U.U.D.M. Project Report 2018:29
Examensarbete i matematik, 15 hp
Handledare: Anders Israelsson
Examinator: Martin Herschend
Juni 2018
Department of Mathematics
Uppsala University
On Singular Integral Operators
ABSTRACT
We introduce the subject of harmonic analysis through the perspective of singular integral theory. The Hilbert transform is motivated through problems regarding convergence of Fourier series and the study of boundary values of harmonic functions. The boundedness of the Hilbert transform in Lp, 1 ă p ă 8, is proved through real-variable methods using Plancherel’s Theorem and the
Calder´on Zygmund decomposition, along with Marcinkiewicz Interpolation Theorem. The same methods can be used to prove the boundedness of certain singular integrals on Rn, and another
approach using the Riesz transforms in Rn gives similar results for homogeneous singular integrals
Acknowledgements
I would like to thank Wulf Staubach for his great suggestion that I write about this subject. I would also like to thank my advisor Anders Israelsson, for his support and guidance while
Contents
0 Introduction 4
1 Convergence of Fourier Series 8
1.1 Properties of Fourier Coefficients 8
1.2 A Criterion for Pointwise Convergence 9
1.3 Convolutions and Kernels 11
1.4 Examples of Summability 14
1.5 Convergence in Norm 17
1.6 The Hilbert Transform on T 18
1.7 Fourier series in L2 20
2 Boundedness of the Hilbert Transform 22
2.1 The Calder´on-Zygmund Decomposition 22
2.2 The Hilbert Transform in L2 23
2.3 Existence of the Hilbert Transform 26
2.4 The Marcinkiewicz Interpolation Theorem 27
2.5 Boundedness of the Hilbert Transform 30
3 Singular Integrals on Rn 33
3.1 The Hilbert Transform on R 33
3.2 The Riesz Transforms 34
3.3 Boundedness for Odd Kernels 36
3.4 The Fourier Transform 38
3.5 The Inversion Formula 40
3.6 Fourier Transforms in L2 41
3.7 Singular Integrals in L2 42
3.8 Boundedness for Even Kernels 45
4 Appendix I: Some Integrals over Rn and Σn´1 48
Introduction
Throughout this text we will study various singular integral operators. The simplest example of such an operator is the Hilbert transform H. For f P L1
pp´12,1
2qq, it is given by the expression
Hf pxq “ ż 12 ´12 f ptq tan πpx ´ tqdt. (1) 1
tan πt is not integrable near the origin (since 1 tan πt
-1
πt is bounded), so the existence of the integral
in (1) is a problem. For example, if |f ptq| ě C for all t P px ´ δ, x ` δq for some C, δ ą 0, then it does not exist in the Lebesgue sense, since the integrand would be bounded below by |tan πpx´tq|C near x. However, this can be handled by considering the integral in the principal value sense. That is, we define the Hilbert transform by
Hf pxq “ p.v. ż 12 ´1 2 f ptq tan πpx ´ tqdt “ limÑ0` ż ď|x´t|ă1 2 f ptq tan πpx ´ tqdt, (2) and the truncated Hilbert transform is defined by
Hf “ ż ď|x´t|ă1 2 f ptq tan πpx ´ tqdt. (3)
By a change of variables we can write Hf “ ż ď|t|ă1 2 f px ´ tq tan πt dt “ ´ ż ď|t|ă1 2 f px ` tq tan πt dt “ ż 12 f px ´ tq ´ f px ` tq tan πt dt. (4) Note that Hf exists when f P L1pp´12,12qq, since tan πt1 is bounded on r,12q. It is still not obvious
that the limit in (2) exists. However, we see that it holds if f satisfies a H¨older condition of order α ą 0, since then the last integral in (4) is bounded near the origin by a constant multiple oftan πttα , which is integrable on p0,12q. By Dominated Convergence Theorem, we also have
Hf pxq “ ż 12
0
f px ´ tq ´ f px ` tq
tan πt dt. (5)
The limit in (2) also turns out to exist almost everywhere for every f P L1pp´12,12qq. We will prove this in the second chapter.
The Hilbert transform appears in the study of holomorphic functions in the unit disc. Let F “ u`iv be a function holomorphic on the unit disc D “ tz P C : |z| ă 1u and continuous on D. Suppose u “ f on BD where f satisfies a H¨older condition. u is unique by the maximum principle, and it is well known that it is given by the Poisson integral
upre2πiθq “ ż 12 ´12 f pe2πitqPrpθ ´ tq dt, (6) where Prptq “ 1 ´ r2 1 ´ 2r cos 2πt ` r2 (7)
is the Poisson kernel. To see this result, we write F pzq “ 1 2πi ż BD F pwq w ´ zdw, (8)
which holds for all z P D by the Cauchy Integral Formula. The inverse point of z with respect to BD is z˚“ pzq´1. By Cauchy’s Theorem we can write
If we now set z “ re2πiθ and w “ e2πit, we get F pre2πiθq “ ż 12 ´12 p1 ´ r2qF pe2πitq |e2πit´ re2πiθ|2 dt “ ż 12 ´12 F pe2πitq 1 ´ r 2 1 ´ 2r cos 2πpθ ´ tq ` r2dt.
Hence the real part of F pre2πiθ) is given by the expression in (6). Since u is unique, v will be
unique up to a constant. To uniquely determine v, we require that vp0q “ 0. To calculate v, do the same calculation on
F pzq “ 1 2πi ż BD F pwq w ´ zdw ` 1 2πi ż BD F pwq w ´ z˚ dw,
which will result in
vpre2πiθq “ ż 12 ´12 vpe2πitq dt ` ż 12 ´12 f pe2πitqQrpθ ´ tq dt, (10) where Qrptq “ 2r sin 2πt 1 ´ 2r cos 2πt ` r2 (11)
is called the conjugate Poisson kernel. By the mean value property of harmonic functions, the first integral in (10) equals vp0q “ 0, so we have
vpre2πiθq “ ż 12 ´12 f pe2πitqQrpθ ´ tq dt “ ż 12 0 ´ f pe2πipθ´tqq ´ f pe2πipθ`tqq ¯ Qrptq dt.
If we let r Ñ 1´ we see that Q
rptq Ñ tan πt1 . Since F is continuous on D and f satisfies a H¨older
condition we get vpre2πiθq rÑ1 ´
ÝÝÝÝÑ Hf pe2πiθq, which can be shown to hold almost everywhere for all f P L1pp´12,12qq. From this perspective the Hilbert transform maps the boundary function of an harmonic function in D to the boundary function of its harmonic conjugate. For this reason it is sometimes called the Hilbert transform on the circle.
One can do a similar calculation for a bounded holomorphic function f defined on upper half space tz P C : Imz ą 0u. The boundary function of the harmonic conjugate will then be given by the Hilbert transform on the real line,
Hf pxq “ p.v.1 π ż8 ´8 f ptq x ´ tdt. (12)
This linear operator has many interesting properties. For example, it commutes with translations, complex conjugation, positive dilations and differentiation. Moreover, it is a Hilbert space isomor-phism on L2
pRq with H´1“ ´H, and both forms of the Hilbert transform map Lp into Lp when
1 ă p ă 8. They are also continuous on Lp, 1 ă p ă 8, which is a deep result proved by Marcel Riesz. In the second chapter we give a proof of this for the Hilbert transform on the circle. In the first chapter we introduce Fourier series and discuss their convergence, which is deeply connected to the Hilbert transform. The L2-theory of Fourier series will prove the boundedness of the Hilbert transform in L2pp´12,12qq, which together with an interpolation theorem can be used to prove the more general Lp-boundedness when 1 ă p ă 8. Moreover, the Lp-boundedness of the Hilbert
transform will then imply that the Fourier series of f P Lp
pp´12,1
2qq, 1 ă p ă 8, converges to f in
Lp-norm! In fact, assuming the boundedness of the Hilbert transform in Lp, we can prove that
y
Hf pjq “ ´i sgn pjq pf pjq, (13)
where pf pjq are the Fourier coefficients of f . In chapter 1 we will prove this fact, and that if ´i sgn pjq pf pjq is the Fourier series of some rf P Lp
pp´12,1
2qq for every f P L p
pp´12,1
2qq then the
Lp-convergence of Fourier series holds in general.
In the third chapter we generalize the operator in (12) to a more general class of singular integral operators. For example, we have the Riesz transforms Rj, j P t1, 2, ..., nu, defined by
for some constant Cn. The Riesz transforms appear when studying conjugate harmonic functions on
upper half space Rn`1` “ tpx, tq P R n
ˆ R : t ą 0u. Let upx, tq be harmonic on Rn`1` and upx, 0q “
f pxq where u has similar properties to the u in (6). Then we can find a system of conjugate harmonic functions vj, j P t1, 2, ..., nu, such that tu, v1, ..., vnu is a set of harmonic functions satisfying the
generalized Cauchy-Riemann equations, #Bu i Bxj “ Buj Bxi Bu1 Bx1 ` ... ` Bun Bxn “ 0 . (15)
In chapter 3 we find vjpx, tq and show that they converge to Rjf pxq as t Ñ 0`, so the Riesz
transforms generalize the complex analysis approach to the Hilbert transform (note that when n “ 1 and Cn “ π1 in (14), the one Riesz transform is the Hilbert transform on the real line).
We can get an identity similar to (13) for the Riesz transforms if we apply the Fourier transform f ÞÑ pf , where p f pξq “ ż Rn f pxqe´2πix¨ξdx, (16)
which we will prove is a Hilbert space isomorphism on L2
pRnq. For the Hilbert transform we will show that this identity is, for L2-functions f ,
y
Hf pξq “ ´i sgnpξq pf pξq, (17)
and more generally for the Riesz transforms, with the proper choice of Cn,
y
Rjf pξq “ ´i
ξj
|ξ|f pξq.p (18)
From this identity we get
n ÿ j“1 z R2 jf pξq “ ´ n ÿ j“1 ξ2 j |ξ|2 p f pξq “ ´ pf pξq. Hence we get the inversion formula
n
ÿ
j“1
R2
jf “ ´f, (19)
and in the case n “ 1,
H´1
“ ´H. (20)
We also note that (18) would prove the L2-boundedness of the Riesz transforms, since using the fact that the Fourier transform is an isometry on L2pRnq (Plancherel’s Theorem), we get
}Rjf }2“ › › ›Ryjf › › ›2ď › › ›fp › › ›2“ }f }2.
