the limit of m(t) as t -4 -00 exists, so by adding a constant to m if necessary we may assume that this limit equals O. Furthermore, we may assume m is normalized so that it is right continuous at each x E JR. Let dm be the Lebesgue-Stieltjes measure associated with m; then

m(~) = j€ -00

dm(t) = ( X(-oo,.;-)(t) dm(t) = (

J~

J~

X(t,oo)(~) dm(t).

Therefore,

(Tmff(~) = faX(t,oo)(~)j(~) dm(t),

3. The Hilbert Transform

60

and so

Tmf(x)

=

l St,=f(x) dm(t).

By Minkowski's inequality

IITmfilp < lIISt,=fll p Idml(t) < Cpllfllp lldml(t); the integral of finite.

Idml

is the total variation of

m and

by assumption this is D

Given a multiplier, we can construct others from it by translation, dilation and rotation. Proposition 3.9. Ifm is a multiplier on LP(JRn ), then the functions defined by m(~ + a), a E JR n , m()..~), ).. > 0, and m(p~), p E O(n) (orthogonal transformations), are multipliers of bounded operators on LP with the same norm as Tm. The proof of this result follows at once from properties (1.16), (1.17) and (1.18) of the Fourier transform. If m is a multiplier on LP(JR), then the function on JRn given by m(~) = m(6) is a multiplier on LP(JRn ). In fact, if Tm is the one-dimensional operator associated with m, then for f defined on JR n ,

Tmf(x)

=

TmfC X2,· .. ,Xn)(Xl).

Then by Fubini's theorem and the boundedness of Tm on LP(JR),

r

jITln

ITmf(x)IP dx

J. (jITlr < J. J.

=

ITln-l

C

ITln-l

ITl

ITmfC X2,' .. ,xn)(xl)IP dX 1 ) dX2'" dX n If(Xl,'" ,xn) IP dXl ... dxn·

If we take m = X(O,=), then by Proposition 3.6 m is a multiplier on LP(JR) , 1 < p < 00. Hence, by the preceding argument the characteristic function of the half-space {~ E JRn : 6 > O} will be a multiplier on LP(JRn ). Further, by Proposition 3.9 the same will be true for the characteristic function of any half-space (since it can be gotten from the one above by a rotation and translation). The characteristic function of a convex polyhedron with N faces can be written as the product of N characteristic functions of halfspaces, so it is also a multiplier of LP(JRn ), 1 < p < 00. This fact has the following consequence.

Corollary 3.10. If P then

c JRn is a convex polyhedron that contains the origin,

lim IISAPf - flip = 0,

.A.--->=

1
00,

6. Notes and further results

61

where S>.p is the operator whose multiplier is the characteristic function of

AP = {AX: X E P}. In light of the above observation, it is perhaps surprising that when n > 1, the characteristic function of a ball centered at the origin is not a bounded multiplier on LP(lRn ) , p -I- 2. We will examine this and related results for multipliers on lR n in Chapter 8. (Also see Section 6.8 below.)

6. Notes and further results 6.1. References.

The proof of the theorem of M. Riesz first appeared in Sur les fonctions conjuguees (Math. Zeit. 27 (1927), 218-244) but had been announced earlier in Les fonctions conjuguees et les series de Fourier (C. R. Acad. Sci. Paris 178 (1924), 1464-1467). The proof in the text became possible only after the Marcinkiewicz interpolation theorem appeared in 1939. See Section 6.3 below for the original proof by Riesz and another one due to M. Cotlar (A unified theory of Hilbert transforms and ergodic theorems, Rev. Mat. Cuyana 1 (1955), 105-167). Additional proofs due to A. P. Calderon (On the theorems of M. Riesz and Zygmund, Proc. Amer. Math. Soc. 1 (1950), 533-535) and P. Stein (On a theorem of M. Riesz, J. London Math. Soc. 8 (1933), 242-247) are reproduced in Zygmund [22]. Kolmogorov's theorem is in Sur les fonctions harmoniques conjuguees et les series de Fourier (Fund. Math. 7 (1925), 23~28). Another proof due to L. H. Loomis (A note on Hilbert's transform, Bull. Amer. Math. Soc. 52 (1946), 1082-1086) can be found in Zygmund [22] and an elegant proof due to L. Carleson is given by Katznelson [10]. Our proof uses the Calderon-Zygmund decomposition in the same way that it appears in Chapter 5 for more general singular integrals. It is derived from the proof of A. P. Calderon and A. Zygmund (On the existence of certain singular integrals, Acta Math. 88 (1952), 85139). Theorem 3.5 is in the paper by M. Cotlar cited above. 6.2. The conjugate function and Fourier series.

The results in this chapter can also be developed for functions defined on the unit circle. Historically, they evolved in close connection with the theory of Fourier series discussed in Chapter 1. Given a function f E L1(1I') (equivalently, a function in L1([0, 1])), its harmonic extension to the unit disk is gotten by convolution with the Poisson kernel

P (t) r

-

1 - r2 1 - 2r cos(2nt)

+ r2 '

0 < r < 1,

3. The Hilbert Transform

62

and the harmonic conjugate of Pr conjugate Poisson kernel

t _ Qr ()

* f(t)

is gotten by convolution with the

2r sin(27rt) 1 - 2r cos(27rt) + r2 '

O
The analogue of the Hilbert transform is the conjugation operator which is defined by (3.10)

-

f(x) = lim Qr r---71-

* f(x) = E---70 lim

1

f(x - t) - f(x () tan 7rt

E
+ t)

dt.

Both limits can be shown to exist for almost every x. The theorems of M. Riesz and Kolmogorov hold for' the conjugation operator. Hence, by an argument similar to the one in Section 5 it can be shown that the partial sum operators S N f of the Fourier series of fare uniformly bounded on LP, and so S N f converges to f in V. (Cf. Chapter 1, Section 4.) Given a function fELl (1['), its conjugate Fourier series is the trigonometric series 00

S[J](x)

=

L -i sgn(k)j(k)e27rikx; -00

when the Fourier series of jis written as in (1.1), then the conjugate series is gotten by switching the coefficients of the sine and cosine terms and taking their difference instead of their sum: 00

S[J](x)

=

L

ak sin(27rkx) - bk cos(27rkx).

k=O

When JELl then S[J] coincides with the Four~er series of

j.

Even for continuous functions the second limit in (3.10) exists because of subtle cancellation and not because the numerator gets small for t close to zero. It can be shown using the Baire category theory that there exists a continuous function f such that for every x E [0,1]'

r

1/ 2

10

If(x - t) - f(x

+ t)1

dt

= 00.

tan(7rt) The same phenomenon occurs with the Hilbert transform. For these and further results, see the books by Bary [1], Katznelson [10], Koosis [11] and Zygmund [21]. 6.3. Other proofs of the M. Riesz theorem.

The original proof of M. Riesz of the LP boundedness of the conjugate function is based on the analyticity of powers of an analytic function. The easiest case is when p = 2k, k an integer. Given a function f on the unit

6. Notes and further results

63

circle, let u = Pr * f and v = Qr * f; then F(z) = u + iv is analytic in the unit disc and hence so is F(z)2k. By Cauchy's theorem

~ 27r'l

r F(z)2k dz J1z1=r z

=

21f F(re r 27r J o

~

it )2k

dt

=

F(0)2k.

If F(O) = 0 then by taking the real part 9f F(re it ) we get that

21f r Iv(re Jo

it )

12k dt <

t

j=l

(~~) ro21f lu(re J J

it ) 12j

Iv(re it ) 12k - 2j dt.

By applying Holder's inequality (with the appropriate exponent) to each term on the right-hand side, we get

fo2~ Iv(re;t) 12k dt < Ckfo2~ lu(reit ) 12k dt, where C k is independent of the radius of the circle. Taking the limit as the radius tends to 1, u tends to f and v to j. (When F(O) = j(O) =I- 0 replace f by f - j(O).) From the inequality for p = 2k, the whole theorem can be obtained using interpolation and duality, but since interpolation was not available in 1923, Riesz next considered the case when p > 1 is not an integer. In order to get that (u+iv)P is analytic, he assumed u > 0 (which is true if f is non-negative and not identically zero) and then used a clever estimate. For technical reasons, this argument does not work for p an odd integer, and the inequality for these values of p is gotten by duality. In a letter dated November 1923, Riesz explained to Hardy how he arrived at the proof and showed him the essential details (reproduced in M. L. Cartwright, Manuscripts of Hardy, Littlewood, M. Riesz and Titchmarsh, Bull. London Math. Soc. 14 (1982), 472-532). From the fact that (u + iv)2 = u 2 - v 2 + 2iuv is analytic we deduce H(f2 - (H J)2) = 2f . H f, which gives the following formula due to Cotlar:

(3.11)

(H f)2

= f2 + 2H(f . H f).

An alternative way to prove this (proposed by Cotlar in his paper) is to show directly that

HI * HI = j * j -

2isgn(~)(j * HI). It follows from (3.11) that if IIH flip < Cpllfll p,' then IIH fl12p < (Cp +

JC~ + 1)llfI12p.

In fact,

IIHfll~p

=

II(Hf)21Ip

< IIf211p + 211 H (f· Hf)llp

< Ilfll~p + 2Cpllf . H flip

3. The Hilbert Transform

64

and the desired inequality follows by solving a quadratic equation. If we begin with (3.4) and repeatedly apply this inequality we get, for instance, that IIH fl12k < (2k - 1) Ilflbk for integers k > 1. By the RieszThorin interpolation theorem (Theorem 1.19), we get

We get the Riesz theorem for p Theorem 3.2.

< 2 by using duality as in the proof of

An elementary real-variable proof when p transform) is the following:

= 2 (without using the Fourier

and the kernel of HEHE, which is the convolution of y-1 X {IYI>E} with itself, can be computed explicitly and shown to be in L1. (By a dilation argument, it suffices to consider the case E = 1.) This proof is attributed to N. Lusin (1951) by E. M. Dyn'kin (Methods of the theory of singular integrals: Hilbert transform and Calder6n-Zygmund theory, Commutative Harmonic Analysis, I, V. Havin and N. Nikolskii, eds., pp. 167-259, Springer-Verlag, Berlin, 1991). A similar argument had been used earlier by Schur for a discrete version of the operator. Another real-variable proof when p ter 9, Section 1.

=

2 using Cotlar's lemma is in Chap-

6.4. The size of constants. The best constants in the theorems of Kolmogorov and M. Riesz are known. For the strong (p,p) inequality, 1 < p < 00, the best constants are Cp = tan(7r/2p), 1 < p < 2, and Cp = cot (7r/2p) , 2 < p < 00. This was first proved by S. K. Pichorides (On the best values of the constants in the theorems of M. Riesz, Zygmund and Kolmogorov, Studia Math. 44 (1972), 165-179); a much simpler proof was later given by L. Grafakos (Best bounds for the Hilbert transform on LP(Il.~l), Math. Res. Let. 4 (1997),469-471). In the proof of the M. Riesz theorem due to Cotlar given above, we get

J

the estimate C2p < Cp + C; + 1. If we begin with C2 = 1, the resulting bounds for C 2 k are sharp. This fact was first observed by 1. Gohberg and N. Krupnik (Norm of the Hilbert transformation in the space LP, Funct. Anal. Appl. 2 (1968), lSD-lSI).

6. Notes and further results

65

The best constant in the weak (1,1) inequality has a more complex expression:

This was first proved by B. Davis (On the weak type (1,1) inequality for conjugate functions, Proc. Amer. Math. Soc. 44 (1974), 307-311) using Brownian motion. A non-probabilistic proof was later given by A. Baernstein (Some sharp inequalities for conjugate functions, Indiana Univ. Math. J. 27 (1978), 833-852).

6.5. The Hilbert transform on LI. As we noted in Section 3, if fELl then H f need not be. However, unlike the case of the maximal function, there exist fELl such that H fELl. For example, if f = X(O,I) - X(-I,O) then Hf(x) = log(lx 2 - 11/x 2 ) is integrable. A simple necessary condition for H f to be integrable if fELl is that j(O) = J f = o. If f is such that the identity (H ff(~) = -i sgn(Oj(~) holds, then it suffices to note that the Fourier transform of an integrable function is continuous, and j(~) and -i sgn(~)j(~) are both continuous only if j(O) = O. This argument clearly works if fELl n L2 since in this case the Fourier transform identity is valid. But one can show that this identity holds if f, Hf ELI. This would follow if ¢ * (Hf) = H(¢ * f) for every ¢ E S: take the Fourier transform of both sides and simplify. This equality can be proved as is done for more general singular integrals in the paper by A. P. Calderon and O. Capri (On the convergence in LI of singular integrals, Studia Math. 78 (1984),321-327). Similarly, when f, H fELl we have the identity H(H f) = -f. We originally gave this in (3.5) for L2 functions, and it can be extended to LP, 1 < p < (Xl, using the results in this chapter. For f and H f integrable, a proof using complex analysis is due to E. Hille and J. D. Tamarkin (On the absolute integrability of the Fourier transform, Fund. Math. 25 (1935), 329352). The interesting paper by J. F. Toland (A few remarks about the Hilbert transform, J. Funct. Anal. 145 (1997), 151-174) examines this question and provides additional historical information. In general, H f ¢:. LI, so we cannot even define the Hilbert transform of H f. Nevertheless, if we replace the Lebesgue integral with a more general one, then it is possible to define H(H f) and to get the identity. (See Zygmund [21, Chapter 7].) The above-cited paper by Toland uses this extended notion of the integral.

3. The Hilbert 'ITansform

66

6.6. L log L estimates.

The following result characterizes the local integrability of the Hilbert transform; it is analogous to Theorem 2.18 for the maximal function. Theorem 3.11. Let Band C be two open balls such that

(1) If flog+ If I E L1(C), then Hf E L 1(B). (2) Conversely, if f > 0 and Hf E L1(C), then flog+

Bc

C.

If I E L1(B).

The first part of the theorem is due to A. P. Calderon and A. Zygmund (see the paper cited above). It can be sharpened using Orlicz spaces (cf. Chapter 2, Section 8.4, and Zygmund [21, Chapter 4]). The second part is due to E. M. Stein (Note on the class L log L, Studia Math. 32 (1969), 305310). The hypothesis f > 0 can be replaced by a weaker condition which measures (in some sense) the degree to which Pt * f differs from a positive harmonic function. See the article by J. Brossard and L. Chevalier (Classe LlogL et densite de l'integrale d'aire dansIR~+\ Ann. of Math. 128 (1988), 603-618).

6.7. Translation invariant operators and multipliers. Let Th be the translation operator: Thf(x) = f(x - h). An operator T is said to be translation invariant if it commutes with Th: Th 0 T = To Th, h E IRn. The Hilbert transform, for example, is translation invariant. Such operators are completely characterized by the following result. Theorem 3.12. Given a linear operator T which is translation invariant and bounded from LP(IRn) to Lq(IRn ) for any pair (p, q), 1
If such a T is bounded from LP to Lq and not zero, then p cannot be greater than q. An elegant and very simple proof of this fact is due to L. Hormander (Estimates for translation invariant operators in LP-spaces, Acta Math. 104 (1960), 93-139). First note that limh~= Ilf + Thfll p = 21/ Pllfll p· Let IITllp,q be the norm of T as a bounded operator from LP to Lq. Since T commutes with translations,

liT f + ThT fllq <

IITllp,q Ilf

+ Thfll p;

if we let h tend to infinity we get

IITfllq < 21/P-1/qIITllp,qllfllp, and this is possible only when p < q. Clearly, if T f = K * f then T is linear and translation invariant. By the properties of the Fourier transform, (T ff = i( j, which implies that k is a

67

6. Notes and further results

multiplier of T, at least if K E L=, following the definition in Section 5. But if T is bounded on LP, then by duality (since its adjoint is the convolution with kernel K(x) = I«-x)) it is bounded on LP'. Thus by interpolation it is bounded on L2, so the condition that k E L= is necessary, and our assumption that a multiplier m is in L= is justified. It would be interesting to characterize the distributions K that define bounded operators (equivalently, to characterize the multipliers on LP), but a complete answer is known only in two cases.

Proposition 3.13.

(1) T is bounded on L2(IRn) if and only if k E L= and the norm of T as an operator on L2 is the norm of k in L=. (2) T is bounded on L1(IRn) if and only if K is a finite Borel measure, and the operator norm ofT equals the total variation of the measure.

Note that since p. v. l/x is not a measure, Proposition 3.13 gives another proof that the Hilbert transform is not bounded on L1. For all of these results, see Stein and Weiss [18, Chapter 1]. 6.8. Multipliers of Fourier series.

The multipliers of trigonometric series are precisely bounded sequences, {.\( k) hEZ. The associated operator is defined for f E L2 (1I') by

T)J(x) =

L

'\(k)j(k)e27fikx.

kEZ

(For multiple Fourier series the definition is the same except that we take k E 7l n .) T>.. can be written as the convolution of f with a distribution whose Fourier coefficients are {.\ (k )}, and there are results analogous to those in Proposition 3.13.

If we know the boundedness properties of a multiplier on LP(IR), then we can deduce results for multipliers of series by restricting the multiplier to the integers. Theorem 3.14. Given 1 < P < 00, suppose the operator T defined by T f = K * f is bounded on LP(IR). If k is continuous at each point of 7l and .\(k) = k(k), then the operator T>.. associated with this sequence is bounded on LP(1I') and liT>.. I < IITII.

