for some finite constant l%. Although this result is wellknown and can be used to
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prove the boundedness of Qa in L~, t < p < or it is not as easy to demve as results 9proving the boundedness of both Qa and Pa from L~ into Lg, q > p. For other proofs of 1.5, see also [6], [7], [3], [1] and [4]. The second set of results in the paper give interesting cases in which certain operators reduce Qa and Pa to projection operators by means of operator multiplication. 2. Preliminary Lemmas. Here we derive the lemmas which will be employed in proving the main results. Firstly we need Young's inequality, which will be used in proving the easier boundedness theorems. (2.1) Young's Inequality. Let / e L~, g e Lq, 1 ~ p < ~ ,
1 <__q < ~ ,
l i p }
q 1/q >__ 1, and let /,g(x)
= [. / ( t ) g ( x  t ) d t . R
Then/*gel
r, where 1/r .= l i p ~ 1 / q 
1, and
]l 1" g]l, < II 1]1, I] glibFor a proof of this result, see for example [2, Theorem 281] (2.2) Lemma. Let {aj, i ~ 1, 2, ..., n} be a sequence of oositive real numbers and /or real numbers ($1, ~2 . . . . . ~n), let n
(t, ,,) = ~
(~, 
t 
i a,)l.
T h e n / o r e > O,
(i) (l/n) ]~(t, a)e(c 2 + (t   x)2)ldt ~ q~(x, a ~ v), R
(ii) (1/~) ~q~(t, a) (t  x) (c2 + (t  x)~)ldt  iq~(x, a Jr e), R
where cf (t, a ~ r is obtained from the expression/or q~(t, a), by replacing each number al by at + c.
P r o o f . Let the functions ua, ua, a > O, be defined by (2.2.1)
Ua(X) = (1/g)a(a 2 ~ x2)1;
ua(x) = (1/g)x(a ~"+
X2)  1 9
Then, using t h e notation of Lemma 2.1, it is easily seen as in the proof of Lemma 3 of [5] that the following hold: (2.2.2)
Ua * Ub ~
Ua+b ~
  ~ a :~ Ub ;
Ua g<~ b ~
~a+b ;
a
> O,
Now it is easily seen by decomposition by partial fractions that
where ~ = ~[ {(~
 ia~)  (~1 iaj)}l,
/c~ 1 , 2 , . . . , n .
b > O.
u
XXII, 1971
Conjugate harmonic functions
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Hence on using 2.2.2, we see that n
(1/~) S ~ (t, . ) c (c~ + (x  t)2)1 at = ~ ~ ~j uc 9 (  7,~, + i ~ , ) ( 5  ~) = j=l
R
n
= 7~ ~ O~s (   Ua,+c ~ i Ua,+e) (~j   X) ~j=l
= ~ =j (~j 
i aj 
i c  x)i =
]=1
= cf (x, a + c ) .
A similar procedure proves result (fi) of the Lemma. 3. Identities Involving Products of Conjugate Functions. For notational convenience we introduce the operator Oa, a > 0, defined b y (3.1) so that (3.1.1)
Oa(]) = Oa(/) + i P a ( / ) , Oa(]) (x) = (1/zc) f /(t) (t  x  i a )  l dt. R
We shall write Oa (/) (3.1.2)
for O,~(f), so tha% Oa(l)  Qa(f)  i Pa(]) .
The main identity is then given as follows. (3.2) Theorem. Let {fj, i = 1, 2 . . . . . n} be a sequence of ]unctions with fl e L v~, 1 <= 791 < 0% and let {ai, ] = 1, 2 . . . . . n} be a sequence o/79ositive numbers. T h e n / o r c > O, we have
(M ',')
(i) Pc
o~,(
"=
(ii) Qc
=
oa,+c(h), j=l
O.,(/j) = iI~O~+c(b ) . ]=1
P r o o f . B y definition,
II oo, @ (x) = ,~~ I ; 
i=i
R R
1 I1 (el) ... t , ( ~ ) (~1  zl) .. (e,  z,) e g ,
R
where zj = x + iaj, d E = d~l d~z"" den, and the nfold integral can be represented in the ndimensional Euclidean form as f. Hence it is easily seen by applying Fubird's theorem and Lemma 2.2 (i), that n~ Pc
Oa,(b) (x) = z~n f ]i(~l) " ' " / , ( ~ n ) (l/g) f e(c 2 q (t  x)Z)lq~(t, a ) d t d g = It.