The Lp-boundedness, 1 ă p ă 8, can also be proved for the Riesz transforms. We prove this in
chapter 3 by a calculation which shows that the Lp-boundedness of the Hilbert transform implies
the Lp-boundedness of the Riesz transforms. Note that a continuity argument would then extend
(19) and (20) to Lp
pRnq, 1 ă p ă 8.
The Riesz transforms appear in the study of partial differential equations. Consider the Laplacian ∆u where u P C2
cpRnq. Then a simple calculation gives x∆upξq “ ´p2πq2|ξ| 2 p f pξq (recall that { Bf {Bxjpξq “ 2πiξjf pξq). Now, by (18),p { RjRk∆upξq “ ˆ ´iξj |ξ| ˙ ˆ ´iξk |ξ| ˙ x
∆upξq “ ´ p2πiξjq p2πiξkqupξq “ ´p {B2u BxjBxk pξq. Thus we get B2u BxjBxk “ ´RjRk∆u. (21) Hence if u P C2
cpRnq satisfies the Poisson equation ∆u “ f where f P LppRnq for some 1 ă p ă 8,
then B2u{Bx
the Riesz transforms. Of course the Riesz transforms are quite complicated, since xj{ |x|n`1 is
not locally integrable near 0. Other operators with similar complications also appear in the study of partial differential equations, where a more general theory of singular integral operators would have to be considered. We will consider general operators of the type
T f “ p.v. ż Rn f px ´ yqΩp y |y|q |y|n dy, (22)
where Ω is a function on the unit sphere Σn´1
“ tx P Rn : |x| “ 1u. We are particularly interested in the boundedness properties of these operators. For the expression in (22) to make any sense we will need to assume that Ω P L1
pΣn´1q andşΣn´1Ωpy1qdy1“ 0. The Riesz transforms are singular integrals of this type where Ω is an odd function. When Ω is odd the Lp-boundedness of T follows
similarly to how the boundedness of the Riesz transforms will follow. For general Ω we can then consider the decomposition Ω “ ΩO` ΩE, where
ΩO “
Ωpxq ´ Ωp´xq
2 , ΩE“
Ωpxq ` Ωp´xq
2 .
ΩO is an odd function so this part of T is Lp-bounded when 1 ă p ă 8. ΩE is even, so it suffices
to prove boundedness for even operators. For an attempt to prove Lp-boundedness when Ω is even
we will use (19) to write
T “ ´
n
ÿ
j“1
RjpRjT q. (23)
Since the Riesz transforms are bounded on Lp
pRnq, 1 ă p ă 8, it suffices to prove the boundedness of the operators RjT . The strategy here will be to show that each RjT is an operator of the type
(22) with an odd Ω. We give a proof of this in the case Ω P Lr
Convergence of Fourier Series
We introduce Fourier series and discuss their convergence, both pointwise and in norm. In par-ticular, we will show how the Lp-boundedness of the Hilbert transform solves the problem of
Lp-convergence of Fourier series.
Properties of Fourier Coefficients
When studying Fourier series we work with complex functions on the circle group T “ R{Z. A function on T is essentially a 1-periodic function on R. Hence spaces like CpTq, Ck
pTq and LppTq are defined as the spaces of functions f : T Ñ C whose corresponding periodic function belongs to CpRq, Ck
pRq and Lppr´12,12qq, respectively. T is naturally identified with the interval r0, 1q, which gives us definitions of Lebesgue measure and the Lebesgue integral on T. We highlight the fact that the measure on T is translation-invariant. This property will be used frequently throughout the text.
Definition 1.1. A trigonometric polynomial of degree n on T is given by ϕpxq “
n
ÿ
j“´n
cje2πijx, (24)
for some complex coefficients cj where at least one of cn and c´n are non-zero.
To calculate the coefficients cj we use the identity
ż
T
e2πijtdt “"1 j “ 0
0 j ‰ 0. (25)
Multiplying ϕ by e´2πikt and then integrating we obtain
ck“
ż
T
ϕptqe´2πiktdt. (26)
This motivates the definition of Fourier series of functions in L1pTq. Note that this will also define Fourier series of functions in Lp
pTq Ă L1pTq, p ą 1, since T has finite measure. Definition 1.2. For f P L1
pTq, its Fourier series is given by the formal expression f „
8
ÿ
j“´8
cje2πijx, (27)
along with its partial sums
snpf ; xq “ n
ÿ
j“´n
cje2πijx, (28)
where the Fourier coefficients cj“ pf pjq are given by
p f pjq “
ż
T
f ptqe´2πijtdt. (29)
If, for example, f equals a series in (27) that converges uniformly we can perform the same cal-culation as for the trigonometric polynomials above, and conclude that cj “ pf pjq. We also see
that snpf ; xq Ñ f if f is a trigonometric polynomial, both pointwise and in Lp-norm. We expect
snpf ; xq to approximate f in some sense at least when f has certain properties, but in general a
Fourier series need not converge, and it may converge to something else than its function. In fact, we don’t necessarily have pointwise convergence of Fourier series in L1
pTq, not even in CpTq! We proceed to list a few basic properties of Fourier series.
Lemma 1.3. Let f, g P L1
pTq and α P C.
(ii) pf pjq “ pf p´jq, where f is the complex conjugate to f .
(iii) pftpjq “ pf pjqe´2πijt, where ft is the translation map ftpxq “ f px ´ tq.
(iv) If gpxq “ e2πinxf pxq, then pgpjq “ pf pj ´ nq.
(v) pf1pjq “ 2πij pf pjq, for f P C1
pTq
(vi) The Fourier coefficients are bounded and ˇ ˇ ˇf pjqp ˇ ˇ ˇ ď }f }1.
(vii) The coefficients go to zero: pf pjqÝÝÝÝÑ 0.jÑ˘8
(i)-(vi) are all trivial. (vii) is a direct consequence of the following. Lemma 1.4. (Riemann-Lebesgue) Let f P L1
ppa, bqq, ´8 ď a ă b ď 8. Then lim P Ñ8 żb a f pxqeiP tdt “ 0.
Proof. It clearly holds for constant functions and thus also for step functions (see Appendix II). We now use the fact that the step functions are dense in L1. Given ą 0 there is a step function
g such that }f ´ g}1 ă {2, and a P0 such that for P ą P0 we have
ˇ ˇ ˇ şb agptqe iP tdtˇˇ ˇ ă {2. When P ą P0we get ˇ ˇ ˇ ˇ ˇ żb a f ptqeiP tdt ˇ ˇ ˇ ˇ ˇ ď ˇ ˇ ˇ ˇ ˇ żb a pf ptq ´ gptqqeiP tdt ˇ ˇ ˇ ˇ ˇ ` ˇ ˇ ˇ ˇ ˇ żb a gptqeiP tdt ˇ ˇ ˇ ˇ ˇ ă 2 ` 2 “ . The function eiP tmay be replaced by sin P t or cos P t in the lemma above and the same result will hold with the exact same proof. Also note that pa, bq may be generalized to any measurable set X Ă R, since any integrable function f on X can be identified with an integrable function on R, by letting f pxq “ 0 for x R X.
A Criterion for Pointwise Convergence
We rewrite the partial sums of a Fourier series as follows. snpf ; xq “ n ÿ j“´n ˆż T f ptqe´2πijtdt ˙ e2πijx“ ż T f ptq n ÿ j“´n e2πijpx´tqdt “ ż T f ptqDnpx ´ tq dt, where Dn is given by Dnpxq “ n ÿ j“´n e2πijx. (30)
The family tDnu8n“0is called the Dirichlet kernel. If we sum the geometric series in (30) we get
Dnpxq “
sin πp2n ` 1qx
sin πx . (31)
Note that Dn is an even function for every n. Using this we can write
snpf ; xq “ ż T f ptqDnpx ´ tq dt “ ż T f px ´ tqDnptq dt “ ż T f px ` tqDnptq dt (32) “ ż T f px ` tq ` f px ´ tq 2 Dnptq dt. (33) Dn has the following properties,
(i) ş
(ii) ş
|x|ěδ|Dnptq| dt nÑ8
ÝÝÝÑ 0 for every δ ą 0.
(i) follows from (30) and (25), and (ii) follows from (31) and Riemann-Lebesgue Lemma, since ˇ
ˇsin πx1 ˇ
ˇis bounded by sin πδ1 when |x| ě δ.
If we want to make an attempt to prove snpf ; x0q Ñ `, the following calculation using (i) is very
useful.
snpf ; xq ´ ` “
ż
T
pf px ´ tq ´ `q Dnptq dt. (34)
Using (33), we can write snpf ; x0q ´ ` “ ż T ˆ f px0` tq ` f px0´ tq 2 ´ ` ˙ Dnptq dt “ ż T ˆ f px0` tq ` f px0´ tq 2 ´ ` ˙ 1 sin πtsin πp2n ` 1qt dt.
Here we would like to apply Riemann-Lebesgue Lemma. The problem is thatsin πt1 is not integrable on T. It is, however, integrable on |x| ě δ for any δ ą 0, so we only need to consider the integral over |x| ă δ. We can apply Riemann-Lebesgue Lemma again if we have the following condition,
ż |t|ăδ ˇ ˇ ˇ ˇ ˆ f px0` tq ` f px0´ tq 2 ´ ` ˙ 1 sin πt ˇ ˇ ˇ ˇ dt ă 8. (35)
A more natural condition can be obtained by replacing sin πt1 by 1t. This is equivalent since |2t| ď |sin πt| ď |πt|. We can also replace |t| ă δ with 0 ă |t| ă δ since the integrand is even. We have proved the following theorem.
Theorem 1.5. (Dini) Let f P L1
pTq and x0 P T. Suppose that for some δ ą 0 and for some
constant `, żδ 0 ˇ ˇ ˇ ˇ ˆ f px0` tq ` f px0´ tq 2 ´ ` ˙ 1 t ˇ ˇ ˇ ˇ dt ă 8. Then snpf ; x0qÝÝÝÑ `.nÑ8
We define the left and right limits, f px0´q and f px0`q, by f px0´q “ limtÑ0´f px0 ` tq and
f px0`q “ limtÑ0`f px0` tq, respectively. We also introduce the left and right derivatives, f1px0´q
and f1px
0`q. They are defined by
f1 px0´q “ lim tÑ0´ f px0` tq ´ f px0´q t , f 1 px0`q “ lim tÑ0` f px0` tq ´ f px0`q t .
We have two important corollaries to Theorem 1.5. Corollary 1.6.