There is also an analogous n-dimensional result. For a proof see Stein and Weiss [18, Chapter 6]. Starting from Theorem 3.14, one can deduce results for the partial sum operators SN of a Fourier series from the corresponding results for SR in

68

3. The Hilbert Transform

Section 5, and in particular one gets the convergence in LP norm of the partial sums of a Fourier series. (Cf. Chapter I, Section 4.) 6.9. The Hilbert transform of characteristic functions. If A is a subset of IR with finite measure, one can show that

21AI

I{x E IR: IH(XA)(x)1 > A}I = sinh(-rrA)" This result was first proved by E. M. Stein and G. Weiss (An extension of a theorem of Marcinkiewicz and some of its applications, J. Math. Mech. 8 (1959),263-284). A simple proof is given by Zygmund [22, p. 15].

4

Chapter

Singular Integrals (I)

1. Definition and examples

The singular integrals we are interested in are operators of the form

Tf(x) = lim

(4.1)

10->0

r 0 y2f(x - y) dy, J1Y1>f (

1

Y

1

where 0 is defined on the unit sphere in IR n , sn-l, is integrable with zero average and where y' = Y/IYI. With these hypotheses, (4.1) is defined for Schwartz functions since it is the convolution of f with the tempered distribution p. v. O(x')/Ixl n , defined by

(4.2) p. v.

O(x')

1.

.

-X InI (¢) = 10->0 hm =

O(x')

IX In

Ixl>f

¢(x) dx

r 0 x2 i1>(x) -1>(0)] dx + r ~(~21>(x) dx. J1xl<1 J1xl>1 1

(

X

1

X

(The second equality follows since 0 has zero average.) Since ¢ E S, both integrals converge. Note that we could also assume, for instance, that ¢ in (4.2) or f in (4.1) is a Lipschitz function of order a with compact support. Proposition 4.1. A necessary condition for the limit in (4.1) or (4.2) to

exist is that 0 have zero average on

sn-l.

Proof. Let f E S(IRn) be such that f(x)

Ixl < 1, Tf(x)

=

=

1 for any

Ixl <

2. Then for

r 0 y2f(x - y) dy + lim 1 0 y2dy. J1Y1>1 f
1

Y

(

1

1

10->0

y

1

-

69

4. Singular Integrals (I)

70

The first integral always converges but the second equals

0,(Y') dcr(y') . log(l/ E),

lim ( 1':--;0 }

sn-l

and this is finite only if the integral of 0, on

sn-l

is zero.

D

When n = 1 the unit sphere reduces to two points, 1 and -1, and 0, must take opposite values on them. Thus on the real line, any operator of the type in (4.1) is a multiple of the Hilbert transform. In higher dimensions we consider two examples given by A. P. Calderon and A. Zygmund. First, let f(xl, X2) be the density of a mass distribution in the plane. Then its Newtonian potential in the half-space IR.! is

U(XI' X2, X3)

=

J.

JR2

(Xl -

f(yl, Y2) ( )2 YI + X2 - Y2

)2

+ x32 dyl dY2.

The strength of the gravitational field is gotten by taking partial derivatives of the potential. The component in the X3 direction is equivalent to a multiple of the Poisson integral which we have already considered. The other two are similar to one another; for example, for the first we formally have that

. au hm -8 (Xl, X2, X3)

X3--;0

Xl

= -

1 JR2

[(

Xl

f(yl, Y2)(XI - YI) )2 + (X2 - Y2 )2]3/2 dYldY2. - YI

This integral does not converge in general, but it exists as a principal value if f is smooth and is in fact the value of the limit of 8u/8xI. It corresponds to a singular integral of the form (4.1) in IR. 2 with 0,(x') = -xI/lxl. (In polar coordinates this becomes cos( 8).) The second example is gotten from the logarithmic potential associated with a mass distribution f(xl, X2) in the plane (i'.e. the solution of the equation /:1u = f):

U(XI' X2)

=

(

JJR2

f(YI, Y2) log ([(

Xl - YI

)2

+1( X2

- Y2

)2]1/2) dYldY2·

This integral converges absolutely if, for example, f has compact support, and partial derivatives can be taken under the integral sign. However, this is not possible for second derivatives: one can show that 82u/8x18x2 is given by an operator of the form (4.1) with 0,(x') = 2xlx2/lxI2. For more information on this operator, see Section 5.

2. The Fourier transform of the kernel A function

f is homogeneous of degree

a if for any

X E

IR.n and any A > 0,

2. The Fourier transform of the kernel

Given any function cp, define cp)..(x)

=

71

).-ncp().-l x ); then

r f(x)cp)..(x) dx =).a JJRnr f(x)cp(x) dx,

J]Rn

so we can define the homogeneity of a distribution as follows.

Definition 4.2. A distribution T is homogeneous of degree a if for every test function cp it satisfies

A simple computation shows that the distribution given by (4.2) is homogeneous of degree -n; details are left to the reader.

Proposition 4.3. If T is a tempered distribution which is homogeneous of degree a, then its Fourier transform is homogeneous of degree -n - a. Proof. By Definition 1.16 and property (1.18), if cp E S then

T(cp)..) = T(¢()'.)) = ).-nT(¢).._l) = ).-n-aT(¢) = ).-n-aT(cp).

o As a corollary to this proposition we can easily calculate the Fourier transform of f(x) = Ixl- a if n/2 < a < n. For in this case f is the sum of an L1 function (its restriction to {Ixl < I}) and an L2 function, so by Proposition 4.3, f is a homogeneous function of degree a - n. Hence, since f is rotationally invariant, j(e) = ca,nlel a- n . We calculate ca,n using Lemma 1.14 and (1.21): A

r

JJRn

A

e-7rlxI2Ixl-a dx = ca,n

since

1

00

a

d=

-7rr2 e rbr

J.JRn

e-7rlxI2Ixla-n dx;

1- r (1 + b)

-7r

2

l+b 2

--

2'

we see that (4.3) In fact, this formula holds for all a, 0 < a < n. For 0 < a < n/2 it follows from the inversion formula (1.22). Further, since Ixl- a tends to Ixl n/ 2 as a -----)- n/2, and since the Fourier transform is continuous, (4.3) holds with a = n/2.

72

4. Singular Integrals (I)

Theorem 4.4. If 0 is an integrable function on sn-1 with zero average, then the Fourier transform of p. v. O(x')//x/ n is a homogeneous function of

degree

°

given by

m(E)

(4.4)

hn-,

=

Q(u) [log

Cu lE'l) - i; sgn(u· E')1daru).

Proof. By Proposition 4.3 the Fourier transform is homogeneous of degree 0; therefore, we may now assume that /~/ = 1. Since 0 has zero average,

m(~) =

1

lim O(y') e-27riy.~ dy E-+O E
r

= lim

E-+O}sn-l

Thus m(~)

+ 1,1/E e-27riru.~dr] r

1

dcr(u).

h - ih, where

=

r

h = lim

O(u) [11 (e-27riru.~ _ 1) dr E r

E-+O}sn-l

O(u) [11 (cos (27rru E

.~) -

1) dr r

+ h .

=

1

lim O(u) E-+O sn-l

/,'/< cos(21rru . 0 ~ ] datu),

[1

1/E sin(27rru· E

dr]

~)-

r

dcr(u).

By the dominated convergence theorem we can exchange the limit and outer integral. In each inner integral make th~ change of variables s 27rr/u· ~/. We may assume that u· ~ # 0, so in 12 we get

/,

27rlu.~IE

As

E ----t

ds

27rlu.~I/E

0 this becomes sgn(u·~)

1= o

sin(s) sgn(u·

~)-. S

sin(s) ds s

7r 2

= - sgn(u· ~).

After the change of variables in h we get

ds 1,27r IU{I/E ds 1,27rIU.~1 ds cos(s)- -; (cos(s) - 1)- + 27rlu·~IE s I S 1 s

/,

as

E ----t

1

0 this becomes

1 1

o

ds (cos(s) - 1)- +

j=

cos(s)

SIS

ds -log /27r/-log /u· ~/.

3. The method of rotations

73

If we integrate against 0, which has zero average on sn-l, the cop.stant terms disappear and we get the desired formula for m. 0 In formula (4.4) the factor multiplied against 0 has two terms: the first is even and its contribution is zero if 0 is odd; it is not bounded but any power of it is integrable. The second is odd and its contribution is zero if o is even; further, it is bounded. Since any function 0 on sn-l can be decomposed into its even and odd parts, 1

Oo(u) = "2(O(u) - O(-u)),

we immediately get the following corollary.

Corollary 4.5. Given a function 0 with zero average on sn-l, suppose that 0 0 E Ll(sn-l) and Oe E Lq(sn-l) for some q > 1. Then the Fourier transform ofp. v. O(x')/Ixl n is bounded. From (4.4) one can easily find an integrable function 0 such that m is not bounded. Nevertheless, in Corollary 4.5 we can substitute the weaker hypothesis Oe E LlogL(sn-l), that is,

hn-l

10e(u)llog+ 10e(u)1 dCJ(u)

< 00.

(Recall that log+ t = max(O, log t).) The sufficiency of this condition follows from the inequality

AB < A log A For in the region D

In

= {u

0.(u) log

<

In

E

+ eB ,

A > 1, B > O.

sn-l : 10(u)1 > I},

Cu CI) du(u) 1

210(u)llog(210(u)l) dCJ(u)

o is bounded in the complement of D,

+

In

lu· (1- 1 / 2 dCJ(u).

and so the integral is finite.

3. The method of rotations Corollary 4.5 together with the Plancherel theorem gives sufficient conditions for operators T as in (4.1) to be bounded on L2. In this and the following section we develop techniques due to Calderon and Zygmund which will let us prove they are bounded on LP, 1 < p < 00. Let T be a one-dimensional operator which is bounded on

LP(~)

and let

u E sn-l. Starting from T we can define a bounded operator Tu on ~n as follows: let Lu = {AU: A E ~} and let Lt be its orthogonal complement in ll~n. Then given any x E ~n, there exists a unique Xl E ~ and x E Lt such

4. Singular Integrals (I)

74

that x = XIU + X. Now define Tuf(x) to be the value at Xl of the image under T of the one-dimensional function f(- U + x). If Cpis the norm of T in LP(~), then by Fubini's theorem,

!.

ITuf(x)IP dx

=

F

<

(

JLt

!.

IT(f(- U + x)(xl)IP dxldx

R

C: JLtr !. If(- U + x)(xl)I Pdxldx C: JRnr If(x)IP dx. R

=

Operators gotten in this way include the directional Hardy-Littlewood maximal function,

Muf(x) = sup Ih h>O 2 and the directional Hilbert transform, . Huf(x) = -1 hm 7r

jh

If(x - tu)1 dt,

-h

1,

E--+O Itl>E

dt f(x - tu)-.

t

Since the operators Tu obtained from the operator T are uniformly bounded on LP (~n), any convex combination of them is also a bounded operator. Hence, the next result is an immediate consequence of Minkowski's integral inequality. Proposition -4.6. Given a one-dimensional operator T which is bounded

on LP(IR) with norm Cp, let Tu be the directional operators defined from T. Then for any nELl (sn-l ), the operator Tn defined by Tnf(x) =

kn-l

n(u)Tuf(x) d(}(u)

is bounded on £p(IRn) .with norm at most cpllnlh. By using this result we can pass from a one-dimensional result to one in higher dimensions. Its most common application is in the method of rotations, which uses integration in polar coordinates to get directional operators in its radial part. For example, given nELl (sn-l), define the "rough" maximal function

(4.5)

Mnf(x) = sup

R>O

r

IB(~, R)I JB(O,R) 1r2(y')lIf(x -

y)1 dy.

If we rewrite this integral in polar coordinates we get

Mnf(x) =

~~ IB(O,ll)IRn kn-l 1!1(u)1 fuR If(x -

ru) Irn - l drdo-(u)

3. The method of rotations

r

< IB( 1 )1

75

Jsn-l

0,1

10(u)IMuf(x) dO"(u).

Corollary 4.7. If 0 E L1 (sn-1) then MD, is bounded on LP(IR(n)) 1 < p

<

00.

If we let O(u) = 1 for all u, then MD, becomes the Hardy-Littlewood maximal function in IR(n; thus the method of rotations shows it is bounded on V(IR(n), p > 1, starting from the one-dimensional case. Further, it is interesting to note that the V constant we get is O(n) since Isn- 1 1/IB(0, 1)1 = n. (See Chapter 2, Section 8.5.) On the other hand, the method of rotations does not give us the weak (1,1) result. (See Section 7.6.)

o

We can also apply Proposition 4.6 to operators of the form (4.1) when is odd. In fact, if we fix a Schwartz function f, then

T f (x)

!,sn-l 1 !,Sn-l 1.

dr f (x - ru) - dO" (u ) E r 00.

=

lim E-->O

=

lim -1 E-->O 2

0 (u )

O(u)

Irl>E

dr f(x - ru) -dO"(u); r

since 0 has zero average, we can argue as we did in (4.2) to get =

~ lim

+~ 2

1 1.

r

O(u)

r

O(u)

2 E-->O J Sn-l

JSn-l

E
(f(x - ru) - f(x)) dr dO"(u) r

f(x - ru) dr dO"(u).

Irl>l

r

Because f E S, the inner integral is uniformly bounded, so we can apply the dominated convergence theorem to get =

7T

2

r

Jsn-l

O(u)Huf(x) dO"(u).

Since the Hilbert transform is strong (p, p), 1 the following.

< p < 00, we have proved

Corollary 4.8. If 0 is an odd integrable function on sn-l) then the operator T defined by (4.1) is bounded on LP (IR(n)) 1 < p < 00. In Corollary 4.8 we implicitly defined T f for f E LP as a limit in LP norm. However, as with the Hilbert transform, we can show that (4.1) holds for almost every x. By an argument similar to the one above we can show that the maximal operator associated with the singular integral,

(4.6)

T* f(x) = sup E>O

r 0 y2f(x JIYI>E Y (

1

1

y) dy ,

4. Singular Integrals (I)

76

satisfies

T* f(x) <

7r

2

r

In(u)IH~f(x) d(/(u) ,

J

sn-l

where H~ is the directional operator defined from the maximal Hilbert transform. (See Chapter 3, Section 4.) Therefore, we can apply Proposition 4.6 and Theorem 2.2 to get the following.

Corollary 4.9. With the same hypotheses as before, the operator T* defined by (4.6) is strong (p, p), 1 < p < 00. In particular, given f E LP, the limit (4.1) holds for almost every x E jRn. An important family of operators with odd kernels consists of the Riesz transforms,

(4.7)

r

y.

Rjf(x) = Cn p. v. J~n IYI~+l f(x - y) dy,

where

1)

n+ cn=f ( -2The constant is fixed so that for

f

7r

_n+l 2

1 <j< n,

.

E L2,

(4.8) this in turn implies that n

""' 6 R~J =-1 ' j=l

(4.9)

where I is the identity operator in L2. Since the Schwartz functions are dense in LP, p > 1, this identity in fact holds in every LP space. To prove (4.8) one could use Theorem 4.4 directly; instead we will argue as follows. With equality in the sense of distributions,

a I I-n+ 1 _ ax j x. so if we apply (4.3) we get

(

)

xj

1 - n p. v. Ix In+ 1 '

77

4. Singular integrals with even kernel n+l .

=

7[-2-

.

-'l, -;--:--:;-:-

r (nt1)

~j

j~j.

4. Singular integrals with even kernel If the function D in the singular integral (4.1) is even, then the method of rotations does not apply since we cannot represent the singular integral in terms of the Hilbert transform. However, we ought to be able to argue as follows: by (4.9), n

Tf

n

LR](Tf) j=l

= -

LRj(RjT)f, j=l

= -

and the operator RjT is odd since it is the composition of an odd and even operator. If we can show that RjT has a representation of the form (4.1), then by Corollary 4.8, T is bounded on LP. In the rest of this section we will make this argument precise by showing that RjT has the requisite representation. Let D be an even function with zero average in Lq(sn-1), for some q > 1, and for E > 0 let (4.10) Note that KE E L T , 1 < r < q. Thus if Fourier transform we see that

f

E

Cgo(JR n ) then by taking the

(4.11) Lemma 4.10. With the preceding hypotheses, there exists a function K j which is odd, homogeneous of degree -n and such that

lim RjKE(x)

E~O

= Kj(x)

in the L oo norm on every compact set that does not contain the origin. Proof. Fix x i=- 0 and let 0 by Corollary 4.9

RjKE(x) - RjKv(x)

< E < l/ < jxj/2. Then for almost every such x,

= Cn c5~O lim =C

J

J.

]Rn

j

Xj

X -

-jY~lX{IX-YI>c5}[KE(Y) Yn

Kv(Y)] dy

Xj - Yj D(y') dy· ' n E
since D has zero average, = Cn

J

E
(Xj - Yj Xj ) D(y') dy jx - yjn+1 jxjn+1 jyjn .

4. Singular Integrals (I)

78

Therefore, if we apply the mean value theorem to the integrand we get

(4.12)

C

IRjKE(x) - RjKv(x) 1< 1 In+l

1

x

10(y') 1 CIIOlh 1 In-l dy < 1 In+l v. x

E
Hence, for any a > 0, {RjKE} is a Cauchy sequence in the L= norm on { 1xl> a}. So for almost every x we can define by

K;

KJ~(x)

= E-+O lim RjKE(x).

The function Rj KE is odd, so (by modifying necessary) is also an odd function.