R
= :~" f 1~ (~1)'" I . (~.) ~ (z, a + c) dE = R*
n
= ]IF 0a,+c(fi) (x). ,i=i
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A similar procedure proves the identity (ii) of the theorem using Lemma 2.2 (ii). (3.2.1) R e m a r k . On writingfi for h in Theorem 3.2, and taking conjugates of the identities we see that the conclusions of the theorem remain valid if Oaj is replaced by 0a~ throughout. (3.2.2) R e m a r k . Note that, since by the case n = 1 of Theorem 3.2, we have Pc Oa (/) = Oa+c(/) ;
Qc Oa (/) = i Oa+c (/),
the conclusions of the theorem can be written alternatively as Pc
0 r (b) =  i Qc
O+ (h) =
Pc 0 ~ (/i).
(3.3) Corollary. Let / ~ L~, 1 <= p < oo, and let a and e be positive real numbers. T h e n / o r any integer n >= 1,
(i) :Pc(o (/)) = oo+c(l) = {Pc
(l))}n,
(fi) Qc (Oa (/)") = i Oa+c (/)" =  ( i) "+1 {Qc (Oa (/))}" 9 P r o o f . These conclusions are easily seen to follow by setting /t = ]2 . . . . .
/ , ~/,
al ~ a2
.
.
.
.
.
an =
a
in Theorem 3.2 and Remark 3.2.2. (3.3.1) R e m a r k . We note here the interesting fact from Corollary 3.3 that ff the function g in Lv, l < p < o o , is either of the form Q a ( / )  F i P a ( / ) , a > 0 , or Qa (/)  i Pa (/), a > 0, where / ~ L r, 1 < r < co, then for each integer n, (i) Pc(g") = Pc(g)"   iQc(g"), and (ii) Qc(g") =  (  i) n+l Qc(g) n 9 A result which is a direct extension of 3.3 involves analytic functions of harmonic functions. We denote by <~ the unit disc in the complex plane (that is
v={z=re
'~
o<0<2
,
Izl=r<=l}).
(3.4) Theorem. Let 9~(z) be analytic in the domain ~ o/ the complex plane, with qD(O)  O, and let / ~ LP, 1 <= p ~ ~ , a > O, e > O. Then there is a real positive number ~o such that, /or [~[ > 20,
(i) Pc (~ 00a (,~./)) = ~ 00a+c (,~ l) = ~ O Pc Oa (,~ ]), (ii) Qc (q~ o Oa (~ f)) = i qDo Oa+c(+~/), where,/or each complexvalued/unction h (t) on tt with [Ih 11r ~ 1, q~ oh(t)
=
flD(h(t) ) .
P r o o f . I t is clear that ~(z) can be represented as
I=1<1
Vol. XXII, 1971
Conjugate harmonic functions
509
I f ~ (z) is a polynomial (that is, ff 3 integer 2V such t h a t b~  0 for all k ~ N), then the conclusions of the theorem follow for all 2 by taking linear combinations of results given in Corollary 3.3. I n the case where ~ is not a polynomial we simply need to justify t h a t Zb~P~(O~(2/)~) = 1r
Since, as ~
b~O~(M)~ . =
be shown in L e m m a 4.1, there is a constant k10 such t h a t tl O~(l)ll~ =< a1/10 k101]1 tl10,
the required conclusion follows from the uniform convergence of the series involved ff 2o is chosen to be a1/10k10 ][l[I 10(3.4.1) N o t e . I f the function ~ is analytic on all bounded regions of the complex plane, then the conclusions of 3.4 hold for all 2. (3.5) R e m a r k . As we mentioned in the introduction certain special cases of the identities given in Theorem 3.2 reduce in limit to theorems proved b y T ~ I c o ~ r in [8] for the Hilbert transform. Since we shall use these again later we state t h e m here. These involve the special case n = 2 of the theorem. Let I ~ ZP, !l ~ Lq, p >= l, q >= l, O <= l/p q1/q ~ l. Then for a > 0 , b>0, we have (3.5.1)
Pc{Pa(I)Qo(g) ~Qa(l) Pb(g)}= Pa+c(l)Qb+c(g) qQa+c(f)Pb+c(g),
(3.5.2)
P c ( Q a ( / ) Q b ( g )  P . ( / ) P b ( g ) } = Q.+c(/)Qb+~(g) 
(3.5.3)
Qc(Pa(/)Qb(g) + Qa(/)Po(g)} = Qa+e(/)Q~+c(g)  Pa+c(/)Pb+c(g),
(3.5.4)
Qc(Qa(/)Qo(g)
P~(/)P~(g)} = 
P.+~(/)P~+c(g),
(P~+~(/)Qb+,(g) + Q.+~(l) Pb+~(g)}.