(i) If f is differentiable at x0P T, then snpf ; x0qÝÝÝÑ f pxnÑ8 0q. More generally, if f1px0`q and
f1px
0´q exist, then snpf ; x0q nÑ8
ÝÝÝÑ f px0`q`f px0´q
2 .
(ii) If f satisfies a H¨older condition of order α ą 0, then its Fourier series converges uniformly to f .
Proof. (i) is immediate from Dini’s criterion with ` “ f px0`q`f px0´q
2 , and (ii) can be derived directly
from (34) by using Riemann-Lebesgue’s Lemma with ` “ f pxq.
Corollary 1.7.
(i) Suppose f P L1
pTq vanishes on an open interval I. Then snpf q converges to 0 on I.
(ii) Let f, g P L1
pTq and suppose f “ g on an open interval I. Then snpf q and snpgq converge
Example 1.8. The function f pxq “ x2on r´1 2,
1
2q satisfies the hypothesis of Corollary 1.6 (i), so
we know that its Fourier series will converge to f pxq for all x P r´1 2,
1
2s. The Fourier coefficients
can be calculated to pf pnq “ 2πp´1q2nn2 for n ‰ 0 and pf p0q “
1 12. This results in x2“ 1 12` 8 ÿ n“1 p´1qne 2πinx ` e´2πinx 2π2n2 , x P r´ 1 2, 1 2s. If we set x “ 0 we obtain the following identity.
8 ÿ n“1 p´1qn n2 “ ´ π2 12. If we instead set x “ 12 we get
8 ÿ n“1 1 n2 “ π2 6 , which is the solution to the famous Basel problem.
Convolutions and Kernels
The convolution product ˚ is defined by pf ˚ gqpxq “
ż
T
f ptqgpx ´ tq dt. (36)
Unlike pointwise multiplication, this operation is closed in Lp
pTq, by the following. Theorem 1.9. (Young) Let 1 ď p ď 8. Then
}f ˚ g}pď }f }p}g}1. Hence, if f, g P Lp
pTq, then f ˚ g P LppTq and exists almost everywhere on T.
Proof. The cases p “ 1 and p “ 8 are obvious. For general p we use H¨older’s inequality in the following way. |pf ˚ gqpxq| ď ż T |f ptq| |gpx ´ tq| “ ż T |f ptq| |gpx ´ tq| 1 p|gpx ´ tq| 1 q ď }g} 1 q 1 ˆż T |f ptq|p|gpx ´ tq| dt ˙p1 . Now integrate to get
ż T |pf ˚ gqpxq|p dx ď }g} p q 1 ż T ż T |f ptq|p|gpx ´ tq| dt dx “ }g} p q 1 ˆż T |gpx ´ tq| dx ˙ ˆż T |f ptq|p dt ˙ “ }g} p q 1 }g}1}f } p p“ }f } p p}g} p 1.
Hence }f ˚ g}pď }f }p}g}1, and since }g}1ď }g}p we have closure in LppTq.
The following identity shows how convolutions behave with Fourier coefficients. Lemma 1.10. zf ˚ g “ pfpg.
The proof is easily verified using Fubini’s theorem. One may also verify that convolution is com-mutative, associative and distributive (with pointwise addition). Together with Young’s theorem, another way of phrasing this is to say that Lp
pTq is a commutative Banach algebra under point-wise addition and the convolution product. This algebra is not unital, in other words, there is no element e P Lp
pTq such that f ˚ e “ e ˚ f “ f . To see this, assume the contrary. Then fn˚ e “ fn
would be true for the functions fnpxq “ einx. Letting n Ñ 8 pointwise, we get a contradiction by
With the notion of convolutions introduced, we can write the partial sums of a Fourier series as
snpf q “ f ˚ Dn. (37)
We are interested in determining when f ˚ DnÝÝÝÑ f , pointwise and in norm. Seen from anothernÑ8
perspective, we want to determine in what sense Dn is an ”approximate identity”. So far we have
not made an approach to norm convergence and we have only been able to determine pointwise convergence for functions satisfying some smoothness condition. The next definition provides a more general approach to determining when f ˚ KnÝÝÝÑ f for a family of functions tKnÑ8 nu8n“0.
Definition 1.11. A summability kernel is a family tKnu8n“0of integrable functions on T satisfying
the following three conditions. (i) ş
T|Knptq| dt ď C, for some constant C (for all n).
(ii) ş TKnptq dt “ 1. (iii) ş |x|ěδ|Knptq| dt nÑ8 ÝÝÝÑ 0 for every δ ą 0.
We will also encounter families tKuą0and tKru0ďră1. In these cases n Ñ 8 will be replaced by
Ñ 0` and r Ñ 1´, respectively. This will not change any of the results that will be developed
below. The results will also apply if we integrate over Rn
instead of T (and Young’s inequality still applies as well).
Theorem 1.12. Let f P Lp
pTq, 1 ď p ă 8, and tKnu8n“0 be a summability kernel. Then
f ˚ Kn Ñ f in Lp-norm. That is,
lim
nÑ8}f ˚ Kn´ f }p“ 0.
To prove this we need the following lemma.
Lemma 1.13. The translation operation t ÞÑ ft is continuous on LppTq, 1 ď p ă 8. That is,
lim
tÑt0
}ft´ ft0}p“ 0.
Proof. The result is clear for continuous functions. Since the continuous functions are dense in Lp
pTq we can, given ą 0, choose such a function g such that }f ´ g}pă {3. Now choose δ ą 0
such that }gt´ gt0} ă {3 when |t ´ t0| ă δ. For |t ´ t0| ă δ we have
}ft´ ft0}pď }ft´ gt}p` }gt´ gt0}p` }gt0´ ft0}pă 3 ` 3` 3 “ . Proof of Theorem 1.12. We begin as in (34) and use H¨older’s inequality to write
It suffices to prove that the last integral converges to zero. By Lemma 1.13, given ą 0 there is a δ ą 0 such that }ft´ f }pp ă {2C when |t| ă δ. We split into the following two integrals.
I1“ ż |t|ăδ }ft´ f } p p|Knptq| dt , I2“ ż |t|ěδ }ft´ f } p p|Knptq| dt.
Clearly I1 ă {2. Now, by condition (iii) of summability kernels, we can choose n0 such that
ş
|t|ěδ|Knptq| dt ă {p2
p`1}f }p
pq for n ą n0. Hence, for n ą n0, we have I1` I2 ă {2 ` {2 “ ,
which proves that }f ˚ Kn´ f } p p
nÑ8
ÝÝÝÑ 0, and so we are done.
The following useful conditions provide a more restrictive class of summability kernels. (i*) Knpxq ě 0.
(iii*) sup|x|ěδ|Knpxq| nÑ8
ÝÝÝÑ 0 for every δ ą 0.
It is clear that (i*) together with (ii) imply (i), and that (iii*) implies (iii). With the conditions (i*), (ii) and (iii*) we will be able to prove slightly more powerful results. We now turn to pointwise convergence of f ˚ Kn.
Theorem 1.14. Let f P L1 and tK
nu8n“0be a summability kernel satisfying (iii*).
(i) If f is continuous at x0P T then pf ˚ Knqpx0q nÑ8
ÝÝÝÑ f px0q and the convergence is uniform
on every compact set of continuity.
(ii) If Kn is even and f px0˘q exists then pf ˚ Knqpx0q nÑ8
ÝÝÝÑ f px0`q`f px0´q
2 .
Proof. To prove the first part of the theorem, we write |pf ˚ Knqpx0q ´ f px0q| ď
ż
T
|f px0´ tq ´ f px0q| |Knptq| dt.
Now split the problem into the two integrals, I1“ ż |t|ăδ |f px0´ tq ´ f px0q| |Knptq| dt , I2“ ż |t|ěδ |f px0´ tq ´ f px0q| |Knptq| dt,
where δ is chosen so that I1ă {2 for a given ą 0, which is possible since f is continuous at x0.
For I2, we see that I2 ď 2 }f }1sup|t|ěδ|Knptq| and by (iii*) we can choose n0 such that I2 ă {2
for n ą n0, so the proof is complete. The uniform convergence follows from the fact that every
continuous function on a compact set is uniformly continuous, so we can choose δ independently of x0 on any given compact set of continuity of f . Now, for the second part, since Kn is even, we
can write |pf ˚ Knqpx0q ´ `| ď ż T ˇ ˇ ˇ ˇ f px0` tq ` f px0´ tq 2 ´ ` ˇ ˇ ˇ ˇ|Knptq| dt “ 2 ż 12 0 ˇ ˇ ˇ ˇ f px0` tq ` f px0´ tq 2 ´ ` ˇ ˇ ˇ ˇ |Knptq| dt,
and the rest follows by the same proof as for the first part with ` “ f px0`q`f px0´q
2 .
We remark that if we assume that f is bounded in Theorem 1.14 we can remove (iii*), since a closer look at the proof will reveal that (iii) would be sufficient to bound I2 in this case. In
particular we have uniform convergence of f ˚ Knfor arbitrary summability kernels tKnu8n“0when
f is continuous on T. Also note that the existence of f px0˘q can be generalized to the existence of
q
Examples of Summability
In the previous section we proved some very good convergence properties of f ˚ Knfor summability
kernels Kn. We have already seen that Dn satisfies properties (ii) and (iii) of Definition 1.11.
Unfortunately the Dirichlet kernel is not a summability kernel because it does not satisfy (i), by the following lemma.
Lemma 1.15. }Dn}1 nÑ8
ÝÝÝÑ 8.
Proof. By (31) and sin πx ď |πx| we get ż T |Dnpxq| dx ą 2 ż 12 0 ˇ ˇ ˇ ˇ sin πp2n ` 1qx πx ˇ ˇ ˇ ˇdx “ 2 π żn`12 0 ˇ ˇ ˇ ˇ sin πx x ˇ ˇ ˇ ˇ dx ą 2 π n ÿ j“1 1 j żj j´1 |sin πx| dx “ 4 π2 n ÿ j“1 1 j ą 4 π2logpn ` 1q nÑ8 ÝÝÝÑ 8. If we redo the proof with |sin πx| ě |2x| we get
}Dn}1ă 2 π n`1 ÿ j“1 1 j ă 2 πp1 ` logpn ` 1qq, (38)
which shows that Dn grows to infinity like a constant multiple of log n. By Lemma 1.15 we can
not deduce convergence of f ˚ Dn from Theorem 1.12 and Theorem 1.14, but there are other
modes of convergence where these theorems will be useful. One of them comes from a method discovered by Fej´er which shows that the problem of convergence in arithmetic means can be reduced to studying a certain summability kernel. A sequence txnu8n“1 in some normed linear
space is said to convergence in mean if the sequencex1`...`xn
n converges. A series that converges in
mean is sometimes called Ces`aro summable, and its means are called Ces`aro means. The following proposition is readily verified by the reader.