K;

K; on a set of measure zero if

To find the desired function K j , fix A > 0; then again for almost every

x,

K;

K;

Hence, for almost every x, (AX) = A-n (x). The set of measure zero where equality does not hold depends on A, but since is measurable, the set

'D

=

{(x, A) C JRn

X

(0,00) : K;(AX)

K;

-# A-n 1<; (x)}

has measure zero. Therefore, by Fubini's theorem there exists a sphere centered at the origin of radius p, Sp, such that D n Sp has measure zero. We now define '

Kj(x) =

UI:I) K; (I~)

if x

n

-# 0 and px/lxl rt. D n Sp;

otherwise. This function is measurable, homogeneous of degree -n (by definition) and odd. Further, K;(x) = Kj(x) almost everywhere. To see this, let x -# 0 be such that Xo = px/lxl rt. D n Sp. (The set of x such that Xo E D n Sp has measure zero since D n Sp has measure zero.) Then for almost every A,

Kj(Axo) = A-nKj(xo)

=

A-nK;(xo) = K;(AXo).

o Lemma 4.11. The kernel K j defined in Lemma 4.10 satisfies

hn-l

IKj(u)1 d(J(u) <

CqlIOll q·

4. Singular integrals with even kernel

Furthermore, if Kj,E(X) and II~EI11 < C~IIOllq·

= Kj(x)X{lxl>E}' then ~E = RjKE- Kj,E

Proof. By the homogeneity of K j

r

IKj(u)1 dO"(u)

J

79

=

Sn-l

E

L1(JR n )

,

-II/, og

2 1
1

IKj(x)1 dx

~

/,

< log2 1
+ - 12 1 /, og

=

In (4.12) let v

1/2 and Ixi

1
IRjK1/2(X)1 dx.

> 1; then if we take the limit as

~

E

~

0 we get

CII0111

IKj(x) - R j K 1/2(X)1 < Ixln+1·

(4.13)

Therefore, the first integral on the right-hand side above is bounded by CII0111 < CIIOllq. To bound the second integral, note that

IRjK1/2(X)1 dx < CIIRj K 1/21Iq < CIIK1/ 2 1Iq < CIIOllq·

/,

1
To prove the second assertion it will suffice to show that II~llh since ~E = E- n ~1 (E- 1X). But then

II~Iil1 = <

r

IRj K 1(x) - K j,1(x)1 dx

r

IRj K 1(x)ldX+/,

JlR

<

00

n

Jxl<2

IKj(x)ldx+

1
1

r

J1xl>2

l~l(X)ldx.

The first integral is bounded by CIIRj K 111q < CIIK111q < CIIOllq; above we showed that the second integral has the same bound. To see that the third integral is bounded, we argue as we did for (4.13) to get l~l(X)1 < CII0Ihlxl-n-1. 0 Theorem 4.12. Let 0 be a function on sn-1 with zero average such that its odd part is in L1 (sn-1) and its even part is in Lq (sn-1) for some q > 1. Then the singular integral T in (4.1) is bounded on LP (JR n ) , 1 < p < 00. Proof. By Corollary 4.8 we may assume 0 is even. Further, by arguing as we did for the Hilbert transform, it will suffice to establish the LP inequality for functions f E Cgo(JR n ). For such f, Tf = limE_o KE * f. From (4.9) and (4.11) we see that n

I{E

*f = -

~Rj ((RjKE) * f), j=l

4. Singular Integrals (I)

80

and in the notation of Lemma 4.11,

By Lemma 4.10, K j is an odd, homogeneous kernel of degree -n, so by Corollary 4.9,

II Kj,E * flip < C

!sn-l 'Kj(u)1 da(u)lIflip < CIIOllqllfll p·

By Lemma 4.11,

If we combine these estimates and use the fact that Rj is bounded in LP we see that

Since the right-hand side is independent of

E,

by Fatou's lemma we get

o

and this completes our proof.

Another proof of Theorem 4.12 is given below in Chapter 8. (See Corollary 8.21.)

5. An operator algebra Let P(~) = L:a ba~a be a polynomial in n variables with constant coefficients and let P(D) be the associated differential polynomial, that is, the operator given by a

It follows from (1.19) that

(P(D)ff(~)

= P(27ri~)j(~).

Define the operator A to be the square root of the positive operator -b.: (4.14) If P is a homogeneous polynomial of degree m, then

(4.15) where the operator T is defined by

.a

(Tff(~) = im~\~ j(~).

5. An operator algebra

81

The multiplier P(~)/I~lm is homogeneous of degree zero and its restriction to sn-l coincides with p(e), so it is a C= function. (In fact, it can be shown that T is a linear combination of compositions of Riesz transforms.) The operator T is not a singular integral of the type given in (4.1) since P(~) may not have zero average on sn-l. However, we can get such a singular integral by subtracting a constant, that is, a multiple of the identity operator. This particular operator T is a special case of the operators in the following result.

Theorem 4.13. If m E C=(JRn \ {O}) is a homogeneous function of degree 0, and Tm is the operator defined by (Tmff = mj, then there exist a E C and n E c=(sn-l) with zero average such that for any f E 5, Tmf

n(x')

= af +p.v. W

*f.

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-l, Theorem 4.13 is an immediate consequence of the following lemma.

Lemma 4.14. Let m E C=(JR n \ {O}) be a homogeneous function of degree o with zero average on sn-l. Then there exists n E C= (sn-l) with zero average such that m(~) = p. v. n(x')/Ixl n . Proof. Since m is a tempered distribution,

m exists.

Hence,

(~:;) (~) = C~fm(O, where C is a constant. The function anm/axi is homogeneous of degree -n, in C=(JR n \ {O}) and has zero average on sn-l. Furthermore,

anm

anm

a -a = p. v. -a + "" x'l!x'l!~ C a D 6, 2

2

lal:=;k

where 6 is the Dirac measure at the origin, since the difference between anm/a~i and p. v. anm/a~i is a distribution supported at the origin. If we take the Fourier transform of both sides of this equation we get

C~rm(~) =

(p.

v.

~:r;:) (~) + L 2

Ca (2rriO"·

lal:=;k

The left- hand side and the first term on the right-hand side are homogeneous distributions of degree 0, so the polynomial on the right-hand side reduces to a constant. Thus the right-hand side is a homogeneous function

82

4. Singular Integrals (I)

of degree 0 which is in coo(IRn \ {O}). Since this is valid for 1 < i < n, m coincides on IR n \ {O} with a homogeneous function of degree -no We will denote its restriction to sn-l by O. To see that 0 has zero average on sn-l, fix a radial function ¢ E S which is supported on the annulus 1 < [xl < 2 and which is positive on the interior. Then

m(¢) = { O[ (x[2 ¢(x) dx = c ( O(u) da(u), J~n x Jsn-l where c> 0; furthermore, since ¢ is radial and m is homogeneous,

m(¢) = m(¢) = c' (

}sn-l

Finally, to see that ference,

m is

m(u) da(u) = O.

identical to p. v. f2(x')j[x[n, consider their dif-

A

O(x')

m-p.v·W' which is supported at the origin. If we take the inverse Fourier transform of this difference we get a polynomial which must be constant since both m and (p. v: f2(x')j[x[nf are bounded. Further, this constant must be zero since both m and f2 have zero average on sn-l. 0

Theorem 4.15. The set A of operators defined by Theorem 4.13 is a commutative algebra. An element of A is invertibte if and only if m is never zero on sn-l . Proof. To see that A is an algebra, it is enough to note that given m2 with associated operators Tml and Tm2 , Tml 0 Tm2 = Tm1m2 ·

ml

and

Since the identity has the function 1 as its multiplier, Tm is invertible if and only if Tl/m E A, and 1jm E coo(IRn \ {O}) precisely when m(~) -I- 0 for any ~ E sn-l. 0

The operator T in (4.15) associated with the polynomial P is thus invertible if and only if P(~) is never zero on sn-l, that is, if P(D) is an elliptic operator. If the coefficients of P are real, then m must be even and Am = (_il)m/2. Thus the problem of solving P(D)u = f reduces to solving (_il)m/2u = T- 1 f. (For more on this operator, see Section 7.7.)

6. Singular integrals with variable kernel

83

6. Singular integrals with variable kernel Let P(x, D) be a homogeneous differential polynomial with variable coefficients:

L

P(x, D) =

ba(x)Da.

[a[=m Since for

f

E

S,

we have that

P(x, D)f(x) =

r P(x, 27ri~)j(~)e27rix.f; d(

J'J1n

Further, with A as defined by (4.14), this has a representation analogous to (4.15):

P(x, D)f = T(Am f),

(4.16) where T is defined by

T f(x) =

(4.17)

fJ

r

fJ(x, ~)j(~)e27riX.f; d~,

J'J1n (

c)

=

P(x, i~)

1~lm.

X, ':,

The function fJ is homogeneous of degree 0 in the variable ~. If we substitute the definition of j(~) into (4.17) we get .

Tf(x) =

r J'J1

fJ(x,~)

n

r J'J1

f(y)e- 27riy .f; dy e27rix .f; d~.

n

Thus formally, (4.18)

Tf(x) =

r K(x, x - y)f(y) dy,

J'J1n

where (4.19) In other words, for fixed x, K(x, .) is the inverse Fourier transform of fJ(x, .). By Theorem 4.13, for each x there exists a constant a(x) and a function O(x,·) E c=(sn-l) with zero average on sn-l such that

K(x, z) = a(x)5(z)

+ p. v.

O(x, z') Izln .

4. Singular Integrals (I)

84

This example is the motivation for studying singular integrals with variable kernel: (4.20)

= lim

T f (x)

£---->0

1.

Iyl>£

D(x, y')

I In Y

f (x - y) dy.

We can again apply the method of rotations. Theorem 4.16. Let D(x, y) be a function which is homogeneous of degree o in y and such that:

(1) D(x, -y) = -D(x, y); (2) D*(u) = sup ID(x,u)1

Ll(sn-l).

E

x

Then the operator T given by (4.20) is bounded on LP(IRn ), 1 < p

< 00.

Proof. Because D is odd, we can argue as in the proof of Corollary 4.8 (where the kernel does not depend on x) to get

(4.21 )

Tf(x)

=

7r

r

2 }sn-l

D(x,u)Huf(x)da(u),

f E S.

Therefore,

ITf(x)1 <

7r

r

2 }sn-l

D*(u)IHuf(x)1 da(u)

and the desired result follows at once from Proposition 4.6.

o

Theorem 4.17. If in the statement of Theorem 4.16 we replace (2) by

(4.22) for some q, 1

s~p (fsn-' 1!1(x, u)lq da(U)) q = Bq < 00 < q <00,

then T is bounded onLP(IRn ), q'

< p < 00.

Proof. Apply Holder's inequality with exponents q and q' to get

(4.23)

IT f(x)1 <

7r

2

Bq (

r IHuf(x)lq'da(u)) }sn-l

l/q'

If we raise this to the q'-th power and integrate with respect to x we get that

If condition (4.22) holds for some value of q, then it holds for any smaller value, so we obtain the desired inequality for all p, q' < p < 00. 0

7. Notes and further results

85

When n is even there is a result analogous to Theorem 4.17 whose proof resembles that of Theorem 4.12. It can be found in the first article by A. P. Calderon and A. Zygmund cited in Section 7.l. Operators of the form (4.17) are a special class of pseudo-differential operators. For more information, see Chapter 5, Section 6.9.

7. Notes and further results 7.1. References.

The method of rotations was introduced in an article by A. P. Calderon and A. Zygmund (On singular integrals, Amer. J. Math. 78 (1956), 289309). The examples in Section 1 come from an earlier article by them (On the existence of certain singular integrals, Acta Math. 88 (1952), 85-139) which will be discussed in Chapter 5. The proof of Theorem 4.12 for even kernels is taken from course notes by A. P. Calderon (Singular integrals and their applications to hyperbolic differential equations, University of Buenos Aires, 1960). It can also be found in the books by Zygmund [22] and Neri [13]. For a general overview of singular integrals and their applications, see the survey article by A. P. Calderon (Singular integrals, Bull. Amer. Math. Soc. 72 (1966),427-465). 7.2. Spherical harmonics.

A solid harmonic is a homogeneous harmonic polynomial, and its restriction to sn-1 is called a spherical harmonic. The spherical harmonics of different degrees are orthogonal with respect to the inner product on L2(sn-1), and among those of a fixed degree there exists an orthogonal subcollection. Thus L2(sn-1) has an orthogonal basis composed of spherical harmonics. Note that if n = 2, the basis of spherical harmonics for L2(Sl) is just {e ik (} , k E Z}. The functions Yk(x)e-7TlxI2, where Yk is a solid harmonic of degree k, are eigenfunctions of the Fourier transform:

(This is referred to as the Bochner-Hecke formula.) The utility of spherical harmonics for studying singular integrals is suggested by the following formula: for k > 1,

4. Singular Integrals (I)

86

For a singular integral with 0 E L2(sn-l) and even, one can give a different proof of Theorem 4.12 using the decomposition of 0 into spherical harmonics. (See Stein and Weiss [18, Chapter 6].) The properties and uses of spherical harmonics can be found in the books by Stein [15], Stein and Weiss [18] and Neri [13]. Another discussion of spherical harmonics, as part of the general theory of harmonic functions, can be found in the book by S. Axler, P. Bourdon and W. Ramey (Harmonic Function Theory, Springer-Verlag, New York, 1992). 7.3. Operator algebras.

Besides the operator algebra studied in Section 5, Calderon and Zygmund considered others of the same type. (See Algebras of certain singular integral operators, Amer. J. Math. 78 (1956), 310-320.) In particular, let Aq be the set of operators

Tf(x) = af(x)

+ lim E->O

1

O(y') -I-In-f(x - y) dy,

lyl>E

Y

where a E C and 0 E Lq(sn-l). If we define

IITllq = lal + IIOIILQ(sn-l), then there exists a constant Bq which depends only on q such that

IITI T211q < BqllTlllqllT21lq. 0

The set Aq is a semi-simple commutative Banach algebra. For more information on algebras of singular integrals, see the article by A. P. Calderon (Algebras of singular integral operators, Singular Integrals, Proc. Sympos. Pure Math. X, pp. 18-55, Amer. Math. Soc., Providence, 1967). 7.4. More on the method of rotations.

In the proof of Theorem 4.17, we could deduce that from inequality (4.23) provided that

(4.24)

(

In kn-l (

) l/p

P/ql

IHuf(x)lq' dO"(u)

)

liT flip < Gllfllp

dx

< Gllfll p·

In the proof of Theorem 4.17 we used that inequality (4.24) is immediate if p > q'. However, there are also values p < q' for which (4.24) is true, and this allows us to improve the theorem. If f is the characteristic function of the ball, then one can show that (4.24) cannot hold unless

n p

n-l q'

-<--+1.

7. Notes and further results

87

It is conjectured that if 1 < p < 00 and p and q' satisfy this condition, then inequality (4.24) holds. It is also conjectured that (4.24) holds for this range of p and q' if Hu is replaced by the associated maximal operator, H~, or by the directional maximal function, Mu. The following results are known: when n = 2 the conjecture for Hu was proved by A. P. Calderon and A. Zygmund (On singular integrals with variable kernel, Appl. Anal. 7 (1978), 221-238); when n > 3 they proved it for 1 < p < 2. M. Cowling and G. Mauceri (Inequalities for some maximal functions, J, Trans. Amer. Math. Soc. 287 (1985), 431-455) showed that the same range holds for Mu. M. Christ, J. Duoandikoetxea and J. L. Rubio de Francia (Maximal operators related to the Radon transform and the Calder6n-Zygmund method of rotations, Duke Math. J. 53 (1986), 189-209) showed that the conjecture is true for all these operators and for p in the range 1

1) .

n+ < p < max ( 2, -2-

7.5. L log L results. As we noted at the end of Section 2, if !1 E L log L(sn-1) is even then the Fourier transform of p. v. !1(x')/lxl n is bounded. Further, Theorem 4.12 can be extended to the case when the even part of n is in L log L(sn-1). (See the first article by Calderon and Zygmund cited in Section 7.1.) These are the best possible results. In the same paper, Calderon and Zygmund noted that if ¢ is any function such that ¢( t) It log t ---+ 0 as t ---+ 00, then there exists an even function !1 such that ¢(!1) E L1(sn-1) but the Fourier transform of p. v. !1(x')/lxl n is unbounded. Later, M. Weiss and A. Zygmund (An example in the theory of singular integrals, Studia Math. 26 (1965), 101-111) showed that given any such ¢ there exists an even n, ¢(!1) E L1 (sn-1 ), and a continuous function f E LP (I~n) for every p > 1, such that for almost every x,

E~O

2

!11 (y, f(x - y) dy = J1Y1>E Y

limsup.f

00.

7.6. Weak (1,1) inequalities. Proposition 4.6 does not extend to weak (1,1) inequalities because the Lorentz space L 1 ,= is not a normed space. (See Chapter 2, Section 8.3.) Hence, we cannot use the method of rotations to prove weak (1,1) inequalities for singular integrals of the type (4.1) or even for the rough maximal function MO, defined in (4.5). However, the following results are known:

88

4. Singular Integrals (I)

If n E LlogL(sn-l) then Mn is weak (1,1). For n = 2 this was proved by M. Christ (Weak type (1,1) bounds for rough operators, Ann. of Math. 128 (1988), 19-42); for n > 2 this is due to M. Christ and J. L. Rubio de Francia (Weak type (1,1) bounds for rough operators, II, Invent. Math. 93 (1988),225-237). It is unknown whether Mn is weak (1,1) if n E Ll(sn-l). If n E LlogL(sn-l) then the operator T in (4.1) is weak (1,1). This was proved by A. Seeger (Singular integral operators with rough convolution kernels, J. Amer. Math. Soc. 9 (1996), 95-105). Partial results were obtained earlier by M. Christ and J. L. Rubio de Francia (in the above papers) and by S. Hoffman (Weak (1,1) boundedness of singular integrals with nonsmooth kernels, Proc. Amer. Math. Soc. 103 (1988), 260-264). By the counter-examples given above this is the best possible result for T for arbitrary n. However, it is unknown whether T is weak (1,1) when n E Ll(sn-l) and is odd. P. Sjogren and F. Soria (Rough maximal operators and rough singular integral operators applied to integrable radial functions, Rev. Mat. Iberoamericana 13 (1997), 1-18) have shown that for arbitrary n E Ll(sn-l), T is weak (1,1) when restricted to radial functions. 7.7. Fractional integrals and Sobolev spaces.

