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THEOREMS ON POWER SERIES OF THE CLASS
GEN-ICHIRO SUNOUCHI
(Received August 22, 1955 and in revised form, October 29,1955)
1. Introduction. Let
be a function regular for [2| < 1. If, for some p > 0, the expression
0 is bounded as r -> 1, then the function f(z) and its power series are said to belong to the class Hp. It is well known that, if f(z) belongs to the class Hp, then f(z) has a boundary function
(1.3) /(O = limfir^), 0 ^ θ ^ 2π
for almost all θ and f(eiθ) is integrable _Lι_ Moreover if p > 1 a necessary and sufficient condition for the function f(z) to belong to the class Hp^ is that the series
is the Fourier series of its boundary function f(eiθ). Hence, in virtiίre of M. Riesz's theorem, if p > 1, the class Hp^ is isomorphic to the class Lp. In this case, the series (1.4) is summable (C, 6), 6 > 0, to the boundary functions /(£*a> at almost all θ. The problem whether in this result we may replace sum- mability (C, £) by ordinary convergence remains open, but if p = 1, the answer is negative (Sunouchi [7]). On the behaviour of power series of class HD^ on the circle of convergence, important results were obtained by Littlewood and Paley [6] and Zygmund [11] [12]. The main tool of Littlewood and Paley was an auxiliary function
(1.5) g(θ) = g(θ f) = ( j (1 - r) X%r, θ) drj" , 0 g θ ^ 2π
o
where
X(r,θ)= ( —
*> Presented to the Meeting of Mathematical Society of Japan on 23 May 195$ (Tokyo).
126 G. SUNOUCHI
and
But they proved an inequality theorem concerning to g*(θ) only for Hp , p = 2k (k = integer) and their proof is very difficult. Later Zygmund [13] gave a complete and simple proof. In the other papers [10J [11], he used another inequality theorem of Littlewood and Paley concerning to
(1.β) g(θ) = g(θ; f) = ( f (1 - o and gave a simple proof of the main result of Littlewood-Paley and many interesting generalizations. The purpose of this paper is to give the gene- ralized theorem on g*(θ) and systematic treatment and generalization of theorems on the power series of the class Hp.
2. The function _g(θ)_* for the class HP (0 < p ^ 2). The definition of the function g*(θ) is slightly less simple. It is given by the formula 1 2Λ
(2.1) t&ff) = &θ:f)= (J (1-rrdrJ ^ ^ ϊ 0 0 If a = 1, gl{θ) reduces to the function g*(θ) of Littlewood and Paley excepting constant factor. So we don't distinguish between gf (θ) and g*(θ). It is known that g*(θ) is a majorant of many important functions. Especially intervenes for the partial sums of the series (1.4). Let us denote
Sn{θ) = 2
<(θ) = ^ τ 2 A«nz\ sv{θ\ (a > - 1)
<{θ) - ~ r 2 At:] tv{θ\ {a > 0)
where
then (2. 2) τ%(θ) = n{<τ%(θ) - σ«Ύ{θ)}_ = rtίσ ; - 1 ^ ) — cr£(0)}. Further put
(2.3) ft«(β) = W(9; f) = ( 2 "^7 """M Λ
128 G.SUNOUCHI
I \f'W**>)* dφ.
0
is non-decreasing function of r. Thus
0
Consequently
ϋ 0
= / + / 0 1/
by (2.9). Thus we have (2.4).
LEMMA 2. If a > 1/2 <md 0 < r < 1, then
ί 2 1 0 ) /"iϊ^ίs 0 PROOF. Since
(2.1D % where δ = 1 — r, are bound above and below by positive numbers,
ί
dc P < A f
dc P - A f +A f 0 -7t 0 δ
o
THEOREMS ON POWER SERIES OF THE CLASS Hp
for 1 - 2a < 0.
LEMMA 3. If we put
129
(2.12) f*(θ) = sup
(2.13) f*(θ)^ = sup
0 | f t | ^
if h J 0
1 /' h J
(2.14) | ^ | /δ) /or
(2.15) l/ϊr^β+^>) I ^ B/*(^) (1 + I φ I /δ)^fc^ /or /f^) € U (k > 1)
wter^. S = 1 - r, flwrf /ϊ(ί) € Z^2 , ί//(^
θ
) € # flwrf Λ < 2.
This is essentially due to Hardy and Littlewood [3]. Since
rlΊt +<p))=,^ JL^ ί^ f(e i«)+u>>)P(r,u-φ)du,
where P(r,t) is the Poisson kernel, by partial integration, we can easily
get (2.14).