I t is convenient in deducing the identities from Theorem 3.2 to assume t h a t both ] and g are real valued functions. I n this event we set al a, a2   b,/1 /,/~ = g and derive 3.5.13.5.4 b y equating real and imaginary parts of Theorem 3.2(i) and (ii). 4. Boundedness of the Operators Q~ and P , . Before discussing the inequalities stated in 1.5 we consider those results which can be easily deduced from L e m m a 2.1. (4.1) Lemma. Let / e L10, 1 <=p < ~ , and let a > O. Then (i) F o r q < p, we have
[[ P a
(~)[1q ~ a(1/q)(1/1~
k10, q ][] H10,
where k10,q= 2 (l qt2)sd \1/s, 1 / s = l q(1/q)(1]p), and k~,10= l, (il) For q ~ p, we have
tlQa (])I]q <=a(1/q)(1/10)k10,q II/ ][10,
(Y
)
where ~c10,q= 2 ts(1 [t2)sdt \1Is.
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P r o o f . These results are easily seen to follow from L e m m a 2.1. (4.2) R e m a r k . Although LemTna 4.t(i) proves the boundedness of P4 in L~, 1 ~ p < 0% the boundedness of Qa in Lp, 1 ~ p ~ 0% is usually obtained from this b y applying 1.5. We shall now give a procedure for obtaining results of the form given in 1.5 for certain values of p from the identities stated in 3.5. The procedure adopted here is akin to one first used b y Rl~sz in [6] (see also [7]) and also by KOBE~ [5]: These methods depended on the fact that, for certain functions/, and integers n, we have (4.2.1)
S {0~(1)(x)}n gx = 0,
a > 0.
R
We note t h a t 4.2.1 can be obtained from 3.3(i). For, on using the fact t h a t for / e L z, fPc(]) (x)dx = ~](t)dt (c > 0), R
R
we see t h a t for c > 0, a > 0, and integer n, On+ (/) (t)" dt = S Pc {04 (f)n} (x) dx = f. 04 q) (~)ndr, R
R
R
so t h a t the integral is independent of the 1 ~ p < n, it follows on using L e m m a 4.1 Before applying the conclusions of 3.5, follows in a rather straightforward manner
parameter. Hence if f e L~ for some p, t h a t I[04+r ~ 0 (c > oo). we treat the case p 2 of 1.5 which from the conclusions of [5, Theorem 1].
(4.3) Lemma. Let ] e L2. Then, /or a > 0, ]1Qa(]) ][2 = I]Pa (]) I]2. ]Proof. From the case n = 1 of Theorem 3.2, which is a direct consequence of 2.2.2, we have, as stated in 1.3
P4 P4 (/) =  Q4 Q4 (/) = P2a (/). Hence b y the product formulae stated in Theorems 1 and 2 of [5], which follow easily from Fubini's theorem we have R
[Q4 (]) (t) 12dt = ~ Qa (f) (t) Qa (]) (t) dt = R
=  S](t) Q4 Q~ if) (t) dt = R
= S](t) P~ P~ if) (t) dt =
R = ~ Pa ( / ) (t) Pa (/) (t) dt = R
= f [ P4(/) (t) lz ~t. R
This proves the required result. In order to deduce more general cases of 1.5, from 3.5 we may follow a number of procedures of which the following is typical. (4.4) Theorem. Suppose that/or some number p, 1 < p < ~ , there is a constant kp such that
[IQc(/)lI~ __< k~ i]/1I~
(c > 0 , / + 5 ~ ) ,
Vol. XXII, 1971
Conjugate harmonic functions
511
then /or a > O,
[IQa (!)[] 2p < K ~
]] Pa (!) ii2p,
where K~ 1 = (1 § k~)l/2  k~. P r o o f . We assume without loss of generality t h a t ! is realvalued. Then on p u t t i n g g ~ / in 3.5(iii), we see t h a t for c > 0, a > 0,
2Qc(Pa(/)Qa(])) = ]Qa+c(/)12  I'Pa+c(/)[ 2 9 T h e n on using the fact t h a t Qa+c = PcQa, Pa+c ~ PcQa, and L e m m a 4.1 (i), we have
IIPcQa(/)H~ = i] IQa+c(/) [2H~ <=2 ][Qc(Pa(f)Qa(/))[lp § []Pc Pa(/)[llV g 2k~ ][ Pa(/)Qa(/)IIp § II P,(/)lli~ =< < 2 ~ IIQa (!)][ 2~ HPa (/)il 2~ + I[ P~ (!)Ili~. N o w if we assume to s t a r t with t h a t ! ~ L~, then Qa (!) ~ L 2p. Also, since Pc (]) m a y be written alternatively as