If ¢ E S then

in Section 5 we defined the square root of the operator -

~

by

More generally we can define any fractional power of the Laplacian by

Comparing this to the Fourier transform of lxi-a, 0 < a < n (see (4.3)), we are led to define the so-called fractional integral operator Ia (also referred to as the Riesz potential) by .

1

Ia¢(x) = -

"'fa

LI ll~n

¢(y)

X -

I - a dy,

Yn

where

Then with equality in the sense of distributions,

7. Notes and further results

89

If we consider the behavior of Ia on LP, then homogeneity considerations show that the norm inequality

(4.25) can hold only if 1 q

(4.26)

1 p

a

n

In fact, this condition is also sufficient for 1

< p < n/ a.

Theorem 4.18. Let 0 < a < n, 1

I{x E JRn : IIaf(xll > .\}I < C

(II~ll) n/(n-a)

For a proof, see Stein [15, Chapter 5]. Another proof uses the following inequality due to L. Hedberg (On certain convolution inequalities, Proc. Amer. Math. Soc. 36 (1972), 505-510):

Iaf(x) < Callfll~p/nMf(x)l-ap/n,

(4.27)

1
< n/a,

where M is the Hardy-Littlewood maximal function. To prove it decompose the integral defining Ia into two parts, one over {x : Ix - yl < A} and the other over {x : Ix - yl > A}; apply Proposition 2.7 to the first part and Holder's inequality to the second. Now choose A so that both terms are the same size. Theorem 4.18 then follows from (4.27) and from the boundedness ofM. Closely related to the fractional integral operator is the fractional maximal function,

Maf(x) = sup

I 1;_a /n JB{

r>O Br

If(x - y)1 dy,

0

< a < n.

T

Ma satisfies the same norm inequalities as Ia. The weak (1, n/(n - a)) inequality can be proved using covering lemma arguments (see Chapter 2, Section 8.6), and Ma is strong (n/a, (0): by Holder's inequality

IBrl;-a/n

inr If(x - yll dy < (inc If(x - ylln/a dy

t

n

< IlfUn/a

The strong (p, q) inequalities then follow by interpolation. For any positive f it is easy to see that Maf(x) < Claf(x). The reverse inequality does not hold in general, but the two quantities are comparable In norm.

90

4. Singular Integrals (I)

Theorem 4.19. For 1 Ca,p such that

00

and 0

< a < n, there exists a constant

This result is due to B. Muckenhoupt and R. Wheeden (Weighted norm inequalities for fractional integrals, Trans. Amer. Math. Soc. 192 (1974), 261-274). Also see D. R. Adams, A note on Riesz potentials (Duke Math. J. 42 (1975), 765-778) and D. R. Adams and L. Hedberg (Function Spaces and Potential Theory, Springer-Verlag, Berlin, 1996). An important application of Theorem 4.18 is in proving the Sobolev embedding theorem. For 1 < p < 00 and k > 1 an integer, define the Sobolev space L~(IRn) to be the space of all functions f such that f and all its derivatives (defined as distributions) up to order k are in LP. More precisely, let L~ be the closure of Cr;: under the norm

IIfIIL~ =

L

!IDa flip·

lal~k

Theorem 4.20. Fix 1 < P <

00

and k > 1 an integer. Then

(1) if p < nlk, then L~(JRn) C LT(JRn ) for all r, p 1/q = lip - kin; (2) if p = nlk, then L~(JRn) C LT(JRn ) for all r, p < r

< r < q, where <

00;

(3) if p > nlk and f E L~(JRn), then f differs from a continuous function on a set of measure o.

This result is due to S. L. Sobolev (On a theorem in functional analysis, Mat. Sb. 46 (1938), 471-497; English translation in Amer. Math. Soc. Transl. ser. 2, 34 (1963), 39-68). For all the results in this section, see Stein [15, Chapter 5], the book by Adams and Hedberg cited above, and the books by R. Adams (Sobolev Spaces, Academic Press, New York, 1975) and W. Ziemer (Weakly Differentiable Functions, Springer-Verlag, New York, 1989).

Chapter 5

Singular Integrals (II)

1. The Calder6n-Zygmund theorem In the previous chapter we used the Hilbert transform to study singular integrals. In this chapter we are going to consider singular integrals whose kernels have the same essential properties as the kernel of the Hilbert transform. This will let us generalize Theorem 3.2 to get the following result. Theorem 5.1 (Calder6n-Zygmund). Let K be a tempered distribution in ffi.n which coincides with a locally integrable function on ffi.n \ {O} and is such that

Ik(~)1 < A,

(5.1) (5.2)

1

Then for 1 < p

ixi>2iyi

<

IK(x - y) - K(x)1 dx < B,

y E ffi. n .

oo!

and

We will show that these inequalities are true for f E S, but they can be extended to arbitrary f E LP as we did for the Hilbert transform. Condition (5.2) is usually referred to as the Hormander condition; in practice it is often deduced from another, stronger, condition called the gradient condition.

-

91

5. Singular Integrals (II)

92

Proposition 5.2. The Hormander condition (5.2) holds if for every x =I- 0

(5.3)

This follows from the mean value theorem; details are left to the reader. Proof of Theorem 5.1. Since this proof is (essentially) a repetition of the proof of Theorem 3.2, we will omit the details.

Let f E S and let Tf = K *f. From (5.1) it follows that IITfl12 < A11J112. It will suffice to prove that T is weak (1,1) since the strong (p,p) inequality, 1 < p < 2, follows by interpolation, and for p > 2 it follows by duality since the adjoint operator T* has kernel K*(x) = K( -x) which also satisfies (5.1) and (5.2). To show that f is weak (1, 1), fix A > 0 and form the Calderon-Zygmund decomposition of f at height A. Then, as in Theorem 3.2, we can write f = g + b, where g E L2 and is bounded by 2n A and b is the sum of functions which are supported on disjoint cubes Qj and have zero average. The argument now proceeds as before, and the proof reduces to showing that

r

(5.4)

}~n\Q;

ITbj(x)1 dx < C

r Ibj(x)1 dx,

}Qi

where Qj is the cube with the same same center as Qj and whose sides are 2fo times longer. Denote their common center by Cj. Inequality (5.4) follows from the Hormander condition (5.2): since each bj has zero integral, if x tj Qj

Tbj(x)

=

r

}Qi

K(x - y)bj(y) dy

=

r [K(x -

}Qi

y) - K(x - cj)]bj(y) dy;

hence,

r

}~n\Q~J

ITbj(x)1 dx <

r Ibj(y)1 ( }~n\Q* r IK(x - y) - K(x - cj)1 dX) dy.

}QJ'

J

Since }R.n \

Qj c {x

E}R.n :

Ix - cjl

>

21Y - cjl},

the term in parentheses is bounded by B, the constant in (5.2).

D

We now consider singular integrals with a homogeneous kernel of degree -n, such as we studied in the previous chapter: K(x) = Q(x')/Ixl n . What must we assume about Q for the Hormander condition (5.2) to hold? Clearly,

1. The Calder6n-Zygmund theorem

93

c

n E 1 (sn-l) is sufficient by Proposition 5.2. But much weaker conditions are sufficient. For example, define

woo(t)

=

sup{ln(Ul) - n(u2)1 : lUI - u21 < t, Ul, U2 E sn-l}.

Then we have the following result. Proposition 5.3. If n satisfies

1 1

(5.5)

woo(t) d

o

t

t<

00,

then the kernel n(x')/lxl n satisfies (5.2). Condition (5.5) is referred to as a Dini-type condition. Proof.

IK(x - y) - K(x)1

= n((x - y)')

n(x') Ixln

Ix - yin

< Ifl((x

I I fl(x') I + Ifl(x') y)') x_yn

1

Ix - yin

1 --

Ixl n

Condition (5.5) implies that n is bounded, so on the set {Ixl > 21yl} the second term is bounded by Clyl/lxl n+1 . Thus its integral on this set is finite. On this set we also have that

I(x so

y)' -

x'i < 4 :~:,

1

1

In((x - y)') - n(x')1 d w oo (4lYl/lxl) d x< x Ixl>2lyl Ix - yin - Ixl>21YI (lxI/2)n

= 2nlsn-11

r2 woo(t) dt

Jo

t


For similar results, see Section 6.2. From Proposition 5.3 and Corollary 4.5 we immediately get the following corollary to Theorem 5.1. Corollary 5.4. If n is a function defined on sn-l with zero integral and satisfying (5.5), then the operator

Tf(x) = p. v. is strong (p,p), 1 < p

n(y') J'rR lYF f(x n

< 00, and weak

(1,1).

y) dy

5. Singular Integrals (II)

94

Since (5.5) implies that n is bounded, we already had the strong (p, p) inequality via the method of rotations. Nevertheless, we now also have the weak (1,1) inequality. 2. Truncated integrals and the principal value In Theorem 5.1 we assumed that the operator was bounded on L2 (via the hypothesis that k E LOO). In this section we give conditions on K which imply this property and which make the associated operator bounded on LP, 1 < p < 00. Given K E Lfoc(IRn \ {O}), define KE,R(X) = K(X)X{E<\X\
Proposition 5.5. Let K E Lfoc(lRn \ {O}) be such that

( K (x) dx < A, Ja<\x\
(5.6)

(

(5.7)

Ja<\x\<2a

0

< a < b<

IK (x) I dx < B,

a

( IK(x - y) - K(x)1 dx < C, J\x\>2\y\

(5.8)

00;

> 0; y E JR n .

Then for all ~ E JR n , 1K:fi(~) I < C, where C is independent of E and R.

Condition (5.7) is equivalent to

1

(5.9)

IxIIK(x)1 dx < Bla,

a> O.

\x\
To see this, note that

and

{ IK(x)1 dx < ( EJ1K(x)1 dx < 2B'. Ja<\x\<2a J\x\<2a a Proof. If we fix ~ then whenever

K:;(~)

=

1 1

E

<

I~I-l

< R,

K(x)e- 27rix .t; dx

E<\x\
=

«\x\<\t;\-l

=h+h.

K(x)e- 27rix .t; dx

+ { J\t;\-l <\x\
K(x)e- 27rix .t; dx

2. Truncated integrals and the principal value

(If 1~1-1 <

95

or if 1~1-1 > R, then it suffices to consider just one of the two integrals.) We treat h first:

h

E

=

1

K(x) dx

+

E
1

K(x)(e-27rix.~ -

1) dx,

E
so

Ihl <

1

K(x) dx

E
+ 27r1~ll

E
IxIIK(x)1 dx < C(A + B).

To evaluate h, let z = ~~1~1-2, so that exp(27riz·~) = -1; then if we make the change of variables x f--+ x - z in h we get .

r

h =-

JI~I-l
K(x -

z)e-27riX.~ dx.

Hence,

2h =

r

JI~I-l
K(x)e-27rix.~ dx -

r

JI~I-l
K(x -

z)e-27rix.~ dx;

this in turn implies that

21hl <

r

JI~I-l
IK(x) - K(x - z)1 dx

r IK(x)1 dx + r IK(x)1 dx. J R- ~ I~I-l
+

J~ 1~I-l
The first integral is bounded by C since 1~1-1 = 21zl. The other two integrals are bounded by 2B: since 1~1-1 < R, R+~I~I-1 < 3(R-~I~I-1). D Corollary 5.6. If K satisfies the hypotheses of Proposition 5.5, then

IIKE,R

* flip < Cpllfllp,

1


00,

and

I{x

E}Rn :

IKE,R

C1

* f(x)1 > '\}I < >:llfI11'

with constants independent of E and R. When p = 2 this result follows immediately from Proposition 5.5; for the rest note that (5.8) implies the Hormander condition,

1

IKE,R(X - y) - KE,R(X) I dx < C, Ixl>2lyl with C independent of E and R. The details are left as an exercise for the reader.

96

5. Singular Integrals (II)

When the kernel is homogeneous, K(x) = Q(x')/Ixl n, conditions (5.6) and (5.7) are equivalent to Q E L1(sn-1) and

hn-l

Q(u) da-(u)

=

O.

Since KE,R E L1, KE,R * f is well defined for f E LP. Ideally we could define the singular integral T by the pointwise limit

Tf(x) = lim KE,R * f(x), E-+O R-+= but this limit need not exist even if f E S. In this case we have convergence as R ---+ 00, so the problem reduces to determining when the principal value distribution of K, p. v. K(¢)

=

lim ( K(x)¢(x) dx, E-+O JI-xI>E

¢

E

S,

exists.

Proposition 5.7. Given a function K which satisfies condition (5.7)7 the tempered distribution p. v. K exists if and only if lim! K(x)dx E-+O E
exists. Proof. Suppose that the tempered distribution exists. If we fix ¢ E S which is identically 1 on B(O, 1), then p. v. K(¢)

=

lim! K(x) dx E-+O E
+ (

Jxl.>1

K(x)¢(x) dx.

1

The second integral exists since

(

Jxl>1

IK(x)¢(x)1 dx < Illxl¢ll= f2- k / , k=O

1

IK(x)1 dx < 2Blllxl¢II=·

2k
Therefore, the limit of the first integral must also exist. Conversely, suppose the limit exists; denote it by L. Then p. v. K(¢)

= ¢(O)L +

1

K(x)(¢(x) - ¢(O)) dx +

Ixl
(

Jxl<1 1

1

K(x)¢(x) dx.

Ixl>l The first integral exists since

IK(x)II¢(x) - ¢(O)I dx < IIV¢II= { IxIIK(x)1 dx < 2BIIV¢II=· Jxl<1 1

o

2. Truncated integrals and the principal value

97

Corollary 5.8. If we add to the hypotheses of Proposition 5.5 the assumption that

lim! K(x) dx E
E---->O

exists, then Tf(x)

= lim E---->O

1

lyl>E

K(y)f(x - y) dy

can be extended to an operator which is bounded on LP, 1 weak (1,1).

00,

and is

Example 5.9 (An application to K(x) = Ixl-n-it). The function Ixl- n - it is locally integrable on }Rn \ {O} and satisfies the hypotheses of Proposition 5.5. In fact,

1

dx a
=

{

Isn-II bit - .a- it < ~ Isn-Il -~t - t ' dx

=

Isn- I ll og 2

Ja
'

In + itl IDK()I v x < Ixl n+I ' Hence,

IIKE,R

* flip

< Cpllfllp with Cp independent of E and R.

However, the limit of the truncated integrals does not exist almost everywhere since

1

dx. E
=

Isn-I11 - E- it -it '

and the limit of E- it as E -+ 0 does not exist. The limit does exist if we choose an appropriate subsequence {Ek}, for example, Ek = e- 27rk / t . If we take this sequence then

.

hm KEbR

k,R---->=

* f(x) =

1.

f(x-y)-f(x) IY In+it dy lyl<1

+

1.

f(x-y) I In+it dy, Iyl>l Y

and this defines an operator which is strong (p,p), 1 < p < 00, and weak (1,1). This operator is the convolution of f with the tempered distribution which we will refer to as p. v.lxl- n - it , even though it depends on our choice of the sequence {Ek}: 1 p. v. Ixln+it

(¢)

=

1

Ixl
¢(x) - ¢(O) Ixln+it

dx

+

1

¢(x)

Ixl>1 Ixln+it dx.

5. Singular Integrals (II)

98

Despite its appearance this distribution is not homogeneous of degree -n - it (see Definition 4.2); that is, it does not satisfy 1 ( ) \-n-it 1 (A-.) p. v. Ixln+it ¢)... = /\ p. v. Ixln+it 'f' ,

where ¢)...(x) = A-n¢(A-IX). This can be shown by a straightforward computation, but here we are going to give an indirect argument which also yields its Fourier transform. Given z such that Re z < n, the function Ixl- z is locally integrable and homogeneous of degree -z. It defines a tempered distribution which is also homogeneous of degree -z:

We can rewrite this as _1 (¢) Ixl z

¢(x) - ¢(O) dx + ¢(O) {

= { J1xl<1

¢(x) -/(0) dx +

= { J1xl<1

~

+ {

J1xl<1 Ixl z

Ixl z

{

J1xl>1 Ixl z

¢(x} dx + Isn-Il ¢(O);

J1xl>1 Ixl

Ixl

¢(x) dx

n - z

this expression makes sense for Re z < n+ 1 except if z = n. This distribution is homogeneous of degree and if we let z = n + it, we see that it differs from the one we call p. v.lxl- n - it by a multiple of the Dirac delta. (This is homogeneous of degree -n, which implies that p. v.lxl-n-it cannot be homogeneous.) This new distribution is thus the unique distribution that is homogeneous of degree -n - it and which coincides away from the origin with the locally integrable function Ixl- n- it . Given the Fourier transform of Ixl- z (see (4.'3)),

-z,

(

1 ) ~

.r;F

(~) =

n

z 7r -

2

r (n1 r (;) I~I-z+n' Z )

if we let z = n + it we get the Fourier transform of the homogeneous distribution, which in turn gives us the Fourier transform of p. v.lxl- n - it :

1) (t) _ ( Ixln+it ~

p. v.

<:,

-

2

(-it)

~+it rr (ntit) I~ lit + ~Isn-Il 7r it .