Proposition 1.16. If txnu8n“1 converges, then it converges in mean to the same value.
Note that the converse is not true. For example, the sequence tp´1qn
u converges to 0 in mean. Hence convergence in mean extends convergence to a larger collection of sequences.
We denote by σnpf ; xq the Ces`aro means of snpf ; xq and proceed to calculate σnpf ; xq.
σnpf q “ 1 n ` 1 n ÿ j“0 sjpf q “ 1 n ` 1 n ÿ j“0 f ˚ Dj “ f ˚ 1 n ` 1 n ÿ j“0 Dj“ f ˚ Fn, where Fn is given by Fn“ 1 n ` 1 n ÿ j“0 Dj. (39)
Lemma 1.17. The Fej´er kernel Fn can be written explicitly as
Fnpxq “ 1 n ` 1 ˆ sin πpn ` 1qx sin πx ˙2 . Proof. Fn is the imaginary part of a geometric series, since
n ÿ j“0 Djpxq “ n ÿ j“0 sin πp2j ` 1qx sin πx “ 1 sin πx n ÿ j“0 Im ´ eπip2j`1qx ¯ .
From here we just sum the geometric series to verify the result. The kernel tFnu8n“0 is called the Fej´er kernel. We see that }Fn}1 “ 1, by (30) and (39). By
Lemma 1.17, Fn is even and positive, and also |Fnpxq| ă pn`1q sin1 2πδ when |x| ě δ, so we have sup|x|ěδ|Fnpxq|ÝÝÝÑ 0 for all δ ą 0. Hence the Fej´nÑ8 er kernel satisfies all the hypothesis of Theorem
1.12 and Theorem 1.14. We restate these theorems for Fn, along with three important corollaries
Theorem 1.18. (Fej´er) Let f P L1
pTq. (i) σnpf q Ñ f in Lp-norm, 1 ď p ă 8.
(ii) If f is continuous at x0P T then σnpf ; x0qÝÝÝÑ f pxnÑ8 0q and the convergence is uniform on
every compact set of continuity.
(iii) If qf px0q “ limtÑ0`f px0`tq`f px2 0´tq exists then σnpf ; x0q nÑ8
ÝÝÝÑ qf px0q.
Corollary 1.19. Let f P L1
pTq. (i) If the Fourier series of f P Lp
pTq converges in Lp-norm, 1 ď p ă 8, then it must converge to f . A similar result holds if we replace Lp
pTq by CpTq.
(ii) If the Fourier series of f converges at x0 P T and x0 is a point of continuity of f , then
it must converge to f px0q. More generally, if qf px0q exists, then the Fourier series must
converge to qf px0q.
Corollary 1.20. Let f, g P L1
pTq. (i) If pf “ 0 then f “ 0 a.e. (ii) If pf “pg then f “ g a.e.
Corollary 1.21. The trigonometric polynomials on T are dense in CpTq and Lp
pTq, 1 ď p ă 8. For all f P L1pTq, σnkpf q Ñ f almost everywhere for some subsequence nk, by Theorem 1.18 (i). It even turns out that the almost everywhere convergence of σnpf q holds for every f P L1pTq, as
stated by the theorem below.
Theorem 1.22. σnpf ; x0qÝÝÝÑ f pxnÑ8 0q if almost everywhere on T.
Proof. It suffices to prove it for every Lebesgue point x0P T, by Lebesgue Differentiation Theorem
(see e.g. [5, Chapter 7]). As always, we write
|σnpf ; x0q ´ f px0q| ď 2 ż 12 0 ˇ ˇ ˇ ˇ f px0` tq ` f px0´ tq 2 ´ f px0q ˇ ˇ ˇ ˇ |Fnptq| dt “ 2I. we split I into I1“ żδ 0 ˇ ˇ ˇ ˇ f px0` tq ` f px0´ tq 2 ´ f px0q ˇ ˇ ˇ ˇ |Fnptq| dt , I2“ ż 12 δ ˇ ˇ ˇ ˇ f px0` tq ` f px0´ tq 2 ´ f px0q ˇ ˇ ˇ ˇ |Fnptq| dt.
For I2, we use |Fnptq| ă pn`1q sin1 2πδ ă
1 pn`1qp2δq2 on rδ, 1 2q to get I2ă 2 }f }1 pn ` 1q sin2πδ ă }f }1 2pn ` 1qδ2 nÑ8 ÝÝÝÑ 0 if pn ` 1qδ2 nÑ8
ÝÝÝÑ 8, hence if we choose δ “ n´14. We split I1 into the integral over r0, n´1s, say I3, and the integral over pn´1, n´
which converges to 0 since x0 is a Lebesgue point. Lastly, for I4we write I4ď 1 4pn ` 1q żn´ 14 n´1 ˇ ˇ ˇ ˇ f px0` tq ` f px0´ tq 2 ´ f px0q ˇ ˇ ˇ ˇ 1 t2dt “ 1 4pn ` 1q żn´ 14 n´1 Φ1ptq t2 dt “ 1 4pn ` 1q Φptq t2 ˇ ˇ ˇ ˇ n´ 14 n´1 ` 1 2pn ` 1q żn´ 14 n´1 Φptq t3 dt.
The first term converges to 0 since x0is a Lebesgue point. For the last term, pick ą 0. Then for
sufficiently large n, we have F ptqt ă , by Lebesgue Differentiation Theorem, and
1 2pn ` 1q żn´ 14 n´1 Φptq t3 dt ă 2pn ` 1q żn´ 14 n´1 1 t2dt “ pn ´ n14q 2pn ` 1q ă 2.
Hence lim supnÑ8|σnpf ; x0q ´ f px0q| ă . Since was arbitrary the result follows.
Corollary 1.23. blabla
(i) If a Fourier series converges on a set E Ă T, then it converges to f almost everywhere on E. (ii) If a Fourier series converges to 0 almost everywhere on T then all its coefficients must be 0. Another useful summability method is the Abel summability of Fourier series. This is described by viewing the Fourier series of f P L1
pTq as the values on the unit circle tz : |z| “ 1u of the function F pzq “ pf p0q ` 8 ÿ j“1 ´ p f pjq ` pf p´jq ¯ zj,
which is holomorphic on the unit disc tz : |z| ă 1u since pf is bounded. If we let z “ re2πix we get
F pre2πixq “ pf p0q ` 8 ÿ j“1 ´ p f pjq ` pf p´jq ¯ rje2πijx“ ż T f ptqPrpx ´ tq dt “ pf ˚ Prqpxq, where Prpxq “ 1 ` 8 ÿ j“1 rje2πijx` 8 ÿ j“1 rje´2πijx“ 1 ` re 2πix 1 ´ re2πix ` re´2πix 1 ´ re´2πix “ 1 ´ r2 1 ´ 2r cos 2πx ` r2
is the Poisson kernel. It is not hard to verify that the Poisson kernel satisfies (i*), (ii) and (iii*) of summability kernels, so Theorem 1.12 and Theorem 1.14 will apply here as well. First, to show (i*), we write Prpxq “ 1 ´ r2 1 ´ 2r cos 2πx ` r2 ě 1 ´ r2 p1 ` rq2 ě 0. To show (ii), we write
ż T Prptq dt “ ż T ˜ 1 ` 8 ÿ j“1 rje2πijt` 8 ÿ j“1 rje´2πijt ¸ dt “ ż T dt “ 1, and for (iii*), when |x| ě δ, we have
Prpxq ď
1 ´ r2 1 ´ 2r cos 2πδ ` r2
rÑ1´
ÝÝÝÝÑ 0.
Hence the statements in Theorem 1.18 also work for the Poisson kernel. The Fourier series of f is said to be Abel summable if f ˚ Pr converges pointwise as r Ñ 1´. More generally, the series
ř8
j“0cj is said to be Abel summable with limit ` if limrÑ1´ř8
j“0cjr j
“ `. A question of interest is if Abel summability extends convergence of series. This turns out to be true, and it even extends Ces`aro summability. We leave the proof for the reader to verify (or see e.g. [2, Chapter VII]). Theorem 1.24. Ifř8
j“0cjis Ces`aro summable, then it is also Abel summable with the same limit.
Convergence in Norm
We now turn to a general discussion of convergence in norm for Fourier series. We begin by proving an equivalent statement using results from functional analysis.
Theorem 1.25. Let 1 ď p ă 8. Then snpf q converges to f for every f P LppTq if and only if the
partial sum operators sn are uniformly bounded, that is, if }snpf q}pď Cp}f }p for some constant
Cp independent of n and f .
Proof. If snpf q converges in Lp-norm then snpf q is bounded in LppTq for each f P LppTq. By
Banach-Steinhaus Theorem (see Appendix II) we immediately get the result. For the converse, given ą 0 choose a trigonometric polynomial ϕ such that }f ´ ϕ}pă {pCp` 1q (this is possible
by Corollary 1.21). For n ą deg ϕ,
}snpf q ´ f }pď }snpf q ´ ϕ}p` }ϕ ´ f }p“ }snpf ´ ϕq}p` }ϕ ´ f }pď pCp` 1q }f ´ ϕ}pă .
We are now interested in the boundedness of the linear operators sn in various Banach spaces. By
(37) and Theorem 1.9, we have
}snpf q}pď }f }p}Dn}1 (40)
when 1 ď p ď 8, so
}sn}LppTqď }Dn}1, (41)
where }sn}LppTq is the operator norm of the partial sum operators sn : LppTq Ñ LppTq. Hence we
know that }sn}LppTq“ Oplog nq. We claim that (41) is an equality in the cases p “ 1 and p “ 8. Lemma 1.26. The following equalities hold:
(i) }sn}L1pTq“ }Dn}1
(ii) }sn}L8pTq“ }Dn}1
(iii) }sn}CpTq“ }Dn}1
Proof. Since }sn}CpTqď }sn}L8pTq, (41) holds for CpTq as well. Thus there is only one inequality left to prove. We begin with (i). First, note that snpFmq “ σmpDnqÝmÑ8ÝÝÝÑ Dn (by Theorem 1.18
(ii)), so for every ą 0 there is an m such that }snpFmq}1ą }Dn}1´ . Hence,
}sn}L1pTq ě }snpFmq}1ą }Dn}1´ ,
so (i) is proved. Next, for L8
pTq, let fnptq “ sgn Dnptq. Clearly }fn}8 “ 1, and fnDn “ |Dn|.