To deduce (2.15) from (2.14) it is enough to note that, by Jensen's in
equality
H i
**l/(έ?ίC
+t)) '
P(r 'V
dt '
k S i " / IΛ«
(ί+t> )I _P(r,t)dt._*
o o
THEOREM 1. Iff(z) € Hp^ (0 < £ S 2), and a > 1/p, then
This was given in my previous paper [9]. We shall give a slightly simpler
proof.
PROOF. We shall begin with the case p = 2.
(1) p = 2. Then for 2α > 1,
THEOREMS ON POWER SERIES OF THE CLASS Hp^ 131
dφ
o
Thus
J {ga(θ f)}»dθ S F^J {F(θ)}-»{g(θ F)}* dθ
0 0 2τc 2 * ί Γ Λ{2-P)j2 ( Γ
^ FJC a { I (F(θ)y dθ\ II (g(θ
\J ί \J 0 0
by the inequality of Holder. From the maximal theorem of Hardy and
Littlewood, we have
Γ Γ
I {Fΐ(θ)}^2 dθ t^A^l I Fie^16 ) ^2 dθ
J J
0 0
and so by the case (1),
P.- / W)^9 dθ.
0 0 0
Thus we proved Theorem 1 completely.
COROLLARY 1. Iff(z) belongs to the class H
p
(0 < p S 2),
then
zjohere
J
2 0
[fit > IIP)
H*(6) = { sup — 2 ki'W) -
This is a maximal theorem concerning with strong summability.
PROOF. Let us put nβ is the index n when Hζ(θ) attains the supremum,
then
lay (2.3). Thus Corollary is immediate from Theorem 1 and Lemma 1.
COROLLARY 2. If f(z) belongs to the class Hp^ (0<p<L 2), then'
V7 K(0)l
a
132 G^ SUNOUCHI
is convergent for almost all θ.
This is immediate from Theorem 1 and Lemma 1. This Corollary has
been ever proved by Chow [1].
From this, we can get the following Corollaries by the well known
method.
COROLLARY 3. Iff{z) belongs to Hp^ (0<pS 2), then the series 2 *>nCnenίθ,
w^ere^l (λw)a/w converges, is summable \C, a,(a > lip) for almost all θ.
For the proof, see Chow [1].
COROLLARY 4. If f{z) belongs to Hp^ (0 < p<, 2), then for almost all θ, the
sequence {n} can be divided into two complementary subsequences {rik} and
{m/c}, depending in general on θ, and such that σf^iθ) tends to f(θ) and the
series 2 */wfc converges, where a >l/p.
For the reduction of this Corollary, see Zygmund [10].
3. The function g*(θ) for the class II
P
(oo > p > 2).
For the class H
p
(oo > p > 2), we have the following theorem. This is
essentially due to Zygmund [13]. The proof is also repetition of his argu-
ment.
THEOREM 2. If f(z) € Hp^ (oo > p > 2) and a > 1/2, then
(3.1) J (tfXθWdθ^Ap,.] \f(e
iθ
)\»dθ.
0 0
PROOF. Let μ be a positive number such that
pβ μ
and let ξ{θ) be any positive function such that
(3.2) ί J ξ»(θ)dθV'
μ ^ 1. 0-
Then it is known that
(3.3) / (g(θ))dθ\ = {/ (g*
2
Pi
dθ\
o o
= sup
and the inner integral
134 G. SUNOUCHI
Thus the left-hand side of (3.3)
f {^(θ)}'^ j
_i B(θ)dθ^Faj Bdrj \f(n»)*ξ{φ)dφ_* 0 0
\fμ. f Γ [J 0
0 ϋ from the maximal theorem of Hardy-Littlewood and the theorem of Lit- tlewood-Paley [6] (simple proof; Zygmund [14]). Thus we get 2τr Γ
2ic ( Γ 0 0 0 and the theorem is proved. From this, we can derive easily that if f(z) belongs to Hp^ (oo >p > 2), and a > 1/2, then
2τr ^
and
/ I sup — V I σ-ίf-W - /(βίθ) Ia^ i d<9 g £ 2 >. β I |/(βί9) I * dθ
4. A proof of the theorem of Littlewood and Paley. From Theorems 1 and 2, we have especially, THEOREM 3. If f(z) belongs to Hp (Kp< oo), then
(4.1) I {g*(θ))v dθSApl \f(eiθ ) | p dθ. J J 0 0
From this, we have the following
{S(θ)}p dθ S BP J l/(^ίa ) Ip ί», (ί > 1) 0 0 and
THEOREMS ON POWER SERIES OF THE CLASS HP 135
,a 2**
{μ(θ)}pdθ^CP I |Λeίθ) *dθ (P>D J 0 0 where S(θ) is the function of Lusin and μ(θ) is the function of Marcinkiewicz. The function of g\θ) is essenitally a majorant of these functions. The reduction of (4. 2) is done by a moment's consideration, but the reduction of (4.3) is somewhat difficult. For the detailed definitions and proofs, see Zygmund [13]. The main theorem of Littlewood-Paley is condensed in the following theorem. THEOREM 4. If f(z) belongs to l J P (1< j l p< ^ (^) 00), W then f o o 2/r
(4.4) APf o
(4.5) BP<a,βJ 0
(4.6) CP^β \f(^ψdθ^ {Σ\MΘ)*W>dθ^Cpaβ
(4. 4) is a consequence of Theorem 3 and Lemma 1*>. The left-halves of (4.5) and (4.6) are proved by the following results of Zygmund [11]. That is
" p/a
for 1 < p < oo. For the proof of the reverse inequalities, we need
LEMMA 4. Let {fn(z)} (n = 1,2 ) δe « sequence of the function of Hp (Kp< oo), dwd /^ί sn, JC(Θ) denote the k-th partial sum of the boundary series offniz). Then
0 w^ =^10 A comparatively simple proof was given by Zygmund [10], using Rade- macher's function. PROOF OF THEOREM 4. From Lemma 4, we have *> We suppose that the left-half of (4.4) is proved by another method.