3. Generalized Calderon-Zygmund operators Up to this point the operators we have been studying could be written as convolutions with tempered distributions. One practical advantage of this hypothesis is that we could use the Fourier transform to deduce the £2 boundedness of the operator. If we assume this then the boundedness on

3. Generalized Calder6n-Zygmund operators

99

the remaining LP spaces only depends on the Hormander condition, which can be adapted to operators which are not convolution operators. Let.6. be the diagonal offfi.n x ffi.n: .6. = {(x,x): x E ffi.n}. By a proof nearly identical to that of Theorem 5.1 we can show the following. Theorem 5.10. Let T be a bounded operator on L2(ffi.n)) and let K be a function on ffi.n X ffi.n \ ~ such that if f E L2(ffi.n) has compact support then

=

Tf(x)

r

K(x, y)f(y) dy, Jrr:r.n Further) suppose that K also satisfies

x

~ supp(f).

r IK(x, y) - K(x, z)1 dx < C, r IK(x, y) - K(w, y)1 dy < J1x-yl>2Ix-wl .

(5.10)

.

J1x-yl>2Iy-zl

(5.11)

C.

Then T is weak (1,1) and strong (p,p)) 1 < p <

00.

The given representation holds only for functions in L2 of compact support, but this suffices for the proof since it is only applied to the functions bj gotten from the Calder6n-Zygmund decomposition. condition (5.10) is used to prove the weak (1,1) inequality for T, and condition (5.11) is used to prove the same inequality for its adjoint. Following the terminology of Coifman and Meyer [2], we say that K : ffi.n X ffi.n \ ~ ~ C is a standard kernel if there exists <5 > 0 such that C

(5.12)

IK(x,y)1 < 1x_yn I'

y) -

(5.13)

IK(x,

(5.14)

' y) IK (x,

K(x, z)1 < K( w, y )1

c I;y--yil:,

b < C xIx-wl _ y n+b 1

1

if

Ix - yl > 21Y - zI,

l'f

1X - Y1> 21Ix -

w 1.

Standard kernels dearly satisfy the Hormander conditions (5.10) and (5.11). The following are examples of operators which have this type of kernel.

(1) The Cauchy integral along a Lipschitz curve. Let A be a Lipschitz function on ffi. (i.e. A' = a E L oo ) and let r = (t, A(t)) be a plane curve. With this parameterization we can regard any function f defined on function of t and conversely. Given f E S(ffi.), the Cauchy integral

roo

f(t)(l + ia(t)) dt 27ri J-00 t + iA(t) - z defines an analytic function in the open set

Cr f(z) = _1

D+ = {z = x

+ iy E C : y >

A(x)}.

r

as a

5. Singular Integrals (II)

100

Its boundary values on

r, lim Cr f(x

E-+O

are given by

~

[f(X)

2

+ i(A(x) + E)),

1

+ ~ lim

f(t)~l + ia(t)). dtj. Ix-tl>EX-t+'l,(A(x)-A(t))

7rE-+O

This leads us to consider the operator

Tf(x) = lim E-+O

1

Ix-YI>E

f(y) dy, X - Y + i(A(x) - A(y))

whose kernel, 1

(5.15)

K(x, y) = x - y + i(A(x) - A(y)) ,

satisfies (5.13) and (5.14) with 8

= 1.

(2) Calderon commutators. If as a geometric series:

K(x,y)

= _1_

IIA'lloo

then we expand the kernel (5.15)

~ (iA(x) -

A(y))k x-y

x-yD k=O

It is, therefore, natural to consider the following operators: given a Lipschitz function A on IR. and an integer k > 0, define

Tkf(x) = lim .

E-+O

1

(A(X)-A(y))k f(y) dy.

Ix-YI>E

X -

Y

x - y

The associated kernels,

Kk(x,y) = (A(X) -A(Y))~_l_, x-y x-y are also standard kernels with 8 = 1. Definition 5.11. An operator T is a (generalized) Calder6n-Zygmund operator if

(1) T is bounded on L2(IR.n);

(2) there exists a standard kernel K such that for support,

T f(x) =

r K(x, y)f(y) dy,

JlRn

f

E

L2 with compact

x tf- supp(f).

Theorem 5.10 implies that a Calder6n-Zygmund operator is bounded on LP,l < p < 00, and is weak (1, 1). Hence, given an operator with a standard kernel, the problem reduces to showing that it is bounded on L2. We will return to this question in Chapter 9.

4. Calder6n-Zygmund singular integrals

101

4. Calder6n-Zygmund singular integrals The examples in the previous section suggest that for any Calder6n-Zygmund operator, T f(x)

=

lim ( K(x, y)f(y) dy, J1x-yl>E

E---O

at least for f E S. However, this is not necessarily the case. From (5.12) we can deduce that TEf(x) =

K(x, y)f(y) dy

(

J1x-yl>E makes sense for f E S(Il~n). Nevertheless, the limit as E ---+ 0 need not exist or may exist and be different from T f (x). An example of the first is the operator we defined in Section 2 as convolution with p. v.lxl- n - it , whose kernel, K(x, y) = Ix - yl-n-it, is standard. (Recall that this operator is strong (2,2) and so a Calder6n-Zygmund operator.) By an argument identical to that in Proposition 5.7 we can prove the following result. Proposition 5.12. The limit

(5.16) exists almost everywhere for f E

C~

if and only if

lim ( K(x, y) dy JE
E---O

exists almost everywhere.

The existence of the limit (5.16) does not imply that it is equal to Tf(x): for example, consider the identity operator I. This is a Calder6n-Zygmund operator associated with the kernel K(x,y) = O-clearly, If(x) = 0 if x rfsupp(f)-but (5.16) equals O. Furthermore, this example shows us that an operator is not characterized by its kernel since the zero operator also has the zero kernel. In general, so does any pointwise multiplication operator Tf(x) = a(x)f(x), a E Loo. However, these are the only ones; this is shown by the following result whose proof is left to the reader. Proposition 5.13. If two Calder6n-Zygmund operators are associated with the same kernel) then their difference is a pointwise multiplication operator.

It is important to assume that a Calder6n-Zygmund operator is bounded on L2 since without this or a similar hypothesis, Proposition 5.13 would be

5. Singular Integrals (II)

102

false. For example, the derivative is an operator with kernel 0 (f' (x) x FJ supp(f)) but it is not a pointwise multiplication operator.

= 0 if

A Calder6n-Zygmund singular integral is a Calder6n-Zygmund operator that satisfies

Tf(x)

(5.17)

=

lim TEf(x).

E~O

To determine when this equality holds for f E LP(n~n), we will proceed as we did for the Hilbert transform and examine the maximal operator

T* f(x)

=

sup ITEf(x)l. E>O

By Theorem 2.2, if T* is weak (p,p) then the set

{f E LP : lim TEf(x) exists a.e.} E~O

is closed in LP. Hence, if (5.17) holds for a dense subset, say it holds for any f E LP.

f

E C~, then

Theorem 5.14. If T is a Calder,'6n-Zygmund operator then T* is strong (p,p), 1 < p < 00, and weak (1,1). The proof of Theorem 5.14 depends on the following result which is analogous to Cotlar's inequality (Lemma 3.5).

Lemma 5.15. If T is a Calder6n-Zygmund operator then for any lJ < 1, and for any f E C~,

(5.18)

T* f(x) < C v (M(IT fIV)(x)l/V

+ M f(x))

lJ,

0

<

.

To prove this we need the following result due to Kolmogorov.

Lemma 5.16. Given an operator S which is weak (1,1), lJ, 0 < lJ < 1, and a set E of finite measure, there exists a constant C depending only on lJ such that

Proof. By (2.1) and the weak (1,1) inequality,

LISf(x)IV dx = v 1000 AV-ll{x .< v =

v

E

E : ISf(x)1 >

10= )."--1 min (lEI, ~ Ilf111)

l

C11f"d'E'

o

A}I dA d)'

x/-lIEI dA + v 100 CIIfildlEI

CA v -

2

11f1h dA.

4. Calder6n-Zygmund singular integrals

103

The desired inequality follows from this immediately.

D

Proof of Lemma 5.15. We will prove that

with C independent of E. Our argument will be translation invariant and so actually yields (5.18). Fix E > 0; let Q = B(O, E/2) and 2Q and 12 = f - h· Then

T 12(0) =

!

= B(O, E), and define h =

fX2Q

K(O, y)f(y) dy = TEf(O).

IYI>E If z E Q then, since K is a standard kernel,

JTh(z) - Th(O)1

=!

lyl>E

< Cl z l8

-

< -

<

ci

(K(z,y) - K(O,y))f(y) dy

!

f

If(y)1 dy

IYI>E lyln+8

r

k=O J2 k E
lJ(y) 1 dy lyln+8

efT" (2k1 )n!lyl<2 + k=O

E

k

If(y)1 dy 1E

< C8 M f(0). It follows from this inequality that

ITEf(O)1 < CMf(O)

(5.19) If TEf(O)

0< >.

+ ITf(z)1 + ITh(z)l·

then there is nothing to prove. If not, fix < ITEf(O)1 and let = 0

Q1 = {z E Q : ITf(z)1 > >./3}, Q2 = {z E Q : ITh(z)1 > >./3}, and if CM f(O) <

>'/3, if CM f(O) > >./3.

Then Q = Q1 U Q2 U Q3, so IQI < IQ11

IQ11 < }

+ IQ21 + IQ31.

However,

fa ITf(z)1 dz < }IQIM(Tf)(O),

>. such that

5. Singular Integrals (II)

104

and by the weak (1,1) inequality for T,

IQ21 = If Q3

I{z

E

Q : ITh(z)1 >

3C ( 3 C TIQIMf(O).

A/3}1 < T i Q If(z)1 dz <

= Q then A < 3CM f(O); if Q3 = 0 then

IQI ~ IQll + IQ21 < 3~IQI(M(TJ)(0) + Mf(O)). Hence, in every case we have that

A < C (M(TJ) (0)

+ M f(O)).

This is true for any A < ITEf(O)1 and so (5.18) holds when v

=

l.

If 0 < v < 1, then it follows from (5.19) that

ITEf(O)I V < CMf(ot

+ ITf(z)IV + ITh(z)l v . If we integrate in z over Q, divide by IQI and raise to the power l/v, we get IT,1{O)I" < C ( Mf{O)

+ M{ITfl"){O)1/" +

(I~I fa ITh{z)l" dZ) 1/") .

By Lemma 5.16, (

1

(

lQJ i Q ITh(z)I Vdz

) l/v

< CIQI-lllhih < CMf(O),

and this completes the proof.

D

Proof of Theorem 5.14. Inequality (5.18) with v = 1 immediately implies that T* is strong (p, p) since both T and Mare.

To show that T* is weak (1, 1) we could argue as we did in the proof of Theorem 3.4; instead we will give a different proof which uses (5.18) with v < l. From this inequality we have that

I{x

E}Rn :

T* f(x) >

A}I


E}Rn :

Mf(x) > A/2C}1

+ I{x E}Rn : M(ITfIV)(x)l/V > A/2C}I, and the first term on the right-hand side satisfies the desired estimate. As for the second term, by Lemma 2.12

where Md is the dyadic maximal operator (2.9). Let E = {x E Md(ITfIV)(x) > AV}; then E has finite measure if f E C~. But then

lEI <

:v LITf(Y)I Vdy;

}Rn :

5. A vector-valued extension

105

by the proof of Theorem 2.10 it suffices to take the integral over E instead of all of ]Rn. Therefore, by Lemma 5.16 lEI < C,\vIEI 1 - v llfllr· The desired inequality now follows if we replace

,\1/

o

by 4 -n ,\1/.

5. A vector-valued extension Let B be a separable Banach space. A function F from]Rn to B is (strongly) measurable if for each b' E B* (the dual of B) the map x r-+ (F(x), b') is measurable. If F is measurable then it follows that the scalar function x r-+ IIF(x)IIB is also measurable. Therefore, we can define LP(B) as the space of (equivalence classes of) measurable functions from ]Rn to B such that

IIFIILP(B) =

(IF!' IIF(x)ll~ dX) IIp <

and Ilflloo = sup{IIF(x)IIB : x E Banach space.

00,

1

00,

The space LP(B), 1 < p <

]Rn}.

00,

is a

If f is a scalar function in LP and b E B, define the function f . b from ]Rn to B by (f· b)(x) = f(x)b. This function is in V(B) and its norm is IlfllpllbllB. The subspace of LP(B) consisting of finite linear combinations of functions of this type, denoted by LP ® B, is dense if 1 < P < 00.

Given F B given by

=

L:j fJ . bj

L

E

L1 ® B, define its integral to be the element of

F(x) tk

=

L

(in !j(X) dX) bj

J

The map F r-+ f F(x) dx extends to L1(B) by continuity. For a function F E L1(B), this integral is the unique element of B that satisfies

(in F(x) dx, b'J L(F(x), b') dx =

for all b' E B*. If F E LP(B) and GELpI (B*), then (F, G)(x) = (F(x), G(x)) is integrable; furthermore,

IIGlid (B» = sup {

.L

From this we see that Lpl (B*) but is, for example, if 1 < p <

(F(x), G(x)) dx : IIFIILP(B) < 1 } .

c (V(B))*. Equality is not true in general, 00

and B is reflexive.

Let A and B be Banach spaces and let £(A, B) be the space of bounded linear operators from A to B. Suppose that K is a function defined on

5. Singular Integrals (II)

106

~n X ~n \

Ll which takes values in £(A, B) and T is an operator which has K as its associated kernel: if f E DX)(A) and has compact support, then

T f(x)

=

r K(x, y) . f(y) dy,

x tJ. supp(f)·

J~n

For such operators there is a vectorial analogue of Theorem 5.10. Theorem 5.17. Let T be a bounded operator from LT(A) to LT(B) for some r, 1 < r < 00, with associated kernel K. If K satisfies

(5.20) (5.21)

r

IIK(x, y) - K(x, z) II£(A,B) dx :::; C,

1

IIK(x, y) - K(w, y)II£(A,B) dy < C,

J1x-yl>2Iy-zl Ix-yl>2Ix-wl

then Tis bounded from £p(A) to LP(B), 1 < p <

00,

and is weak (1, I), that

'is,

I{x

E

~n:

IITf(x)IIB > A}I <

~llfll£l(A).

The proof of Theorem 5.17 initially follows the outline of the scalar result: inequality (5.20) together with the boundedness from LT(A) to LT(B) yields the weak (1,1) inequality; then the Marcinkiewicz interpolation theorem (which can easily be shown to be true in these spaces) shows that T is bounded for 1 < p < r. To prove that T is bounded for r < p < 00 we must pass to the adjoint operator. However, this presents a problem since, as we noted above, Lpf (A*) need not be equal to LP(A)*. When A is reflexive (and it will always be so in our applications in Chapter 8) they are equ<;1l, and in that case it is enough to note that the kernel associated with the adjoint operator T* is K(x, y) = K*(y, x) E £(B*, A*). Inequality (5.21) for K is equivalent to (5.20) for K, so by repeating the above argument we get that T* is bounded for 1 < p < r'o When Lpf (A*) 0:/= LP(A)* we must first consider the finite dimensional subspaces of A. Given such a subspace A o, let To : LT(Ao) ~ LT(B) be the restriction of T to functions with values in Ao. The kernel associated with To isKo E £(Ao, B), the restriction of K to Ao· Since IIKol1 < IIKII, inequalities (5.20) and (5.21) hold for Ko with constants independent of the subspace Ao. Therefore, arguing as before,

T; : Lq(B*)

~

Lq(Ao),

1 < q < r',

with a constant independent of Ao. So by duality, To is bounded from LP(A o) to LP(B), r < p < 00, with a constant independent of Ao. That it is bounded on all of LP (A) now follows since LP 0 A is dense in LP (A) .

6. Notes and further results

107

We leave it to the reader to fill in the details of the outlined proof of Theorem 5.17 by following the scalar model.

6. Notes and further results 6.1. References.

The Calderon-Zygmund decomposition and its application to proving the weak (1,1) inequalities for singular integrals appeared in the classic article by both authors (On the existence of certain singular integrals, Acta Math. 88 (1952), 85-139). In this article they used the gradient condition (5.3); L. Hormander first observed that (5.2) is sufficient (Estimates for translation invariant operators in £P -spaces, Acta Math. 104 (1960), 93-139). Also see Stein [15]. Proposition 5.5 is due to A. Benedek, A. P. Calderon and R. Panzone (Convolution operators with Banach-space valued functions, Proc. Nat. Acad. Sci. U.S.A. 48 (1962), 356-365). The tempered distribution p. v.lxl- n- it and its generalizations were first considered by B. Muckenhoupt (On certain singular integrals, Pacific J. Math. 10 (1960), 239-261). For more on generalized Calderon-Zygmund operators, consult Coifman and Meyer [2], Journe [8] and Stein [17]. The Cauchy integral has a long history: see, for example, the book by N. T. lMushkelishvili (Singular Integral Equations, P. Noordhoff, Groningen, 1953). The associated commutators were first studied by A. P. Calderon (Commutators of singular integral operators, Proc. Nat. Acad. Sci. U.S.A. 53 (1965), 1092-1099). Commutators are discussed again in Chapter 9. For more on vector-valued singular integrals see Garcia-Cuerva and Rubio de Francia [6, Chapter 5], the article by J. L. Rubio de Francia, F. J. Ruiz and J. L. Torrea (Calder6n-Zygmund theory for operator-valued kernels, Adv. in Math. 62 (1986), 7-48) or the monograph by J. L. Torrea (Integrales Singulares Vectoriales, Notas de Algebra y Analisis 12, Univ. Nacional del Sur, Bahia Blanca, 1984). The survey article by C. Fefferman (Recent progress in classical Fourier analysis, Proceedings of the I.C.M. (Vancouver, B.C., 1974), Vol. 1, pp. 95118, Canad. Math. Congress, Montreal, 1975) contains a succinct discussion of singular integrals and their connection to many other problems in analysis.