Hence,
}sn}L8pTq ě }snpfnq}8ě |snpfn; 0q| “
ż
T
fnptqDnptq dt “ }Dn}1,
so (ii) is proved. Since the functions fn are not continuous we have not yet proved (iii), but we can
easily fix this problem. Dn has exactly 2n zeros on T and we know that fn is discontinuous only
at these zeros. For each (sufficiently small) ą 0, we change the values of fn in neighbourhoods of
length {2n of each zero such that each fn becomes continuous and }fn}8 remains equal to 1 (e.g.
draw straight line segments). Now,
}sn}CpTqě }snpfnq}8ě |snpfn; 0q| “
ż
T
fnptqDnptq dt ą }Dn}1´ ,
so (iii) is also proved.
By Lemma 1.15 we see that each of these norms are unbounded and so, by Theorem 1.25 and Lemma 1.26, snpf q does not converge to f for every f P L1pTq. If we examine the proof of
Theo-rem 1.25 we see that the first direction will hold in any Banach space of integrable functions on T. Hence we can also conclude that norm convergence does not hold in L8
pTq and CpTq (however, the case L8
information we need is that the trigonometric polynomials are dense in the space. In particular, this means that Theorem 1.25 holds for CpTq as well.
Part (iii) of Lemma 1.26 is particularly interesting. In its proof we saw that the bounded linear functionals f ÞÑ snpf ; 0q are not uniformly bounded on CpTq. By Banach-Steinhaus Theorem,
sup
n
|snpf ; 0q| “ 8
for some f P CpTq. In other words, there is a continuous function whose Fourier series diverges at 0, even a whole dense Gδ of continuous functions (see Appendix II). We get the same result for
any x P T if we redefine fn in the proof of Lemma 1.26 by fnptq “ sgn Dnpx ´ tq. We have proved
the following.
Theorem 1.27. Given x P T there is a dense Gδ-set Ex of continuous functions on T whose
Fourier series diverge at x.
The complement of a dense Gδ is meagre (a countable union of nowhere dense sets). Hence
divergence at a given point is in some sense typical. We may take this even further. Pick out countably many points xjP T. The set E “Ş8j“1Exj is a dense Gδ, by Baire Category Theorem. If we look at the set of points where the Fourier series of f P E diverges unboundedly, we see that
" x : sup n |snpf ; xq| “ 8 * “ 8 č m“1 8 ď n“1 tx : |snpf ; xq| ą mu .
Since snpf ; xq is continuous, this is a Gδ-set, and we may choose the points xj such that the set is
dense. In this case it turns out that the Fourier series must diverge at more points.
Theorem 1.28. In a complete metric space with no isolated points every dense Gδ is uncountable.
Proof. Suppose txju8j“1is a countable dense Gδ. Then it is equal to
Ş8
i“1Ui for some open dense
sets Ui. Let Vi “ Ui´
Ťi
j“1txju. Then each Vi is also open and dense, and
Ş8
i“1Vi “ ∅, which
contradicts Baire Category Theorem.
The Hilbert Transform on T
The Hilbert transform on T is given by the principal value convolution Hf pxq “ p.v.
ż
T
f ptq
tan πpx ´ tqdt “ limÑ0`Hf pxq, (42) where the truncated Hilbert transform Hf is defined by
Hf pxq “
ż
|x´t|ě
f ptq
tan πpx ´ tqdt. (43)
We can write Hf “ f ˚ K, where Kpxq “ tan πx1 1t|t|ěupxq. Since K is bounded for all ą 0,
Theorem 1.9 implies that H is a bounded linear operator on LppTq, 1 ď p ď 8. The most
important result regarding the Hilbert transform is that H is a bounded linear operator, stated in the following theorem.
Theorem 1.29. (M. Riesz) There is a bounded linear operator H : LppTq Ñ LppTq, 1 ă p ă 8, such that }Hf ´ Hf }pÑ 0 as Ñ 0` for all f P L
p
pTq. In fact, if }Hf ´ Hf }p
Ñ0`
ÝÝÝÝÑ 0, then H is bounded. To see this, first note that it would imply that sup}Hf }pă 8 for every f P LppTq. By Banach-Steinhaus Theorem, we have sup}H}LppTqď M for some M . Hence, }Hf }p ď }Hf ´ Hf }p` }Hf }p, and taking the limit we see that }Hf }p ď
M }f }p. The goal of chapter 2 will be to prove Theorem 1.29. We will do this by first proving
that H exists as the pointwise almost everywhere limit of Hf , for all f P L1pTq, then prove that
this linear operator is bounded on Lp
Theorem 1.30. Suppose there is a linear operator H on Lp
pTq such that }Hf ´ Hf }p Ñ0` ÝÝÝÝÑ 0 for all f P LppTq, 1 ď p ď 8. Then H is a multiplier operator on LppTq with multiplier ´i sgn pjq. That is,
y
Hf pjq “ ´i sgn pjq pf pjq. Proof. By Lemma 1.10, yHf “ xKf . We now compute xp K.
x Kpjq “ ż |t|ě 1 tan πte ´2πijtdt “ ´i ż |t|ě sin 2πjt cos πt sin πt dt.
Since we see that xKp´jq “ ´ xKpjq and xKp0q “ 0 we only need to consider positive j. By (31),
we can rewrite ż |t|ě sin 2πjt cos πt sin πt dt “ ż |t|ě pDjptq ´ cos 2πjtq dt.
By Dominated Convergence Theorem, lim Ñ0` ż |t|ě pDjptq ´ cos 2πjtq dt “ ż T pDjptq ´ cos 2πjtq dt “ 1. Hence xKpjq Ñ0`
ÝÝÝÝÑ ´i sgn pjq. We also have ˇ ˇ ˇHyf pjq ´ yHf pjq ˇ ˇ ˇ ď }Hf ´ Hf }p Ñ0` ÝÝÝÝÑ 0, so yHf pjq Ñ0`
ÝÝÝÝÑ yHf pjq. Thus yHf pjq “ ´i sgn pjq pf pjq and we are done. Definition 1.31. Let B be a Banach space of integrable functions on T. B is said to admit conjugation if for every f P B, there is a function rf P B, such that
r f „
8
ÿ
j“´8
´i sgn pjq pf pjqe2πijx.
The series is called the conjugate Fourier series of f P L1
pTq and rf is called the conjugate function of f .
Lemma 1.32. If B admits conjugation and }¨}1ď }¨}B then f ÞÑ rf is a bounded linear operator.
Proof. Suppose fn Ñ f and rfn Ñ g in B. By Closed Graph Theorem, it is sufficient to prove
g “ rf . This follows by Corollary 1.20 (ii), since p gpjq “ lim nÑ8 x Ă fnpjq “ ´i sgn pjq lim nÑ8xfnpjq “ ´i sgn pjq pf pjq “ pf pjq.r The property of conjugation gains interest when we find out that it is related to norm convergence of Fourier series.
Theorem 1.33. Norm convergence of Fourier series in LppTq, 1 ď p ă 8 holds if and only if LppTq admits conjugation.
Proof. If LppTq admits conjugation then there is a function g P LppTq such that g „ř8j“0f pjqep 2πijx, and g “ 1
2f p0q `p 1 2f `
i
2f .r Conversely, if such a g exists then L
p admits conjugation since
r
f “ ´2ig ` if ` i pf p0q. Also note that f ÞÑ g is a bounded linear operator, by Lemma 1.32. If such a g exists we say that LppTq admits projections and we denote its partial sums by Pnpf q. The
strategy will be to show that LppTq admits projections if and only if the operators snare uniformly
bounded. From there the result follows by Theorem 1.25 and the previous remarks. First, suppose }snpf q}pď Cp}f }p independent of n and f . By Lemma 1.3 (iv) we see that
By assumption we get }P2npf q}p ď Cp}f }p. Given f P LppTq, pick a polynomial ϕ such that
}f ´ ϕ}pď {2Cp. Then
}P2npf q ´ P2npϕq}pď Cp}f ´ ϕ}pď {2.
For n, m ě 12deg ϕ we now get
}P2npf q ´ P2mpf q}pď }P2npf q ´ P2npϕq}p` }P2mpϕq ´ P2mpf q}pď .
Hence P2npf q is a Cauchy sequence in LppTq and thus it converges to some g P LppTq. Hence
p
gpjq “ limnÑ8P{2npf qpjq “ pf pjq1tjě0upjq. In other words, g „ ř8
j“0f pjqep
2πijx. Now for the
other direction, assume g P LppTq satisfiespgpjq “ pf pjq1tjě0upjq. Then, as previously discussed, there is constant Cp such that }g}p ď Cp}f }p. Similarly, there is an h P L
p
pTq such that phpjq “ p
f pj ` 2n ` 1q1tjě0upjq and }h}pď Cp}f }p, since by Lemma 1.3 (iv), pf pj ` 2n ` 1q are the Fourier
coefficients of e´2πip2n`1qxf pxq (which is an Lp-function). Now,
P2npf ; xq “ gpxq ´ e2πip2n`1qxhpxq,
which can be seen by calculating the Fourier coefficients of both sides. This shows that }P2npf q}pď
2Cp}f }p. We also have
snpf ; xq “ e´2πinxP2npe2πinxf ; xq.
Hence }snpf q}pď 2Cp}f }p, so the theorem is proved.
Once Theorem 1.29 is proved, Theorem 1.30 and Theorem 1.33 prove norm convergence of Fourier series in Lp
pTq, 1 ă p ă 8. Lemma 1.26 has already ruled out L1pTq and CpTq. Hence L1pTq does not admit conjugation, and the proof of Theorem 1.33 applies to CpTq just as well, so CpTq also does not admit conjugation. By Theorem 1.30, Hf cannot converge in norm in L1pTq and CpTq.
Fourier series in L
2We will now solve the problem of norm convergence in L2
pTq. This space turns out to be the best suited for studying Fourier series. Recall that L2
pTq is a Hilbert space under the inner product xf, gy “
ż
T
f ptqgptq dt.
Let ϕjpxq “ e2πijx. The set tϕjujPZ is orthonormal, by (25), and its linear span is the set of
trigonometric polynomials on T. Let ϕ “ řn
j“´ncjϕj be a trigonometric polynomial. Then
cj“ xϕ, ϕjy by (3), and by Pythagoras Theorem,
}ϕ}22“ n ÿ j“´n |xϕ, ϕjy| 2 .