THEOREMS ON POWER SERIES OF THE CLASS Hp^ 137
COROLLARY 6. If {£&} is any sequence of numbers of which each has one of the values 1, — 1, and if f ^ Hp^ (p > 1), β > n^λjnk >a>l, then
Γ"* ί J l , p 0 re=J-
- The poΛver series of Jϊ-elass. Concerning with the power series of ϋΓ-class, A.Zygmund [11] proved that
(5.1) J hι(θ) dθ^Aj \f(eiθ) Ilog+ \f{eiθ) | dθ + A
and
(5.2) J {h(θ)) μ dθ<Bμ(J 1/(^)1^) , (0</»<l). 0 0 So, from Lemma 1, we get THEOREM 5. Iff(z) belongs to the class H, then
(5.3) J g*(θ)dθ^AJ |/(«")|log
|/r*")|rf0 + A'
r27C^ r~π (5.4) J (g*φ)rdθ^B^J \f(«P)\dθy, (0 < μ < 1)
The present author has not ever a simple and direct proof of this theprem.
THEOREM 6. If f(z) belongs to H, then
(5- 5) J {μ(θ)}dθSΛJ I (feiθ ) |log+ \f(eiθ )\dθ + A' 0 0
(5. 6) J { ^ F dθ S Br (j \f(e«) I dθj , (0<r< 1) 0 0
11/ j 0 and
138 G. SUNOUCHI
This is immediate from the fact μ(θ)<Cg*(θ) which was proved by Zygmund [13].
- Power series of the class JBP(O < p < 1). For the class Hpφ < p < 1), we have a more precise results than Theorem 2.
THEOREM 7. If f(z) belongs to Hp^ (0 < />< 1) and a = lip, then
f 0
ί
\f(e«) I * log- \f(eiθ)\dθ + A;
PROOF. This case is reduced to the case p = 1. If we take 0 < p < 1, a^ljp and p G(z) = {f(z)}p
then
and
fix) = a{G(z)y-iG'(z),
(l-rY«dr _ί __
AG
From (2.14) of Lemma 3,
where
the right-hand side is smaller than
{Θ) ί l + L | l l
G*(0)= sup
(S = 1 - r)
_(θ)}«-vj_* (1 - ry " drj (^) (1 _ ^fiΐ)^"^)^
140 G SUNOUCHI
<7.D / {snp\σl{θ)\Ydθ<AthJ
J 0<n<oo J 0 0
THEOREM 10. Iff(z) belongs to Hp(0 <p^ 1/2), and a=l/p- 1, then
<7.2) { sup I (^) σ % β ) \ y d θ % B (^) p \ \fi<f*) ) * log
\f(e iθ 0 )\dθ + B'p 2τc
(7. 3) Γ { sup I <(0)| »}^ <# Si C,J f \f{e«)\» dθX, (0<μ< 1).