6.2. A more general Dini-type condition. The Hormander condition (5.2) holds for kernels of the form K(x) n(x')/Ixln under weaker hypotheses than those in Proposition 5.3. Given P E O(n), let Ilpll

= sup{lu - pul : u

E

sn-l}.

5. Singular Integrals (II)

108

If we define W1(t)

=

r

sup

Ilpll~t } sn-l

1r2(pu) - r2(u)1 dO"(u),

then the Dini-type condition

1

1 W1(t) d

o

- - t
implies that (5.2) holds for K. This condition also implies that r2 E L log+ L. This result is due to A. P. Calderon, M. Weiss and A. Zygmund (On the existence of singular integrals, Singular Integrals, Proc. Sympos. Pure Math. X, pp. 56-73, Amer. Math. Soc., Providence, 1967). Later, Calderon and Zygmund proved that this condition is also necessary for the Hormander condition to hold (A note on singular integrals, Studia Math. 65 (1979), 77-87). 6.3. N onisotropic dilations. The Euclidean norm

I . I on ]Rn is associated with a family of dilations 8t (x)

in the sense that

=

(tX1,'"

18t (x)1 = tlxl, t> 0,

,txn )

x E ]Rn.

Given a1,' .. ,an > 0, we can define a family of dilations that act on each variable differently: 8t (x) = (tal Xl, . .. ,tanxn ).

II . II such that 118t (x) II = tllxll Ilzll = 1. Define II . II as follows: given

Then for each 8t there exists a quasi-norm

and if z E sn-1 then x E ]Rn, there exists a unique x' E sn-1 such that 8t (x ' ) = x for some t > O. Let Ilxll = t. This is not an actual norm since the triangle inequality is only satisfied up to a constant L > 1:

Ilx + yll < L(llxll + Ilyll)· Given]Rn with this quasi-norm we can get results analogous to Theorem 5.10 by considering conditions like

r

hx-YII~21Iy-zll

IK(x, y) - K(x, z)1 dx


and the symmetric condition gotten by changing the order of the variables. This kind of singular integral appears primarily in connection with the parabolic operators introduced by E. Fabes and N. Riviere (Singular integrals with mixed homogeneity, Studia Math. 27 (1966), 19-38). Also see C. Sadosky ( On some properties of a class of singular integrals, Studia Math.

6. Notes and further results

109

27 (1966), 73-86) and M. de Guzman (Singular integral operators with generalized homogeneity, Rev. Acad. Ciencias Madrid 64 (1970), 77-137). In Chapter 8, Section 7, we will use the method of rotations to study the kernels associated with this structure.

6.4. Spaces of homogeneous type. Singular integrals can be studied in very general spaces. R. Coifman and G. Weiss considered this question in their book Analyse harmonique nonconmutative sur certains espaces homogenes (Lecture Notes in Math. 242, Springer-Verlag, Berlin, 1971). For a discussion in a slightly less general setting, see Stein [17, Chapter 1]. Let X be a space with pseudo-metric p, that is, a function p : X x X [0,00) such that

= 0 if and (2) p(x, y) = p(y, x); (1) p(x, y)

(3) there exists L

only if x

~

= y;

> 0 such that p(x, z) < L(p(x, y) + p(y, z)).

A space of homogeneous type is a topological space X with a pseudo-metric p such that the balls B(x, r) form a basis of open neighborhoods of X, and there exists a positive integer N such that for any x E X and r > 0, the ball B(x, r) contains at most N points Xi such that p(Xi' Xj) > r /2. The second property holds if there exists a Borel measure fJ- on X such that

0< fJ-(B(x,r)) < AfJ-(B(x,r/2)) < 00. (Such a measure fJ- is called a doubling measure. We first discussed doubling measures in Chapter 2, Section 8.6.) If X is a space of homogeneous type, then given a kernel K(x, y) E L2(X X X, dfJ- 0 dfJ-) we can define the operator

Tf(x) If

=

Ix

IITfllq < Cqllfllq for some q > 1,

(5.22)

(

K(x, y)f(y) dfJ-(y). and if for any y, z E X,

IK(x, y) - K(x, z)1 dfJ-(x) < C,

} p(x,y)?2Lp(y,z)

then T is strong (p,p), 1 < p

< q, and weak (1,1).

Very recently, in connection with the study of the Cauchy integral, the question has arisen of extending the theory of singular integrals to nonhomogeneous spaces: topological spaces X with a pseudo-metric p and a measure fJ- which is not doubling. F. Nazarov, S. Treil and A. Volberg (W~ak type estimates and Cotlar inequalities for Calderon-Zygmund operators on nonhomogeneous spaces, Int. Math. Res. Let. 9 (1998), 463-487) have shown

5. Singular Integrals (II)

110

that the above result remains true if: p is a metric; there exists a constant v > 0 such that for all x E X and r > 0, p,(B(x, r)) 0 such that

p(y, z)O IK(x,y) - K(x,z)1 < C p(x,y)v+o'

if p(x,y) ~ 2p(y,z).

6.5. The size of constants.

In the proof of Theorem 5.1, if we form the Calder6n-Zygmund decomposition at height A-lA, then the weak (1,1) constant becomes Cn(A+B), and by interpolation the strong (p, p) constant is dominated by C~(A

+ B)p2

1

p-l

00.

In general, this is the best possible constant asymptotically as p ~ 1 or p ~ 00. To see this, suppose the kernel K satisfies the gradient condition with constant C (5.3) and also satisfies c

IK(x)1 > Ixl n '

Ixl > O.

Let B = B(O, 1) and let B* = B(O, R), where R is such that C / R Fix f = IBI-ljpXB; then Ilfllp = 1. Further, for x tJ. B*,

ITf(x)1 > IBll-ljp -IBI-ljp

>

clBll-ljp

-

-

Ix In

> -

clBll-ljp 2 Ixl n .

< c/2.

L

IK(x - y) - K(x)1 dy

CIBll+ljn-l jp --,--,------=---

Ixl n+l

It follows from this that for p small, IITfllp = O((p_1)-1). If we apply the same argument to the adjoint operator, then by duality we get that for p large, IITfllp = O(p).

This proof can easily be adapted to the case when K is the kernel of one of the Riesz transforms, and a similar argument holds for Calder6n-Zygmund operators. (See Stein [17, p. 42].) 6.6. More on truncated integrals.

Given a Calder6n-Zygmund singular integral T, that is, a CalderonZygmund operator that satisfies (5.17), it is natural to ask when TEl converges to T f in LP. If 1 < p < 00, this is immediate: if f E LP then by Theorem 5.14, T* f E LP, so convergence follows from the dominated convergence theorem. This argument fails when p = 1, but nevertheless it can be shown that if f, T fELl, then TEf ~ T f in Ll. This result is due to

6. Notes and further results

111

A. Calderon and O. Capri (On the convergence in £1 of singular integrals, Studia Math. 78 (1984),321-327). 6.7. Vector-valued singular integrals and maximal functions. The vector-valued extension of the theory of singular integral operators in Section 5 has wide application, since many of the operators in harmonic analysis can be viewed as vector-valued singular integrals. This approach is due to Rubio de Francia, Ruiz and Torrea (see the reference given in Section 6.1). Here we will show that the Hardy-Littlewood maximal function can be treated as a vector-valued singular integral. We first extend Theorem 5.17 to include the case when T is a bounded operator from LOO(A) to LOO(B). The proof is essentially the same. Fix f, form the Calderon-Zygmund decomposition of IlfilA at height >., and decompose f as g+b, where IIg(x)IIA < 2n>. a.e. Then, since T is bounded from LOO(A) to LOO(B), for some constant /3 > 0,

{x ElRn

:

IITf(x)liB > /3>'} C {x E lRn

:

IITb(x)liB > >.}.

The proof then continues as before. Now fix a non-negative function ¢ E S such that ¢ Define the maximal operator

MqJ(x)

=

sup I¢t t>O

> 1 on

B(O,l).

* f(x)l,

where ¢t(x) = t-n¢(x/t). Since ¢ E S, it suffices to take the supremum over rational t. By our choice of ¢, M f(x) < M¢>(lfl)(x). (The reverse inequality was proved in Proposition 2.7.) Thus, to prove norm inequalities for M it will suffice to prove them for M¢>. Let A = lR and B = ,eOO(Q+), and define the vector-valued function K(x) = {¢t(X)}tEQ+' Then K takes values in £(A,B) and we can define an operator T to be convolution with K. Since

. IITf(x)liB = M¢>f(x), it follows that T : LOO(A)

----+

LOO(B) is bounded. Further, since ¢

ely I

sup I¢t(x - y) - ¢t(x)1 < I In+1' t>O

x

E

S,

Ixl > 21YI > 0,

which implies the Hormander condition (5.20) for K. Hence, by Theorem 5.17 (as extended above), T : £P(A) ----+ LP(B); equivalently, M¢> is bounded on LP. The boundedness of M¢> was originally proved by F. Zo (A note on approximation of the identity, Studia Math. 55 (1976), 111-122).

5. Singular Integrals (II)

112

6.8. Strongly singular integrals.

In Theorem 5.1, given an arbitrary tempered distribution with bounded Fourier transform, the Hormander condition (5.2) seems to be the weakest hypothesis possible to get LP boundedness. It is of interest, however, to see how this hypothesis can be weakened if stronger conditions are assumed on the Fourier transform. The prototypical operator motivating this is the convolution operator Ta ,b = Ka ,b * f, 0 < a < 1 and b > 0, where ~

eil~la

Ka,b(e) = ¢(e)W' and ¢ is a Coo function which vanishes near the origin and is identically equal to one outside of some compact set. This integral arises in the study of the LP convergence of multiple Fourier series (cf. Chapter 1, Section 10.2). Clearly,

A IKa,b(e) I ::; (1 + lel)b' A

and it can be shown that for x close to the origin, B

IKa,b(X)1 ~ Ixln+8'

where 8 = (naj2-b)j(1-a). Hence, this kernel does not satisfy the gradient condition (5.3). (It also fails to satisfy the Hormander condition; this fact follows from the next result.) Howeve.r, these two growth conditions are sufficient to determine the behavior of Ta ,b on LP. Theorem 5.18. The operator Ta ,b is bounded on LP if

(5.23)

1

1

2-]; <

(bjn)(nj2 + 8) b+8'

8'

naj2 - b I-a'

and is unbounded if the reverse inequality holds. If equality holds in (5.23), then Ta,b maps LP into the Lorentz space LP'P' .

The LP boundedness of Ta,b was first shown by I. Hirschmann (On multiplier transformations, Duke Math. J. 25 (1959), 221-242) when n = 1 and by S. Wainger (Special trigonometric series in k dimensions, Mem. Amer. Math. Soc. 59 (1965)) in higher dimensions. Also see the survey article by E. M. Stein (Singular integrals, harmonic functions and differentiability properties of functions of several variables, Singular Integrals, Proc. Sympos. Pure Math. X, pp. 316-335, Amer. Math. Soc., Providence, 1967). The endpoint inequalities were proved by C. Fefferman (Inequalities for strongly singular convolution operators, Acta Math. 124 (1970), 9-36). He showed that they are consequences of the following general result, which

6. Notes and further results

113

further illustrates the delicate relationship between the conditions on K and/(. Theorem 5.19. Let K be a tempered distribution on ~n with compact support which coincides with a locally integrable function away from the origin. Suppose there exists 0 < < 1, such that

e,

e

A IK(e)1 < (1 + lel)-nB/2' ~

r

J xl>2Iyll-a

eE ~n,

IK(x - y) - K(x)1 dx < B,

Iyl < 1.

1

Then the operator T f = K a weak (1,1) inequality.

*f

is bounded on LP, 1 < p <

00,

and satisfies

This result was later generalized by P. Sjolin (LP estimates for strongly singular convolution operators in ~n, Ark. Mat. 14 (1976), 59-64). 6.9. Pseudo-differential operators.

Pseudo-differential operators are generalizations of differential operators and singular integrals. They are formally defined as in equation (4.17), that IS,

where CJ, the symbol of T, is a complex-valued function defined on ~n X ~n. Conditions must be imposed on CJ to ensure that T f is well defined for

f

E cgo(~n).

If CJ(x, e) = m(e) is independent of x, then T is the multiplier associated with m; if CJ(X, = b(x) is independent of T corresponds to pointwise multiplication by b. Although in these cases T is bounded on L2 if and only if m or b are bounded functions, this is not true in general. For example, CJ(x, e) = b(x )m(e) with m, b E L2 can be unbounded, but by applying Holder's inequality it is immediate that T is always bounded on L2.

e)

e,

Symbols are classified according to their size and the size of their derivatives. The simplest classification is motivated by the study of elliptic differential operators: given m E Z, we say that CJ E if

sm

ID~DrCJ(x, e)1 < Ca ,jJ(1 + lel)m-1a l . Pseudo-differential operators can be rewritten as

(5.24)

Tf(x)

=

r K(x, x - y)f(y) dy,

J~n

where K is given by (4.19). When CJ E SO then one can show that K satisfies a Hormander condition (cf. (5.10) and (5.11)) and that T is bounded on L2.

5. Singular Integrals (II)

114

Hence the theory developed in this chapter applies and T is bounded on LP, 1 < p < 00. The composition of pseudo-differential operators yields a corresponding symbolic calculus. In the simplest cases (pointwise multiplication and multipliers) the symbol of the composition of two operators is the product of the symbols. (Cf. Theorem 4.15.) In general, given pseudo-differential operators T1 and T2 with symbols in sml and 8 m2 , their composition is a pseudo-differential operator with symbol in sml +m2 . Further, the symbol of T1 0 T2 has an asymptotic expansion whose dominant term is the product of the symbols of T1 and T2 and the remaining terms are symbols of pseudo-differential operators in 8 m1 +m 2 - j , j > O. If (J E 8 m is an elliptic operator (that is, it has a lower bound 1(J(x,~) I > C(l + 1~I)m), then there exists a symbol in such that the composition of the corresponding operators is the identity (which is in 8 0 ) plus an error term with symbol in 8- 1 . Thus, the theory of pseudo-differential operators with symbols in is readily applicable to the problem of the inversion of elliptic operators.

s-m

sm

When considering the corresponding problem for other differential operators, a more general class of symbols arises. We say that (J E s~o if ID~D~(J(x, ~)I

< A a,/3(l + 1~I)m-plal+oIJ31.

This class includes the previous one: with this notation 8 m becomes Slo. , If T is a pseudo-differential operator with symbol in S~o' then it can again be written in the form (5.24). One can show that the kernel K satisfies the standard estimates (5.12), (5.13) and (5.14) if and only if m = 0 and p = 6 < 1. A remarkable result due to A. P. Calderon and R. Vaillancourt (A class of bounded pseudo-differential operators, Proc. Nat. Acad. Sci. U.S.A. 69 (1972), 1185-1187) gives us that a pseudo-differential operator with symbol in 82,p for some 0 < P < 1 is bounded on L2, and so we can again apply the techniques of this chapter to show that it is bounded on LP. The proof of Calderon and Vaillancourt uses Cotlar's lemma (see Chapter 9, Section 1). Their result is false when p = 1: there exist symbols in 8~,1 (and so in L OO ) whose associated pseudo-differential operators are unbounded on L2.

When m =f- 0, the properties of pseudo-differential operators whose symbols are in S~o are studied using a scale of spaces which measure the regularity of functions; for example, the Sobolev spaces (see Chapter 4, Section 7.7) or the Lipschitz spaces (see Stein [15, Chapter 5]). For further results on pseudo-differential operators and additional references see Stein [17], Journe [8] and CoHman and Meyer [2].

Chapter 6

HI and BMO

1. The space atomic HI In Chapter 3, Section 3, we saw that the Hilbert transform of an integrable function is not, in general, in Ll, and that a necessary condition for this is that the function have zero integral. Here we are going to define a subspace of Ll whose image under any singular integral is in Ll. We begin by defining its basic elements-atoms.

Definition 6.1. An atom is a complex-valued function a defined on which is supported on a cube Q and is such that

!aa(X)dx=O

}Rn

1

Iiall oo < 1Qf'

and

Proposition 6.2. Let T be an operator as in the hypotheses of Theorem 5.10 whose kernel satisfies condition (5.10). Then there exists a constant C such that, given any atom a,

IITalh < C.

Proof. Since a E L2, Ta is well defined. Let Q* be the cube with the same center as Q, cQ, and side length 2fo times larger. Then, since T is bounded on L2 , 1/2

!a. ITa(xll dx < IQ'1

1/ 2 (

!a. ITa(xW dx

)

-

115

6. HI and BMO

116

< ClQI'/2

(10 la(xll 2dX) '/2

~C.

Further, since a has zero average, by condition (5.10)

r

J~P\Q*

ITa(x)1 dx

r r K(x,y)a(y) dy dx = r r [K(x, y) - K(x, cQ)]a(y) dy J'Rn\Q* JQ =

J~n\Q* JQ

rr

<

JQ J'Rn\Q*
dx

IK(x, y) - K(x, cQ)1 dx la(y)1 dy

o We now define the space atomic HI, denoted by Hlt, by

H~t(lRnl

= {

~>jaj

: aj

atoms,

Aj E

C,

~ IAjl <

00 } •

Clearly, Hlt is contained in Ll. Define a norm on Hlt by

IlfIIH~,

= inf {

~ IAjl : f L =

Ajaj } .