With the inner product notation we may rewrite the partial sums snpf q as
snpf q “ n
ÿ
j“´n
xf, ϕjy ϕj. (44)
Hence snpf q is the orthogonal projection of f onto the subspace spanned by tϕju n
j“´n. That is,
snpf q is the best approximation of f by trigonometric polynomials of degree n. Moreover, by
Pythagoras Theorem, }snpf q ´ f } 2 2“ }f } 2 2´ n ÿ j“´n |xf, ϕjy| 2 , (45)
so we get the following inequality of Bessel,
Lemma 1.34. Suppose fnÑ f in L2pTq. Then for every g P L2pTq,
lim
nÑ8xfn, gy “ xf, gy
Proof. By Cauchy-Schwartz inequality,
|xfn, gy ´ xf, gy| ď }fn´ f }2}g}2Ñ 0.
Lemma 1.35. Let f P L2pTq. Suppose f “ř8j“´8cjϕj in L2-norm. Then cj“ pf pjq.
Proof. By Lemma 1.34, p f pjq “ xf, ϕjy “ lim nÑ8 C n ÿ k“´n ckϕk, ϕj G “ cj. Theorem 1.36. (Riesz-Fischer) Supposeř8
j“´8|cj| 2
ă 8. Then there is an f P L2pTq such that f “ř8
j“´8cjϕj in L
2-norm, and c
j “ pf pjq.
Proof. Given ą 0, choose N such thatř
|j|ěN|cj| 2 ă . Let sn “ řn j“´ncjϕj. For n ą m ą N , }sn´ sm} 2 2“ ÿ mă|j|ďn |cj| 2 ă .
Hence sn converges to some f P L2pTq, and by Lemma 1.35, cj“ pf pjq.
By Bessel’s inequality,ř8
j“´8|xf, ϕjy| 2
ă 8 for all f P L2pTq, so by Riesz-Fischer Theorem there is a g P L2pTq such that g “ ř8j“´8xf, ϕjy ϕj and xg, ϕjy “ xf, ϕjy. Hence g “ f by Corollary
1.20 (ii), so we have proved norm convergence of Fourier series in L2, and by (45) we get the following identity of Parseval,
8
ÿ
j“´8
|xf, ϕjy|2“ }f }22.
More generally we have the following result. Theorem 1.37. (Parseval) Let f, g P L2
pTq. Then xf, gy “ 8 ÿ j“´8 xf, ϕjy xg, ϕjy Proof. By Lemma 1.34, xf, gy “ lim nÑ8 C n ÿ j“´n xf, ϕjy ϕj, g G “ 8 ÿ j“´8 xf, ϕjy xg, ϕjy. These results on Fourier series in L2are summarized by the following Theorem.
Theorem 1.38. The map f ÞÑ pf is a Hilbert space isomorphism between L2
pTq and `2pZq. Proof. It is injective by Corollary 1.20 (ii), surjective by Riesz-Fischer Theorem, and preserves
Boundedness of the Hilbert Transform
The goal of this chapter is to prove the boundedness of the Hilbert transform in LppTq, 1 ă p ă 8. There are a few different proofs of this, and while the one presented in this chapter is not necessarily the most elementary one, the same methods will also be useful for proving boundedness properties of more general singular integral operators.
The Calder´
on-Zygmund Decomposition
We begin the chapter with a decomposition method which will be useful to us later. We fix a number λ ą x|f |y, where xf yE “ mpEq1 şEf is the average value of f on E Ă T. If E “ T we may write xf y instead of xf yT. Subdivide T “ r´1
2, 1 2q into I “ p´ 1 2, 0q and J “ p0, 1 2q. Note that x|f |yI` x|f |yJ “ 2 ż I |f | ` 2 ż J |f | “ 2 ż T |f | “ 2 x|f |y ă 2λ,
so at least one of x|f |yI and x|f |yJ are still greater than λ. We also have x|f |yI ă 2λ as well as x|f |yJ ă 2λ. If x|f |yI ą λ we subdivide I into two new open intervals by removing the middle point
of I, and if λ ď x|f |yI ă 2λ we separate I. Do the same for J . We now treat the new intervals
similarly, so we get an infinite process going, which separates a countable collection of intervals tQju, with λ ď x|f |yQj ă 2λ, and ÿ j mpQjq ď ÿ j 1 λ ż Qj |f | ď 1 λ ż T |f | .
We now define a function g by
gpxq “ f pxq1pŤ jQjq cpxq ` ÿ j xf yQ j1Qjpxq. Clearly gpxq ă 2λ when x PŤ
jQj. We are interested in a similar bound for other x. If x R
Ť
jQj
and x is not an endpoint of any of the intervals then there is a sequence of intervals Ij Q x with
limjÑ8diam Ij“ 0 and
Ş
jIj“ txu, such that for all j,
1 mpIjq
ż
Ij
|f | ă λ.
Letting j Ñ 8 we get |f pxq| ď λ almost everywhere on ´
Ť
jQj
¯c
, by Lebesgue Differentiation Theorem. Hence }g}8ď 2λ. From this we conclude }g}ppď p2λqp´1}f }1, since
ż T |g|pď p2λqp´1 ż T |g| ď p2λqp´1 ż T |f | “ p2λqp´1}f }1.
Now we define a function b by
bpxq “ f pxq ´ gpxq “ÿ j ´ f pxq ´ xf yQj ¯ 1Qjpxq. Clearly the support of b lies inŤjQj and xbyQj “ 0 for all j. We also have
x|b|yQ j “ 1 mpQjq ż Qj |b| ď 2 mpQjq ż Qj |f | ă 4λ, and }b}1ď 2 }f }1.
We have now constructed the Calder´on-Zygmund decomposition f “ g ` b for f P L1
pTq at height λ ą x|f |y. The function g is referred to as the good part of f , while b is the bad part. We saw that g has good boundedness properties, while b contains the part of f where the function behaves worse. However, the construction ensures that b has average value 0 over each of our intervals Qj,
Theorem 2.1. (Calder´on-Zygmund Decomposition) Suppose f P L1
pTq and λ ą x|f |y. Then there is a decomposition f “ g ` b and a sequence of open and disjoint dyadic intervals tQju such
that
(i) λ ď x|f |yQj ă 2λ for all j.
(ii) mpŤ jQjq ď 1 λ ş Ť jQj|f |. (iii) |f pxq| ď λ for almost every x P
´ Ť jQj ¯c . (iv) gpxq “ f pxq1pŤ jQjq cpxq `ř jxf yQj1Qjpxq. (v) }g}8ď 2λ. (vi) }g}ppď p2λqp´1}f }1, 1 ď p ă 8. (vii) bpxq “ř j ´ f pxq ´ xf yQj ¯ 1Qjpxq. (viii) şQ jb “ 0 for all j. (ix) x|b|yQ j ă 4λ for all j. (x) }b}1ď 2 }f }1.
The Hilbert Transform in L
2We can apply the results about Fourier series in L2 to derive properties of the Hilbert transform.
These results will play an important part in the proof of Theorem 1.29.
Theorem 2.2. There is a constant C such that }Hf }2ď C }f }2 for all f P L2pTq and ą 0.
Proof. By Parseval, }Hf }2“ › › ›Hyf › › › 2“ supj ˇ ˇ ˇKxpjq ˇ ˇ ˇ › › ›fp › › › 2“ supj ˇ ˇ ˇKxpjq ˇ ˇ ˇ }f }2.
Using |tan πx| ě |πx|, we get ˇ ˇ ˇKxpjq ˇ ˇ ˇ “ ˇ ˇ ˇ ˇ ˇ ż |t|ě sin 2πjt tan πt dt ˇ ˇ ˇ ˇ ˇ ď ˇ ˇ ˇ ˇ ˇ ż |t|ě sin 2πjt πt dt ˇ ˇ ˇ ˇ ˇ ď 2 π ˇ ˇ ˇ ˇ ˇ żπ|j| 2π|j| sin s s ds ˇ ˇ ˇ ˇ ˇ . Now, a simple argument using integration by parts shows that
ˇ ˇ ˇ ˇ ˇ żb a sin s s ds ˇ ˇ ˇ ˇ ˇ ď 4 for all a, b ě 0. Hence sup}Hf }2ď
8 π}f }2. Theorem 2.3. › › ›Hf ´ rf › › › 2 Ñ0` ÝÝÝÝÑ 0 for all f P L2pTq.
Proof. We begin by proving it for trigonometric polynomials ϕpxq “ řn
j“´ncje2πijx. First, it
follows pointwise, since Hϕpxq “ ż |t|ě ϕpx ´ tq tan πt dt “ n ÿ j“´n cje2πijx ż |t|ě e´2πijt tan πt dt Ñ0` ÝÝÝÝÑ n ÿ j“´n ´i sgn pjq cje2πijx“ϕpxq,r
where the limit is taken of the same integral as in the calculation in the proof of Theorem 1.30. Hϕ is bounded independent of , since
By Dominated Convergence Theorem we have }Hϕ ´ϕ}r 2 Ñ0`
ÝÝÝÝÑ 0. From here, it follows by a simple density argument. For each trigonometric polynomial ϕ, we write
› › ›Hf ´ rf › › ›2ď }Hf ´ Hϕ}2` }Hϕ ´ϕ}r 2` › › › rϕ ´ rf › › ›2.
The first term is bounded by C }f ´ ϕ}2, where C is as in Theorem 2.2, and by Parseval the last term equals }ϕ ´ f }2´
ˇ ˇ ˇpϕ ´ f qp0q{ ˇ ˇ ˇ 2
ď }ϕ ´ f }. Now, given η ą 0, choose small enough so that the second term is less than η{2, and choose ϕ such that }f ´ ϕ}2 ă η{2pC ` 1q. Then
› ›
›Hf ´ rf › ›
›2ă η and the proof is complete.
In the proof above we derived a result worth stating as a separately. Lemma 2.4. When 1 ď p ă 8 we have }Hϕ ´ϕ}r p
Ñ0`
ÝÝÝÝÑ 0 for all trigonometric polynomials ϕ. We define Hf “ rf for all f P L2
pTq. By Theorem 2.3 and Theorem 1.30 we now know that y
Hf pjq “ ´i sgn pjq pf pjq. From this we can easily derive many properties of Hf for f P L2pTq (and also f P LppTq, 1 ă p ă 8, once we have proved the general boundedness result). For example, we can obtain an inversion formula for the Hilbert transform in Lp
pTq, 1 ă p ă 8, by the following calculation. z HHf pjq “ ´i sgn pjq yHf pjq “ p´i sgn pjqq2 p f pjq “ " 0 j “ 0 ´ pf pjq j ‰ 0. (47) By Corollary 1.20 (ii) we get the inversion formula for H,
HHf “ xf y ´ f a.e. (48)
We can define a new linear operator rH by r
Hf “ i xf y ` Hf. (49)
Clearly rH is bounded if and only if H is bounded. By the calculation in (47), z
r
H rHf “ ´ pf , (50)
and the we thus get the nicer inversion formula, r
H rHf “ ´f a.e. (51)
By similar arguments we may derive the following selected properties of H (under suitable condi-tions on f and g).