J 0<n<oo \J / 0 0 Theorem 9 is a generalization of classical results of Hardy-Littlewood {4] and Gwilliam [2]. (7.2) of Theorem 10 is an affirmative answer of a problem of Zygmund [12J. From (7.3) we can easily see that σ%(θ) (a = 1/p — 1) converges to f(eiθ) for almost all θ. This was proved by Zygmund [12] for 0 < p < 1. For the case 1/2 < p < 1, the maximal theorem is left open, but convergence of σ%{θ) is proved in the next section. The present author [9] deduced Theorem 9 and Theorem 10 from Lemma 1 and Theorem 8 with the aid of the following lemma. LEMMA 5. Iff(z) = g\z) and a > 0, then
, ., J.
where σ?τ(θ; f) is the a-th Cesaro mean of the boundary series of f(z). For the detailed argument, see my previous paper [9J.
8. Strong summability and ordinary summability of the power series. In Corollary 1, we have proved the maximal theorem of the strong summability of σ%-\θ) (a > 1/p) for Hp (Kp<L 2). But if we give up the maxinal theorem, then we can prove the more precise result. THEOREM 11. If f{z) belongs to Hp (1 <Lp < 2) and a = 1/p, then
fc= for almost all θ, where 0<q<p/(p — 1).
This was proved in the author's paper [8]. The method of the proof depends closely upon the paper of Zygmund [12]. In his paper, Zygmund proved the strong summability theorem of the function of L and the Cesaro summability theorem of the series of the class IP (0 < p < 1). The proofs
THEOREMS ON POWER SERIES OF THE CIASS Hp 141
of both theorems have many features in common, but the the details of proofs are different. After proving Theorem 11, we can deduce from that the following Cesaro summability theorem. THEOREM 12. If f(z) belongs to H» (1/2 <p<l) and a = l/p± 1, then the series 2 cneinθ^ is summable (C, a) to f(eίθ) for almost all θ.
PROOF. If we put
then g(z) belongs to H\ = 2p, 1 < λ < 2), and (a + l)/2 = l/2p = 1/λ. From the formula in [8], p. 225, we have n 2 Al> I al^1 *-^1 {θ g) - a-lHΘ g) |* = o(«»'*+1). a. e. fc = l that is n 2 ; g) - < r ^ + 1 ) / 2 ( 0 ; g)* = o(wα + 2 ), a. e. fc=l By Abel's transformation,
From Lemma 5, we have
Since evidently σ £+ 1 ) / 2^ (#; r/) tends to g(eiθ) for almost all θ, if we take polynomial fe(z) near to f(z), then we can conclude <r%(fi ί /) _-f(βίθ),_ a. e. as /2 ->- oo. Thus we get Theorem 12 from Theorem 11.
9. The affirmative answer to a problem of Zygmund. On another conjecture of Zygmund, we can.prove the following theorem.
THEOREM 13. Iff(z) belongs to Hp, then
Γ
|a ΪP/
(9. 2) / IQsup L^»Wl-.__ J J0 ^ B (^) p J |/(^) I * <*0, (0 < ί ^ 1, α =
(9. 2) is deduced from (9.1) by the usual method. (9. 2) is an answer to the problem raised by Zygmund [12]. But there is a slip in his original paper, so I proved (9. 2) in the case 1/2 < p<l, since this case is better than Zygmund's original conjecture. After-that, I noticed his correction [I4J in such a form as (9.2), so we will prove Theorem [13J completely. For the
THEOREMS ON POWER SERIES OF THE CLASS H" 143
^ Aa 2 (v + I)^1 '^21 τ?l*(θ) 1 1 log (1 - r) 1 »• r F+«?) (by (B) of Lemma 2)
(
1 / 2 (^) °° 1 / 2
2(^ + i)
So
Φ(r, fl) ^ CΛFKΘ)y\l - r)-M log(l
By the formula of Lemma (A)
0
κ-
Consequently ϊit
0 0
(F\θ)f dθ J 1 ' 1 ί Γ _{F\θ)f dθ_* V " 0
0
by Lemma (B).
The case 1/2 < p < 1. put
then G(<ε) € -ίΓ. Denote by s(^), _σ(θ)_ and τ*(^) the corresponding partial sums, Cesaro means and their differences of the boundary series of G(z). Then we have
144 G.SUNOUCHI
r
0 " J by the case proved.
Moreover we have
0 and let us put
then we can prove analogously to Lemma 3 in [9]
_Wt(θ)_ ^ l o g ( (^) W + "
and
r r 0 I I(θ)dθ^J 0 \G(έ»)\dθ. On the other hand, since a = 1/p, and G(z) = {f(z)}p, we have f(z) =. and by applying Holder's inequality,
W n{log(n-- l)}2lP^ )
— dy J — _dcp _ 0 0
— [J J \ \l—retφXλ^ j o o
| l - (^) r ^ | a ( a - « ) 0
, ί ( \G(r<?<-»)*. y«(l - r)
- r ) | 3 J '2τr \J |l_r ^|2/(2-«) ^ ) J