\Vith this norm Hlt is a Banach space. (Details are left to the reader.) Further, it follows immediately from Proposition 6.2 that singular integrals are bounded from Hlt to Ll. Corollary 6.3. Let T be an operator as in Proposition 6.2, and let f E Hlt .

Then

IITflh < CllfllH~t· The space Hlt is the largest subspace of Ll(I~n) for which Corollary 6.3 holds in the following sense: let R l , ... ,Rn be the Riesz transforms in ~n (the Hilbert transform if n = 1), and define the space Hl(~n) =

{f

E Ll(~n)

: Rjf

E Ll(~n), 1 <j

with the norm n

IIfllHl =

IIfill

+L j=l

Then the following is true.

IIRjflll.

< n}

117

2. The space BMO

We will not prove Theorem 6.4; a proof can be found in either GarciaCuerva and Rubio de Francia [6, Chapter 3] or Stein [17, Chapter 3].

2. The space BMO Given a function f E Lfoc(lRn) and a cube Q, let fQ denote the average of f on Q:

fQ =

I~I far

Define the sharp maximal function by

(6.1)

M# f(x)

= sup IQ11 Q3x

i

r If - fQ!, Q

where the supremum is taken over all cubes Q containing x. Each of these integrals measures the mean oscillation of f on the cube Q; we say that f has bounded mean oscillation if the function M# f is bounded. The space of functions with this property is denoted by BMO:

BMO = {f E Lfoc : M# f E LOO}. We define a norm on BMO by

11111*

= IIM# flloo.

This is not properly a norm since any function which is constant almost everywhere has zero oscillation. However, these are the only functions with this property, and it is customary to think of BMO as the quotient of the above space by the space of constant functions. In other words, two functions which differ by a constant coincide as functions in BMO. On this space 11·11* is a norm and the space is a Banach space. It is easy to see that we have the pointwise inequality

M# f(x) < CnMf(x) with the Hardy-Littlewood maximal function. The constant depends on the dimension n, but if we replace M by M", the variant of the maximal operator where the supremum is taken over all cubes containing x (see (2.6)), the constant is 2. Proposition 6.5.

(6.2)

(6.3)

1 . 1 -Ilfll* < sup mf -IQI 2 Q aEC

1 Q

If(x) -

al dx < Ilfll*;

M#(lfl)(x) < 2M# f(x).

6. HI and BMO

118

Proof. The second inequality in (6.2) is immediate. To prove the first, note that for all a,

k

If(x) - fQldx:;

k

If(x) - al dx +

Now divide both sides by supremum over Q.

IQI,

k

la -

k

If(x) - al dx

take the infimum over all a and then the

Inequality (6.3) follows from (6.2) (with a

I~I

fol dx < 2

k

Ilf(x)I-I!QII dx <

I~I

=

IfQI) since

k

If(x) - fQI dx.

o

"*'

Inequality (6.2) defines a norm equivalent to " . one which provides a way to show that f E BMO without using its average on Q: it suffices to find a constant a (that can depend on Q) such that

I~I

k

If(x) - al dx < C

with C independent of Q.

It follows from (6.3) that if f E BMO then If I is also in BMO. However, the converse is not true. This confirms what the definition suggests: being in BMO is not just a question of size. Clearly Loo C BMO, but there are also unbounded BMO functions. The typical example on IR. is

f(x)

log ( -1) = {0 Ixl

Ixl < 1,

IX,I > 1. But it is easy to see that the function sgn(x)f(x) ¢ BMO even though f is its absolute value. The next result shows the connection between BMO and singular integrals. Theorem 6.6. Let T be an operator as in the hypotheses of Theorem 5.10 whose kernel satisfies condition (5.11). Then if f is a bounded function of compact support, Tf E BMO and

Proof. Fix a cube Q in IR. n with center cQ, and let Q* be the cube centered at cQ whose side length is· 2-.Jii, times that of Q. Decompose f as f = h + 12, where h = fXQ*·

119

2. The space BMO

Let a

= T 12 (cQ). Then, since (5.11) holds and since

L2 ,

I~I

k

ITf(x) -

<

T is bounded on

al dx

kITh{x)ldx + I~I k C~I k f2 I~I

ITj,(x) - Tj,(cQ) I dx

<

I

T h(x)1 2 dx

+ IQ11 ( {

JQ J'Rn\Q*


C~I

k.

[K(x, y) - K(cQ, y)]j(y) dy dx

If(x)12 dx )

+ IQ11 ( {

JQ J'Rn\Q*

1/2

IK(x, y) - K(cQ' y)1 dy dx· 11J1100

< CllJlloo. D Theorem 6.6 is a first step towards our goal of finding a space which contains the images of bounded functions under singular integrals. However, the set of bounded functions with compact support is not dense in L oo , so we cannot extend T to the whole space by continuity. Instead, we will extend the definition of T to one which holds on all of Loo. Let f be a bounded function and let Q be a cube in ]Rn centered at the origin. Define Q* as above and again decompose f as h + 12, h = fXQ*· Since h is bounded and has compact support, This well defined as an L2 function; hence T h (x) exists for almost every x. For x E Q define

(6.4)

Tf(x)

=

Th(x)

+ (

J'Rn

[K(x, y) - K(O, Y)]12(Y) dy.

This integral converges since it is bounded by

Ilflloo (

J'Rn\Q*

IK(x, y) - K(O, y)1 dy,

provided K satisfies (5.11). N ow let Q be sOII?-e other cube centered at the origin which contains Q. Then we have two definitions of Tf(x) for x E Q. Let 11 = fX{r and 12 = f X'Rn \{j*; then the difference between the two definitions is

6. HI and BMO

120

r [K(x, y) - K(O, y)Jf(y) dy - r _[K(x, y) - K(O, y)Jf(y) dy = - (

T(h - JI)(X)

+

JJRn\Q*

JJRn \Q*

K(O, y)f(y) dy,

J Q*\Q*

and this is independent of x. Hence, the two definitions coincide as functions in BMO since in BMO we identify functions which differ by a constant. We can, therefore, take (6.4) as the definition of the image of f under T. Further, by arguing as we did in Theorem 6.6, we can show that if f E L oo then T f E BMO. Finally, note that we do not have to take cubes centered at the origin but can take any cube. If f is a bounded function of compact support, then (6.4) coincides with our original definition since for Q large enough, f = h. More generally, if f E S then the two definitions agree as B M 0 functions: since

r

K(O, y)f(y) dy

JJRn\Q*

converges absolutely to a value independent of x, the two definitions differ by a constant. Example 6.7. Let f(x) = sgn(x); we will find the image of f under the Hilbert transform, whose kernel is K(x, y) = [7f(x - y)]-I. Let Ixl < a/2; then

. ja sgn(y) dy +

7fHf(x) = p. v.

-a X -

Y

lim

j-a + iN (1 - - + -1)

N-+oo

-N

a

X -

Y

Y

sgn(y) dy

= 2log Ixl - 2Iog(a). Hence, ignoring the constant,

H(sgn(x))

=

2

-log Ixl. 7f

This is an indirect proof that log Ixl is a function in BMO. This result also agrees with our original motivation for the Hilbert transform since there exists an analytic function in the upper half-plane whose real part tends to sgn(x) and whose imaginary part tends to ~ log Ixl as we approach the real axis. We can .give such a function explicitly: i log(iz) As y

-+

= i log( -y + ix) = i log(x 2 + y2)1/2 + arctan(x/y).

0 the limit of this function is 7f

.

- sgn(x) + dog Ixl. 2 .

3. An interpolation result

121

3. An interpolation result In applying interpolation theorems (such as the Marcinkiewicz interpolation theorem) we often use the fact that an operator is bounded from L= to L=. Here we show that we can replace this with the weaker condition of boundedness from L= to BMO. Theorem 6.8. Let T be a linear operator which is bounded on LPO for some Po, 1 < Po < 00, and is bounded from L= to BMO. Then for all p, Po < p < 00, T is bounded on LP.

The proof of Theorem 6.8 requires two lemmas. Lemma 6.9. If 1
(6.5)

00

and f E LPO then

r Mdf(x)P dx < c JlRnl M# f(x)P dx,

JlRn

where Md is the dyadic maximal operator (2.9). By Lemma 2.12 we can replace the dyadic maximal function in (6.5) with the Hardy-Littlewood maximal function. Thus, while the reverse of the pointwise inequality M# f(x) < CM f(x) is not true in general, Lemma 6.9 provides a substitute norm inequality. The proof of Lemma 6.9 uses a technique known as a "good-A inequality"; more precisely, it depends on the following result. Lemma 6.10. If f E LPO for some Po, 1
<

00,

then for all "f > 0 and

A> 0, I{x E JRn : Mdf(x) > 2A, M# f(x) < "fA} I < 2n -yl{x E JRn : Mdf(x) > A}I.

Proof. Without loss of generality we may assume that f is non-negative. Fix A, -y > o. If we form the Calder6n-Zygmund decomposition of f at height A, then the set {x E JRn : Mdf (x) > A} can be written as the union of disjoint, maximal dyadic cubes. (See Theorem 2.11; we can substitute f E LPO for the hypothesis fELl in the proof.) If Q is one of these cubes, then it will suffice to show that

Let Q be the dyadic cube containing Q whose sides are twice as long. Then, since Q is maximal, fa < A. Furthermore, if x E Q and Mdf(x) > 2A, then Md(fXQ)(x) > 2A, so for those x's,

Md((f - fo)XQ)(x) > Md(fXQ)(x) - fa > A.

6. HI and BMO

122 By the weak (1,1) inequality for Md (Theorem 2.10),

I{x E Q : Md((f - J(j)xQ)(x) > A}I <

~ 10 IJ(x) - J¢I dx

2n lQI 1 ( < -A-IQI lQ If(x) -

fol dx

2n lQI <- inf M# f(x).

A

xEQ

If the set {x E Q : Mdf(x) > 2A, M# f(x) < ,,(A} is empty there is nothing to prove. Otherwise, for some x E Q, M# f(x) < "(A and so inequality (6.6) holds. 0 Proof of Lemma 6.9. For N

> 0 let

IN = ioN pAP-'I{x E]Rn : MdJ(X) > A}I dA. IN is finite since IN < and Mdf E

~NP-PO

(N POAPO-Il{x E lRn Po lo £PO since f E £Po. Furthermore,

:

Mdf(x) > A}I d,X,

IN = 2 io N p>'P-'I{x E]Rn : MdJ(X) > 2A}1 dA P

!2

{N/2

< 2Plo'

p,Xp-1 (I{x E lR n

:

Mdf(x) > 2A, M# f(x) < "(A}I

+ I{x E lRn 2P 1'YN/2

< 2P+n ,,(IN + -

(6.7)

P I IN < -2 + "(P

M# f(x) > "(A}I)dA

, p,Xp-ll{x E lR n : M# f(x) > A}I d'x.

0

"(P

Fix"( such that 2P+n ,,(

:

= 1/2; then

l'Y

N/ 2

p,Xp-ll{x

E

lR n

:

0

M# f(x) > A}I d'x.

If f (M# f)P = 00 then there is nothing to prove. If it is finite then we can take the limit as N ~ 00 in inequality (6.7) to get (6.5). 0 Proof of Theorem 6.8. The composition M# 0 T is a sublinear operator. It is bounded on LPO since both these operators are, and it is bounded on L oo since

IIM#(Tf)lJoo = IITfll* <

Cllflloo.

Hence, by the Marcinkiewicz interpolation theorem M# LP, Po < P < 00.

0

T is bounded on

4. The John-Nirenberg inequality

123

Now suppose that f E LP has compact support. Then f E LPo and so Tf E LPo. Therefore, we can apply Lemma 6.9 to Tf. In particular, since ITf(x)1 < Md(Tf)(x) a.e.,

r

J~n

ITf(x)I P dx <

r

J~n

Md(Tf) (x)P dx

r M#(Tf)(x)P dx < 0 r If(x)IP dx. J~n <0

J~n

o 4. The John-Nirenberg inequality In this section we examine the rate of growth of functions in B MO. We first consider 10g(I/lxl), which we gave as an example of an unbounded function in BMO. On the interval (-a, a), its average is (I-log a); given A > 1, the set where Ilog(I/lxl) - (1 - log a) I > A -has measure 2ae- A- 1 . The next result shows that in some sense logarithmic growth is the maximum possible for BMO functions. Theorem 6.11 (John-Nirenberg Inequality). Let f E BMO. Then there exist constants 0 1 and O2 , depending only on the dimension, such that given any cube Q in IR n and any A > 0,

(6.8)

I{x E Q : If(x) - fQI > A}I < 0Ie-c2A/llfll* IQI·

Proof. Since (6.8) is homogeneous, we may assume without loss of generality that IIfll* = 1. Then

(6.9)

I~I

k

If(x) - fQI dx < 1.

Now form the Calder6n-Zygmund decomposition of f - fQ on Q at height 2. (This is done as in Theorem 2.11 except that we begin by bisecting the sides of Q to form 2n equal cubes and repeating this for all cubes formed. Inequality (6.9) replaces the hypothesis that f ELI.) This gives us a family of cubes {Ql,j} such that 2 < IQl .1 I,J

r

JQl,j

If(x) - fQI dx < 2n+l

6. HI and BMO

124

and If(x) - fQI < 2 if x

rJ Uj Q1,j.

In particular,

:L. IQ1,jl < ~ JrQ If(x) -

fQI dx <

~IQI,·

J

and

IfQu - fQI

= IQ1

.1

1,J

r

JQ1

(f(x) - fQ) dx < 2n+1. ,j

On each cube Q1,j form the Calder6n-Zygmund decomposition of f fQl,j at height 2. (Again by assumption the average of this function on Q1,j is at most 1.) We then obtain a family of cubes {Q1,j,k} which satisfy the following:

IfQl,j,k - fQl,j I < 2n+1,

UQ1,j,k,

If(x) - fQl,j I <2 if x E Q1,j \

k

:L IQ1,j,kl < "21 IQ1,jl. k

Gather the cubes Q1,j,k corresponding to all the Q1,j'S and collectively rename them {Q2,j}. Then

and if x

rJ Uj .Q2,j,

If(x) - fQI < If(x) - fQl,jl

+ IfQl,j

- fQI < 2 + 2n+1 < 2· 2n+1.

Repeat this process indefinitely. Then for e<;Lch N we get a family of disjoint cubes {Q N,j} such that

If(x) - fQI < N· 2n+1 if x

rJ Uj QN,j

and

Fix A > 2n+1 and let N be such that N2 n+1 < A < (N

I{x

E

Q : If(x) - fQI > A}I <

:L IQN,jl j

1

< 2NIQI = e-Nlog2lQI < e- C2A IQI,

+ 1)2n+1.

Then

4. The John-Nirenberg inequality

125

where C2 = log2/2n+2. If A < 2n+1 then C 2 A < 10gv'2. Hence,

I{x

E

Q : If(x) - fQI > A}I < IQI < elogv!2-C2AIQI = V2e- C2A IQI,

so we can take C 1

o

= v'2.

Corollary 6.12. For all p, 1 < p <

Ilfll.,p =

00,

s~p (I~I 10 If(x) II . Ik

is a norm on BMO equivalent to

Proof. It will suffice to prove that Ilfll*,p < equality is immediate. But by Theorem 6.11

10 If(x) - fQI Pdx

=

1''

I~I 10 If(x) -

Cpllfll*

since the reverse in-

p).p-ll{x E Q : If(x) - fQI > )'}I d)'

< C11QI

Make the change of variables s

fQI PdX) 'ip

10

00

pAP-1e-C2A/llfll* d)".

r10=

= C2A/llfll*; then we get

fQIP dx < C,p

e~~'

r'e- s ds

= c1PC;-Pr(p)lIfll;,

o

which yields the desired inequality.

As a consequence of the proof of Corollary 6.12 we get two additional results. Given the size of the constant Cp , if we expand the exponential function as a power series we immediately get the following. Corollary 6.13. Given f E BMO, there exists).. cube Q,

>

0 such that for any

Further, the proof of Corollary 6.12 (with p = 1) can be readily adapted to show that the converse of the John-Nirenberg inequality (Theorem 6.11) holds. Corollary 6.14. Given a function f, suppose there exist constants C 1 , C2 and K such that for any cube Q and A > 0,

I{x Then f E BMO.

E

Q : If(x) - fQI > )..}I < C1e-C2A/KIQI·

6. HI and BMO

126

5. Notes and further results 5.1. References.

For the development of Hardy spaces and in particular the space HI, see Section 5.2 below. E. M. Stein (Classes HP, multiplicateurs et fonctions de Littlewood-Paley, C. R. Acad. Sci. Paris 263 (1966), 716-719, 780-781) first proved that singular integrals and multipliers are bounded from HI to LI. The space BMO was introduced by F. John and L. Nirenberg while studying PDE's (On functions of bounded mean oscillation, Comm. Pure Appl. Math. 14 (1961), 415-426). There they proved Theorem 6.11. That singular integrals take L= into BMO was first observed by S. Spanne (Sur l'interpolation . entre les espaces £~' ,Ann. Scuola Norm. Sup. Pisa 20 (1966), 625-648), J. Peetre (On convolution operators leaving £p,).. spaces invariant, Ann. Mat. Pura Appl. 72 (1966), 295-304), and E. M. Stein (Singular integrals, harmonic functions, and differentiability properties of functions of several variables, Singular Integrals, Proc. Sympos. Pure Math. X, pp. 316-335, Amer. Math. Soc., Providence, 1967). The sharp maximal function was introduced by C. Fefferman and E. M. Stein (HP spaces of several variables, Acta Math. 129 (1972), 137-193); in the same paper they proved Theorem 6.8. 5.2. The HP spaces.