Hf “ Hf , (52)
Hpf ˚ gq “ Hf ˚ g, (53)
Hf1“ pHf q1. (54)
Lemma 2.5. rH is a unitary operator on L2
pTq. Proof. By Parseval, A r Hf, rHgE“ 8 ÿ j“´8 x Ă Hf pjqxHgpjq “Ă 8 ÿ j“´8 p f pjqpgpjq “ xf, gy . Using the inversion formula, we may write
A r Hf, gE“
A
f, ´ rHgE. (55)
In other words, rH˚“ rH´1“ ´ rH. By (49) we get
xHf, gy “ ´ xf, Hgy , (56)
so H˚
Lemma 2.6. For f, g P L2 pTq, we have ż T Hf ptqgptq dt “ ´ ż T f ptqHgptq dt.
We can also prove the same identity if we replace H with H, since by Parseval’s identity and
Lemma 1.3 (ii) we get ż T Hf ptqgptq dt “ 8 ÿ j“´8 x Hpjqpgpjq “ 8 ÿ j“´8 x Kpjq pf pjqpgp´jq “ ´ 8 ÿ j“´8 p f pjq xKp´jqpgp´jq “ ´ 8 ÿ j“´8 p f pjq yHgp´jq “ ´ ż T f ptqHgptq dt.
We finish the section by proving that our definition of the Hilbert transform in L2
pTq agrees with the pointwise one.
Theorem 2.7. Hf Ñ0`
ÝÝÝÝÑ rf almost everywhere for all f P L2pTq. Proof. Since L2
pTq admits conjugation we know that the conjugate Fourier series of f is Abel summable to rf almost everywhere, and we have
´i 8 ÿ j“1 ´ p f pjq ´ pf p´jq¯rje2πijx“ ż T f ptqQrpx ´ tq dt “ pf ˚ Qrqpxq, where Qrpxq “ ´i ˜8 ÿ j“1 rje2πijx´ 8 ÿ j“1 rje´2πix ¸ “ ´ire 2πijx 1 ´ re2πix ´ ´ire´2πix 1 ´ re´2πix “ 2r sin 2πx 1 ´ 2r cos 2πx ` r2
is the conjugate Poisson kernel. From here, we are done if we can prove that pf ˚ Q1´qpxq ´ Hf pxq Ñ0
`
ÝÝÝÝÑ 0 a.e. We prove this for all f P L1
pTq. First, since Q1pxq “ 1{ tan πx, we can write
pf ˚ Q1´qpxq ´ Hf pxq “ ż |t|ă f px ´ tqQ1´ptq dt ` ż |t|ě f px ´ tq pQ1´ptq ´ Q1ptqq dt.
We begin by proving convergence of the first integral. Note that we have the estimate |Qrpxq| “ ˇ ˇ ˇ ˇ 2r sin 2πx 1 ´ 2r cos 2πx ` r2 ˇ ˇ ˇ ˇ ď 4πr |x| p1 ´ rq2.
Hence Q1´pxq ď 8πp1 ´ q |x| {2ă 4π{ when |x| ă . Now, since Q1´ is odd, we can write
ˇ ˇ ˇ ˇ ˇ ż |t|ă f px ´ tqQ1´ptq dt ˇ ˇ ˇ ˇ ˇ “ ˇ ˇ ˇ ˇ ˇ ż |t|ă pf px ´ tq ´ f pxqq Q1´ptq dt ˇ ˇ ˇ ˇ ˇ ď 4π ż |t|ă |f px ´ tq ´ f pxq| dt, so the first integral converges to 0 at every Lebesgue point. To prove convergence of the second integral, an easy calculaton shows that
Q1´pxq ´ Q1pxq “ 2p1 ´ q sin 2πx 1 ´ 2p1 ´ q cos 2πx ` p1 ´ q2 ´ 2 sin 2πx 2 ´ 2 cos 2πx “ ´ 2Q1pxqP1´pxq, and when |x| ě we have
To estimate P1´pxq, we use that 1 ´ cos 2πx “ 2 sin2πx ě 8x2and write Prpxq “ 1 ´ r2 1 ´ 2r cos 2πx ` r2 ď 1 ´ r2 2rp1 ´ cos 2πxqď 1 ´ r 8x2 .
Hence P1´pxq ď {p8p1 ´ qx2q, so it suffices to prove the convergence of
ż
|t|ě
|f px ´ tq ´ f pxq| 1 t2dt,
which we showed converges to 0 at every Lebesgue point in the proof of Theorem 1.22, so we are
done.
Existence of the Hilbert Transform
We now extend our definition of the Hilbert transform to all f P L1pTq, by the following theorem. Theorem 2.8. The Hilbert transform, H, exists as the pointwise almost everywhere limit of Hf
for all f P L1
pTq. That is,
lim
Ñ0`Hf “ Hf a.e.
Proof. We apply the Calder´on-Zygmund decomposition with λ ą x|f |y. Let g, b, and Qj be as in
Theorem 2.1, and let Ω “Ť
j2Qj, where 2Qjis the interval concentric with Qjof measure 2mpQjq.
As g P L2
pTq, Theorem 2.7 shows that Hg converges almost everywhere in T, so it suffices to prove
the almost everywhere convergence of Hb. It even suffices to prove almost everywhere convergence
of Hb on Ωc, since then the same holds on
Ť kΩ c k “ p Ş kΩkq c
, where Ωk are the sets where the
Qj’s correspond to λk. We can let λkÑ 8, and then
Ş
kΩk has measure zero, by (ii) of Theorem
2.1. The problem is now reduced to showing that Hbpxq is Cauchy for almost every x P Ωc. So
we pick η ą 0 and assume ą δ ą 0. First we write |Hbpxq ´ Hδbpxq| ď ż px´,x´δq ˇ ˇ ˇ ˇ bptq tan πpx ´ tq ˇ ˇ ˇ ˇ dt ` ż px`δ,x`q ˇ ˇ ˇ ˇ bptq tan πpx ´ tq ˇ ˇ ˇ ˇ dt.
We show that the second term is less than η{2 for sufficiently small and δ, and the exact same methods will apply to show the same for the first term. Since the support of b lies inŤjQj, we
only need to consider the integral over the part some Qj intersects. There may be an interval
intersecting x ` δ, and an interval intersecting x ` , but only one interval for each of the points since the Qj’s are disjoint. We first single out the integral over Q X px ` δ, x ` q, where Q is the
interval containing x ` δ. Note that mpQq ă 2δ, since if mpQq ě 2δ then x P 2Q, which contradicts that x P Ωc. Hence Q Ă px, x ` δ ` mpQqq Ă px, x ` 3δq, so Q X px ` δ, x ` q Ă px ` δ, x ` 3δq. From here we can write
ż px`δ,x`qXQ ˇ ˇ ˇ ˇ bptq tan πpx ´ tq ˇ ˇ ˇ ˇdt ď ż px`δ,x`3δq ˇ ˇ ˇ ˇ bptq πpx ´ tq ˇ ˇ ˇ ˇdt ď 1 πδ ż px`δ,x`3δq |bptq| dt “ 1 πδ ż pδ,3δq |bpx ` tq| dt “ 1 πδ ż pδ,3δq |bpx ` tq ´ bpxq| dt, where we usedˇˇtan πt1
ˇ ˇď
ˇ ˇπt1
ˇ
ˇ in the third inequality and bpxq “ 0 in the last equality. Hence we see that this integral is less than η{6 for sufficiently small δ ą 0 when x P Ωc is a Lebesgue point,
and in the case when x ` P Q we get similar bounds with the same methods. Hence, for small enough and δ we have
ˇ ˇ ˇ ˇ ˇ ż px`δ,x`q bptq tan πpx ´ tqdt ˇ ˇ ˇ ˇ ˇ ăη 6` η 6 ` ÿ QjĂpx`δ,x`q ˇ ˇ ˇ ˇ ˇ ż Qj bptq tan πpx ´ tqdt ˇ ˇ ˇ ˇ ˇ .
Note that the collection of all Qj Ă px ` δ, x ` q depends on and δ. We prove that the
sum converges almost everywhere independent of the collection. Let xj be the center point of
If |x ´ xj| ď mpQjq{2 then x P 2Qj, so |x ´ xj| ą mpQjq{2. Hence |x ´ t| ď |x ´ xj| ` |xj´ t| ď
|x ´ xj| ` mpQjq ă 3 |x ´ xj|, and similarly |x ´ xj| ď 3 |x ´ t|. Using this and 2 |x| ď sin πx ď
π |x| we get ˇ ˇ ˇ ˇ ˇ ż Qj bptq tan πpx ´ tqdt ˇ ˇ ˇ ˇ ˇ ď ż Qj bptqπ |t ´ xj| 2 |x ´ t| 2 |x ´ xj| dt ď 3πmpQjq 8 |x ´ xj|2 ż Qj |bptq| dt ď 3πmpQjq 8 |x ´ xj|2 ż Qj |f ptq| dt. It suffices to conclude the convergence of
∆pf ; xq “ÿ j mpQjq |x ´ xj| 2 ż Qj |f ptq| dt
for almost every x P Ωc. We do this by proving that ∆pf ; xq is integrable on Ωc. Interchanging
the order of integration we have ż Ωc ∆pf ; xq dx “ÿ j ż Ωc mpQjq |x ´ xj| 2dx ż Qj |f ptq| dt,
and to bound the first integral we write ż Ωc mpQjq |x ´ xj| 2dx ď ż p2Qjqc mpQjq |x ´ xj| 2dx ď ż yěmpQjq{2 mpQjq y2 dy ď 2. Thus we have ż Ωc ∆pf ; xq dx ď 2ÿ j ż Qj |f ptq| dt ď 2 }f }1,
and the sum converges almost everywhere. Now, for some n we have ÿ j“n`1 3πmpQjq 8 |x ´ xj| 2 ż Qj |f ptq| dt ă η 6.