The Hardy spaces were first studied as part of complex analysis by G. H. Hardy (The mean value of the modulus of an analytic function, Proc. London Math. Soc. 14 (1914), 269-277). An analytic function F in the unit disk JD) = {z E C : Iz I < I} is in HP (JD)), 0 < p < 00, if sup O
j7r IF(reiB)IP de < -7r

00.

A function F which is analytic in the upper half-plane y > O} is in HP(ffi.t) if sup y>O

r IF(x + iy)IP dx <

IIR

ffi.t = {z = x + iy :

00.

When p > 1 it can be shown that HP coincides with the class of analytic functions whose real parts are the Poisson integrals of functions f in £P(1I') or LP(ffi.) , respectively. We can therefore identify HP(lOl) with LP(1I') and HP(ffi.t) with LP(ffi.). This identification does not hold, however, for p < 1. Functions in HP(lOl) and HP(ffi.t) have a rich structural theory based on "factorization" theorems which allow them to be decomposed into well understood components. For a thorough treatment from this perspective, see Koosis [11] or Rudin [14]. Unfortunately, these results cannot be extended to higher dimensions using the theory of functions of several complex variables. E. M. Stein

5. Notes and further results

127

and G. Weiss (On the theory of harmonic functions of several variables, 1: The theory of HP spaces, Acta Math. 103 (1960), 25-62, also see Stein [15, Chapter 7]) defined HP(JR~+I) by considering vector-valued functions which satisfy generalized Cauchy-Riemann equations. This extension is only valId if p > 1 - lin, which does include HI. This space HI is isomorphic (by passing to boundary values) to the space of functions in LI(JRn ) whose Riesztransforms are also in LI(JR n ). (This is the space we called HI (JR. n ) in Section 1.) D. Burkholder, R. Gundy and M. Silverstein (A maximal characterization of the class HP, Trans. Amer. Math. Soc. 157 (1971), 137-153) proved a crucial result for establishing a strictly real-variable theory of HP spaces. Given F = u + iv, analytic in the upper half-plane, they showed that F is in HP if and only if the non-tangential maximal function of its real part, u~(XO)

= sup{lu(x, y) I : (x, y)

E

r a(XO)},

where

r a(XO) = {(x, y)

E JR~ : Ix - xol < ay},

is in LP(JR.). (See Chapter 2, Section 8.7.) Since u(x, y) = P y * f(x), the Poisson integral of its boundary value f, this characterizes the restriction to JR. of the real parts of functions in HP (JR~) . This space of functions on JR., which is customarily denoted by HP(JR.) (or sometimes by Re HP(JR)), is equivalent when p = 1 to the space HI (JR.) defined in Section 1. The original proof of this result used Brownian motion; a complex analytic proof was given by P. Koosis (Sommabilite de la fonction maximale et appartenance a HI, C. R. Acad. Sci. Paris 28 (1978), 1041-1043, also see [11, Chapter 8]). The key feature of this result is that the fact that u is a harmonic function is irrelevant, in the sense that the same characterization of the boundary values holds if we replace the Poisson kernel by any approximation of the identity defined starting from ¢ E S with non-zero integral. This leads to an elegant definition of HP in higher dimensions: given ¢ E S(JR. n ) with non-zero integral, we say a function f E HP(JR. n ) if Mqd(xo)

= sup{l¢y * f(x) I : (x, y)

E

r a(XO)}

is in LP(JR.n ). The resulting space is independent of ¢: in fact, one can replace M¢ by the so-called grand maximal function Mf(x)

= supM¢f(x), ¢

where the supremum is taken over all such ¢. For p > 1 - 1/n this space is isomorphic (again by taking boundary values) to the HP spaces of Stein and Weiss above. This approach is developed in the excellent article by Fefferman and Stein cited above. Also see Stein [17, Chapter 3].

6. HI and BMO

128

The atomic decomposition of functions in HP(lR) is due to R. Coifman (A real variable characterization of HP, Studia Math. 51 (1974),269-274); it was extended to higher dimensions by R. Latter (A decomposition of HP(n~n) in terms of atoms, Studia Math. 62 (1978), 93-101; also see R. Latter and A. Uchiyama, The atomic decomposition for parabolic HP spaces, Trans. Amer. Math. Soc. 253 (1979),391-398). For simplicity we used an atomic decomposition as the basis of our definition of HI (lRn )-the space we called Hlt(lR n ). Using these methods, R. Coifman and G. Weiss (Extensions of Hardy spaces and their use in Analysis, Bull. Amer. Math. Soc. 83 (1977), 569-645) defined HP spaces on spaces of homogeneous type (see Chapter 5, Section 6.4). Their definition only holds for Po < p < 1, where Po is a constant which depends on the space being considered. A monograph on the real theory of HP spaces with applications to Fourier analysis and approximation theory is due to Shanzhen Lu (Four Lectures on Real HP Spaces, World Scientific, Singapore, 1995). Finally, we mention another characterization of HP spaces in terms of the Lusin area integral. If u is a harmonic function on lR~+I, let

where

~~(xo)

=

{(x, y) E lR~+1 :

Ix -

xol

< ay, 0 < y <

h}

and

(This is called the area integral since if n = 1, S (u ) 2 is the area of the image of r~(xo), counting multiplicities, under the analytic function F whose real part is u.) Then u E HP(lR~+I) if and only if S(u) E LP(lRn ) and u(x,y) ---t 0 as y ---t 00. An alternative characterization is gotten by replacing the area integral by a Littlewood-Paley type function which is its radial analogue: g( u) (x)

=

(Joroo l\7u(x, y) 12y dy.) 1/2

This characterization is due to A. P. Calderon (Commutators of singular integral operators, Proc. Nat. Acad. Sci. U.S.A. 53 (1965), 1092-1099). See also Garcia-Cuerva and Rubio de Francia [6], Stein t15], Zygmund [21] and the article by Fefferman and Stein cited above.

5. Notes and further results

129

5.3. Duality.

The results on boundedness from HI to LI and from L= to BMO are dual to one another. This is a consequence of a deep result due to C. Fefferman (announced in Characterizations of bounded mean oscillation, Bull. Amer. Math. Soc. 77 (1971), 587-588). Theorem 6.15. The dual of HI (JR n ) is BMO.

The article by Fefferman and Stein cited above gives two proofs of this result. Three appear in Garcia-Cuerva and Rubio de Francia [6]. The duals of the HP spaces, P < 1, are Lipschitz spaces. This is due to P. Duren, B. Romberg and A. Shields (Linear functionals on HP spaces with o < P < 1, J. Reine Angew. Math. 238 (1969), 32-60) on the unit circle and to T. Walsh (The dual of HP(JR+.+ I ) for P < 1, Can. J. Math. 25 (1973), 567-577) in JRn . Also see [6, Chapter 3]. 5.4. The Hardy-Littlewood maximal function on BMO.

The maximal function is bounded on BMO: if f E BMO then Mf E BMO. This was first proved by C. Bennett, R. A. DeVore and R. Sharpley (Weak L= and BMO, Ann. of Math. 113 (1981), 601-611); simpler proofs were given by F. Chiarenza and M. Frasca (Morrey spaces and HardyLittlewood maximal function, Rend. Mat. Appl., Series 7, 7 (1981), 273-279) and D. Cruz-Uribe, and C. J. Neugebauer (The structure of the reverse Holder classes, Trans. Amer. Math. Soc. 347 (1995), 2941-2960). See also a proof in Torchinsky [19, p. 204]. 5.5. Another interpolation result. 1

In applying the Marcinkiewicz interpolation theorem one often uses that the operator in question is bounded on L= or satisfies a weak (1,1) inequality. In Theorem 6.8 we showed that we can replace the first hypothesis by the assumption that the operator maps L= into BMO. One can also use HI to replace the weak (1,1) inequality. Theorem 6.16. Given a sublinear operator T, suppose that T maps HI into LI and for some PI, 1 < PI < 00, T is weak (PbPI). Then for all p, 1 < P < PI, T is bounded on LP.

This result can be generalized to the other HP spaces. For details and references, see Garcia-Cuerva and Rubio de Francia [6, Chapter 3]. 5.6. Fractional integrals and Sobolev spaces.

For fractional integral operators fa (see Chapter 4, Section 7.7), the critical index is P = n/ a: we would expect fa to map L n / a to L=, but it does not. Instead, we have a result analogous to Theorem 6.6.

6. HI and BMO

130

Theorem 6.17. If f E Ln/a then Iaf E BMO.

The proof of this theorem follows at once from an inequality relating the fractional integral, the sharp maximal function and the fractional maximal function. Theorem 6.18. There exist positive constants C I and C 2 such that for all locally integrable f,

CIMaf(x)

< M#(laf)(x) < C 2 M a f(x).

Both of these results are due to D. R. Adams (A note on Riesz potentials, Duke Math. J. 42 (1975), 765-778). When p = 1, besides the weak (1, n/(n - a)) inequality in Theorem 4.18, we also have an analogue of Corollary 6.3: IIlaflln/(n-a) < CllfllHl. This result is due to Stein and Weiss-see the article cited above. Similar inequalities hold for all the HP spaces: see the article by S. Krantz (Fractional integration on Hardy Spaces, Studia Math. 73 (1982), 87-94). For Sobolev spaces, the corresponding limiting theorem would be that if f E L~/k then f E BMO. However, this is only known when k = 1: see the paper by A. Cianchi and L. Pick (Sobolev embeddings into BMO, VMO and L oo , Ark. Mat. 36 (1998), 317-340). In the general case we have as a substitute an exponential integrability result (cf. Corollary 6.13). Theorem 6.19. Given 0 < a < n and any cube Q, then there exist constants C I and C2 such that if f E Ln/a(Q),

1 IQI

_1

exp (

Q

laf ' PI) < CIllfIILn/a(Q) -

c2 .

Theorem 6.19 was proved by N. S. Trudinger (On imbeddings into Orlicz spaces and some applications, J. Math. Mech. 17 (1967), 473-483) when a = 1 and in general by R. S. Strichartz (A note on Trudinger's extension of Sobolev's inequalities, Indiana Univ. Math. J. 21 (1972), 841842). Sharp constants have been found by J. Moser (A sharp form of an inequality by N. Trudinger, Indiana Univ. Math. J. 20 (1971), 10771092) and D. R. Adams (A sharp inequality of J. Moser for higher order derivatives, Ann. of Math. 128 (1988), 385-398). Also see the books by W. Ziemer (Weakly Differentiable Functions, Springer-Verlag, New York, 1989) and D. R. Adams and L. Hedberg (Function Spaces and Potential Theory, Springer-Verlag, Berlin, 1996).

5. Notes and further results

131

5.7. Vanishing mean oscillation.

. Given a function f E BMO, we say that f has vanishing mean oscillation, and write f E V MO, if

I~I

fa

If(x) - fQI dx

tends to zero as IQI tends to either zero or infinity. As in BMO, there are unbounded functions in V MO. For example,

f(x)

=

{~~g log(l/Ixl), Ixl < lie, Ixl > lie,

is in VMO, and in fact it can be shown that in some sense it has the maximum rate of growth possible for functions in V MO. Clearly, if f E Co (i.e. the space of continuous functions which vanish at infinity) then f E V MO; further, V MO is the closure in BMO of Co.

V MO lets us sharpen Theorem 6.6 as follows: given a singular integral operator T, if f E Co then Tf E VMO. The space V MO was introduced by D. Sarason (Functions of vanishing mean oscillation, Trans. Amer. Math. Soc. 207 (1975), 391-405). Also see the survey article by R. Coifman and G. Weiss (Extensions of Hardy spaces and their uses in analysis, Bull. Amer. Math. Soc. 83 (1977), 569-645). 5.8. Commutators and BMO.

We can give another characterization of BMO in terms of singular integrals. Let T be a singular integral, and for a locally integrable function b, let Mb be the operator given by pointwise multiplication by b: Mbf(x) = b(x)f(x). Define the commutator [b, T] by MbT - TMb. If Tf = K * f then for f E C~ [b, T]f(x)

=

r

JlR

(b(x) - b(y))K(x - y)f(y) dy,

x Fj. supp(f).

n

Clearly, if b E L= then [b, T] is bounded on LP, 1 < p < 00. Since both MbT and T Mb are bounded exactly when b is a bounded function, it would be reasonable to conjecture that this is also the case for [b, T]. However, due to cancellation between the two terms, a weaker condition is sufficient. Theorem 6.20. Given a singular integral T whose associated kernel is standard, then the commutator [b, T] is bounded on LP, 1 < p < 00, if and only ifb E BMO.

Commutators were introduced by R. Coifman, R. Rochberg and G. Weiss (Factorization theorems for Hardy spaces in several variables, Ann. of Math. 103 (1976), 611-635) who used them to extend the classical theory of HP

132

6. HI and BMO

spaces to higher dimensions. They proved that b E B M 0 is sufficient for [b, T] to be bounded and proved a partial converse. The full converse is. due to S. Janson (Mean oscillation and commutators of singular integral operators, Ark. Mat. 16 (1978), 263-270). This paper also contains a simpler proof of sufficiency due to J. O. Stromberg which is reproduced in Torchinsky [19, p.418]. The commutator [b, T] is more singular than the associated singular integral; in particular, it does not satisfy the corresponding weak (1,1) estimate. However, the following weaker result is true. Theorem 6.21. Given a singular integral Tand bE BMO, then

J{x E]Rn : J[b, TIf("')J > A}J < Cllbll*

Ln Jf~)J (1 + log+ (Jf~)J)) dy.

Theorem 6.21 is due to C. Perez (Endpoint estimates for commutators of singular integral operators, J. Funct. Anal. 128 (1995), 163-185). In this paper he also discusses the boundedness of [b, T] on HI. A. Uchiyama (On the compactness of operators of Hankel type, T6hoku Math. J. 30 (1978), 163-171) proved that the commutator [b, T] is a compact operator if and only if b E V MO.

Chapter 'l

Weighted Inequalities

1. The Ap condition In this chapter we are going to find the non-negative, locally integrable functions w such that the operators we have been studying are bounded on the spaces V(w). (These are the LP spaces with Lebesgue measure replaced by the measure w dx.) We will refer to such functions as weights, and for a measurable set E we define w(E) = w.

IE

Our first step is to assume that the Hardy-Littlewood maximal function satisfies a weighted, weak-type inequality, and deduce from this a necessary condition on the weight w. To simplify our analysis, throughout this chapter we will replace our earlier definition of the Hardy-Littlewood maximal function with Mf(x) =

~~~ I~I 10 Ii(y) Idy,

where the supremum is taken over all cubes Q containing x. This is the non-centered maximal operator we originally denoted by Mil (see (2.6)), but recall that it is pointwise equivalent to our original definition (2.3). The weighted, weak (p, p) inequality for M with respect to w is (7.1)

W({XEIRn:Mf(x»A})<

~

AP

r If(x)IP

J'ffln

w (x)dx.

f

be a non-negative fun~tion and let Q be a cube such that f(Q) = f > O. Fix A such that 0 < A < f(Q)/IQI. Then Q C {x E IR n : M(JXQ)(x) > A}, so from (7.1) we get that

Let

IQ

w(Q)

<

~ 10 If(xWw(x) dx.

-

133

134

7. Weighted Inequalities

Since this holds for all such A, it follows that (7.2) Given a measurable set SeQ, let

(7.3)

w(Q)

f = xs. Then (7.2) becomes

G~DP < eweS).

From (7.3) we can immediately deduce the following: (1) The weight w is either identically 0 or w > 0 a.e. For if w = 0 on a set of positive measure S (which we may assume is bounded), then (7.3) implies that for every cube Q that contains S, w(Q) = O.

(2) The weight w is either locally integrable or w = 00 a.e. For if w (Q) = 00 for some cube Q, then the same is true for any larger cube, and so by (7.3), w(S) = 00 for any set S of positive measure. Note that while we had assumed a priori that w was locally integrable, this is actually a consequence of the weighted norm inequality. To deduce the desired necessary conditions, we will consider two cases. Case 1: p

=

1. In this case inequality (7.3) becomes

w(Q) < Cw(S)

IQI -

lSI·

Let a = inf{w(x) : x E Q}, where inf is the essential infimum, that is, excluding sets of measure zero. Then for each E > 0 there exists SE C Q such that ISEI > 0 and w(x) < a + Efor any x ESE. Hence, for all E> 0,

w(Q)

lQf < C(a + E), and so

w(Q)

.

lQf < C ~~b w(x). Therefore, for any cube Q,

(7.4)

w(Q) . lQf < Cw(x)

a.e. x E Q.

This is called the Al condition, and we refer to the weights which satisfy it as Al weights. Condition (7.4) is equivalent to '

(7.5)

Mw(x) < Cw(x)

a.e. x E

]Rn.

Clearly, (7.5) implies (7.4). Conversely, suppose that (7.4) holds and let x be such that Mw(x) > Cw(x). Then there exists a cube Q with rational vertices such that w(Q)/IQI > Cw(x), so x lies in a subset of Q of measure

1. The Ap condition

135

zero. Taking the union over all such cubes we see that Mw(x) > Cw(x) holds only on a set of measure 0 in IRn.

Case 2: 1

< p < 00. In (7.2) let f = wI-p' XQ; then w(Q)

(_1 {wI-PI)P < C { wI-pI. IQI J J Q

Q

Equivalently, (7.6) where C is independent of Q. (Note that to derive (7.6) we have assumed that wI-p' is locally integrable. To avoid this assumption we can replace it by min (wI-pI ,n) and take the limit as n tends to infinity. However, since w > 0 a.e., (7.6) implies that wI-pI is locally integrable.) Condition (7.6) is called the Ap condition, and the weights which satisfy it are called Ap weights. Theorem 7.1. For 1