When is small enough the first n intervals always lie outside of px ` δ, x ` q. Then ÿ QjĂpx`δ,x`q ˇ ˇ ˇ ˇ ˇ ż Qj bptq tan πpx ´ tqdt ˇ ˇ ˇ ˇ ˇ ď 8 ÿ j“n`1 12πmpQjq |x ´ xj| 2 ż Qj |f ptq| dt ă η 6,
and the proof is complete.
The Marcinkiewicz Interpolation Theorem
We have already seen that H is not Cauchy in L1pTq. In general, Hf is not integrable (as can
be seen for example by calculating the Hilbert transform of the indicator function of an interval). However, we can prove that H does map L1pTq into a certain larger space, introduced through the definition below.
Definition 2.9. Let X be a measure space with measure µ. The space weak Lp
pX, µq, 1 ď p ă 8, denoted Lp,8
pX, µq, is the space of all measurable functions f : X Ñ C, for which there is a constant C with
λpµptx P X : |f pxq| ą λuq ď Cp, λ ą 0.
The weak Lp-norm of f , denoted }f }
1,8, (which is not a norm) is defined as the infimum over all
the C’s. Hence we have
}f }1,8“ sup
λą0
λµptx P X : |f pxq| ą λuqp1. In the case p “ 8 we define weak L8
A weak Lp-function, p ă 8, need not even be locally in Lp, which can be verified for the functions
1{x1{p. On the other hand, we have Lp
pX, µq Ă Lp,8pX, µq, by the well-known Chebyshev’s inequality, µptx P X : |f pxq| ą λuq ď 1 λp ż X |f pxq|p dµpxq. (57)
More generally, suppose T : Lp1pX, µq Ñ Lp2pY, νq, p
2ă 8, is a bounded linear operator, that is,
there is a constant C such that }T f }p2 ď C }f }p1 for all f P Lp1pX, µq. Then we also have the weak bound }T f }p
2,8ď C }f }p1, with the same C in both bounds. That is, λνptx P X : |T f pxq| ą λuqp21 ď C }f }
p1 for every λ ą 0 and every f P Lp1pX, µq. To see this, we write
νptx P X : |T f pxq| ą λuq “ ż t|T f pxq|ąλu dνpxq ď ż t|T f pxq|ąλu |T f pxq|p2 λp2 dνpxq ď 1 λp2 }T f } p2 p2 ď Cp2 λp2 }f } p2 p1. (58)
Note that if T is the identity operator on Lp
pX, µq, p ă 8, then this is just Chebyshev’s inequality. The weak Lp-spaces are mainly of interest because of the theorem below, which takes us a great
step further towards proving the Lp-boundedness of the Hilbert transform.
Theorem 2.10. (Marcinkiewicz) Suppose pX, µq and pY, νq are measure spaces, X is σ-finite, 1 ď p1 ă p2 ď 8, and T is a sublinear operator Lp1pXq Ñ Lp1,8pY q and Lp2pXq Ñ Lp2,8pY q
satisfying
}T f }p1,8ď C1}f }p1(all f P Lp1pXqq , }T f }p2,8ď C2}f }p2(all f P Lp2pXqq
for some constants C1, C2. Then T is also a sublinear operator LppXq Ñ LppY q for all p between
p1 and p2, satisfying }T f }pď C }f }p, where
C “ 2 ˆ p p ´ p1 ` p p2´ p ˙1p C 1 p´ 1p2 1 p1´ 1p2 1 C 1 p1´ 1p 1 p1´ 1p2 2 .
The proof of Theorem 2.10 requires the following lemma.
Lemma 2.11. Let pX, µq be a σ-finite measure space. Then we have the identity }f }pp“ p ż8 0 λp´1mptx P X : |f pxq| ą λuq dλ for every f P Lp pXq, 1 ď p ă 8. Proof. p ż8 0 λp´1mptx P X : |f pxq| ą λuq dλ “ ż8 0 pλp´1 ż X 1t|f pxq|ąλupxq dµpxq dλ “ ż X ż8 0 pλp´11t|f pxq|ąλupxq dλ dµpxq “ ż X ż|f pxq| 0 pλp´1dλ dµpxq “ ż X |f pxq|pdµpxq “ }f }pp. Note that the change of order of integration in the calculation above is why we need to assume X is σ-finite.
Proof of Theorem 2.10. Fix λ ą 0 and f P Lp
pXq, p1 ă p ă p2. Consider the decomposition
choose properly. Note that f1 P Lp1pXq and f2P Lp2pXq. By subadditivity |T f | ď |T f1| ` |T f2|, so we have tx P X : |T f pxq| ą λu Ă " x P X : |T f1pxq| ą λ 2 * Y " x P X : |T f2pxq| ą λ 2 * , which implies mptx P X : |T f pxq| ą λuq ď mp " x P X : |T f1pxq| ą λ 2 * q ` mp " x P X : |T f2pxq| ą λ 2 * q. We have to separate the cases p2 ă 8 and p2“ 8. We begin by assuming p2 ă 8. By (58), we
have mp " x P X : |T f1pxq| ą λ 2 * q ď 2 p1Cp1 1 λp1 }f1} p1 p1 , mp " x P X : |T f2pxq| ą λ 2 * q ď 2 p2Cp2 2 λp2 }f2} p2 p2. Now combine this with Lemma 2.11, and write
}T f }ppď p ż8 0 λp´1mp " x P X : |T f1pxq| ą λ 2 * q dλ ` p ż8 0 λp´1mp " x P X : |T f2pxq| ą λ 2 * q dλ ď p2p1Cp1 1 ż8 0 λp´1´p1}f 1} p1 p1 dλ ` p2 p2Cp2 2 ż8 0 λp´1´p2}f 2} p2 p2 dλ “ p2p1Cp1 1 ż8 0 λp´1´p1 ż t|f pxq|ąαλu |f pxq|p1 dµpxq dλ ` p2p2Cp2 2 ż8 0 λp´1´p2 ż t|f pxq|ďαλu |f pxq|p2 dµpxq dλ “ p2p1Cp1 1 ż X |f pxq|p1 ż |f pxq|α 0 λp´1´p1dλ dµpxq ` p2p2Cp2 2 ż X |f pxq|p2 ż8 |f pxq| α λp´1´p2dλ dµpxq “ p2p1Cp1 1 1 p ´ p1 1 αp´p1 ż X |f pxq|p1 |f pxq|p´p1 dµpxq ` p2p2Cp2 2 1 p2´ p 1 αp´p2 ż X |f pxq|p2 |f pxq|p´p2 dµpxq “ ˆ p2p1Cp1 1 1 p ´ p1 1 αp´p1 ` p2 p2Cp2 2 1 p2´ p 1 αp´p2 ˙ }f }pp.
We have thus proved Lp-boundedness when p
2 ă 8 and we may choose α to get the desired
constant C in the statement of the theorem. Now consider the case p2“ 8. Then we have
}T f2}8ď C2}f2}8 ď C2αλ, so if we choose α “ 1{p2C2q we have }T f2}8 ď λ 2. Hence, mptx P X : |T f pxq| ą λuq ď mp " x P X : |T f1pxq| ą λ 2 * q ` mp " x P X : |T f2pxq| ą λ 2 * q “ mp " x P X : |T f1pxq| ą λ 2 * q.
From here we have just half of the previous case, and we have already done the calculation, so }T f }ppď p2p1Cp1 1 1 p ´ p1 1 αp´p1 }f } p p“ p2 pCp1 1 C p´p1 2 1 αp´p1}f } p p,
and the reader may check that this does in fact coincide with the C in the statement of the
An example where Marcinkiewicz Interpolation Theorem can be applied is the Hardy-Littlewood maximal function M. On T it is defined by
Mf pxq “ sup 0ăδď1 2 1 2δ ż px´δ,x`δq |f ptq| dt. (59)
This operator appears for example when studying differentiation theorems for measures (and is used to prove Lebesgue Differentiation Theorem). Note that it maps L8
pTq boundedly into L8pTq, since 1 mpQq ż Q |f ptq| dt ď }f }8
for every measurable set Q Ă T (and the operator norm is equal to 1, since there is no better bound for constant functions). Also note that M is sublinear, and we have weak L1-boundedness,
by the famous Hardy-Littlewood Maximal Theorem:
Theorem 2.12. (Hardy-Littlewood Maximal Theorem) There is a constant C such that for every λ ą 0 and f P L1 pTq, we have λmptx P T : |Mf pxq| ą λuq ď C ż T |f ptq| dt.
Proof. See e.g. [5, Chapter 7] or, for another proof involving the Calder´on-Zygmund decomposition,
see [2, Chapter IV].
Hence M is bounded on Lp
pTq, 1 ă p ă 8, by Marcinkiewicz Interpolation Theorem. We would like to make a similar argument for the Hilbert transform. We already have boundedness of H in the case p “ 2. Now we would like to show weak L1-boundedness. For this, Theorem 2.12 will
turn out to be useful.
Boundedness of the Hilbert Transform
Theorem 2.13. (Kolmogorov) There is a constant C such that for all f P L1pTq and all λ ą 0, λmptx P T : |Hf pxq| ą λuq ď C }f }1,
and for H we have (independent of )
λmptx P T : |Hf pxq| ą λuq ď C }f }1.
Proof. We begin by proving the result for H. It is not hard to see that it suffices to prove the
result for all sufficiently large λ. We attempt to prove it for all λ ą x|f |y, where we can apply the Calder´on-Zygmund decomposition. Let Ω and ∆ be as in the proof the Theorem 2.8. First, we write
mptx P T : |Hf pxq| ą λuq ď mpΩq ` mptx P Ωc: |Hf pxq| ą λuq.
By (ii) of Theorem 2.1 we have mpΩq ď 2řjmpQjq ď λ2}f }1, so it remains to prove a similar
bound for the last term. We now decompose f into g and b, and write mptx P Ωc: |Hf pxq| ą λuq ď mp " x P Ωc: |Hgpxq| ą λ 2 * q ` mp " x P Ωc: |Hbpxq| ą λ 2 * q. For the ”good term”, we use (57), Theorem 2.2, and (vi) of Theorem 2.1, to get
mp " x P Ωc: |Hgpxq| ą λ 2 * q ď 1 `λ 2 ˘2 ż T |Hgptq| 2 dt ď 4C λ2 }g} 2 2ď 8C }f }1 λ .
We are now left with the ”bad term”. We make use of the estimate of |H´ Hδ| we got in the
proof of Theorem 2.8. In the case “ 1{2, this results in (for almost every x P Ωc)