Pc(/) (x) = (1/=) f ( 1 § t2)lf(ct ~ x)dt, R
it is easy to verify, b y a p p l y i n g Lebesgue's convergence t h e o r e m and the continuous form of Minkowski's inequality t h a t
ltPc(/)/JIr~O
(4.4.1)
(c~ 0 §
(!~Lr, r>__l);
see also T h e o r e m 1 of [5]. H e n c e on letting c ~ 0 in the above inequality we see t h a t (1 ~ k~) [[Qa (!)][~v =< (k~ [IQa (])n 2p § [1Pa (])[i2~) 2 , a n d the required conclusion follows. (4.4.2) N o t e . I f we know Qc is bounded in LP, 1 < p < ~ , for some n u m b e r e > 0, then Qa is bounded for all a > e. Since Qa (/) = PacQe (!) b y case n = 1 of T h e o r e m 3.2, the conclusion follows f r o m L e m m a 4.1 (i). (4.4.3) R e m a r k . A n u m b e r of special cases of 1.5 can be obtained b y using cases of Corollary 3.3 more general t h a n t h a t used in the p r o o f of 4.4 (for example as indicated in R e m a r k 4.2). We shall omit further details here. 5. Multiplieative Reduction of Poisson Operators to Projections. As in 2.2.1, the functions ua, ua are defined b y
Ua(x) g1 a (a 2 } x2) 1 ;
ua (x) = 7c:t x (a 2 § x2) 1 .
W e shall write, for convenience
v~ (x) = ~ (x) § iu~ (x) = ~1 (x  i a )  l . Further, we define the operators Ua, ~Ya and Va, b y (5.1) Definitions.
U~ (1) (z) = u~ (x) / (x); ~7~ (/) (x) = ?z~ (z)/(~); V~ (!) (x) = v~ (x) ! (x).
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We shall now prove results which give projection operators from L~, 1 ~ p ~ oo onto (Ua, Ua, Va} b y suitable compositions of the Poisson operators a n d the operators defined in 5.1. The main result is the following (5.2) Theorem. Let ] e L p , 1 ~ p ~ 0% and let a ~ 0, b ~ O, c ~ O. Then
(i) ~ VbO~(/) = vb+~ S v~+b (/) (~) d~ + O~+~ Vb+~(/), R
(ii) Qc Vb Oa (]) = i v~+c ~ Va+b (/) (~) d~  i Oa+c Vb+c (/), R
(iii) Oc Vo Oa (]) ~ 2 i vb+c ~ Va+b (/) (~) d~. R
P r o o f . The proofs of (i) a n d (ii) follow b y the straightforward application of Fubini's t h e o r e m and the case n = 2 of L e m m a 2.2. The result (ii) is obtained b y suitable combination of (i) and (il). W e shall prove (i), the proof of (ii) being similar. I t follows b y Fubini's t h e o r e m and L e m m a 2.2 t h a t
Pc Vb Oa (]) (x) = ~3 ~ c (c 2 H (t  x)~) 1 ~ / (~) (t  i b)i (~  t  i a)~ d~ dt ~ It
R
=  ~~ ]](~)d~ ]c(c ~ + (t  x)2) ~ (t  ib)~ (t  ~ + ia)~dt = R
R
I
x
i(b + c)
g~R~ Va+b (]) (~) d~ x~ § ~ + c)e ~ x~ + (b + c)~  ~ i(a + c) I (a + c)~ + (x  ~)~ § (a + c)~ + (x  ~)~ '
and the required conclusion is easily seen to follow from this. (5.3) R e m a r k . On expanding the identities into real and i m a g i n a r y parts, a n d equating these in the case where ] is realvalued, we obtain the following results. (5.3.1) Theorem. L e t / e L P ,
t ~ p ~ 0% let a ~ O, b ~ O, c > O, and let
)~a = ~ ~ (t) ua (t) dr,
~a = ~ / (t) ~a (t) dt .
R
Then (i) Pc U~ Pa (]) (x) + Qc Ub Qa (]) (x) =
(ii) Pc Ub Pa (]) (x) § Qc Ub Qa (]) (x) ~
~+b ub+c(x)

~+~ ~+c (x),
~+b ub+~ (x) + ;,~+~ ~b+c (x),
(iii) Qc Ub Pa (/) (x)  Pc U~ Qa (]) (x)   ~a+b ub+c (x)  2a+b Ub+c(X) , (iv) Qc UbPa(/) (x)  Pc UbQa(]) (x) =
~a+bUb+c(x)  ~a+b~t~+c(x).
Relerenees
[1] [2] [3] [4]
A. P. CALDEI~,OI~] ", Singular Integrals. Bull. Amer. Math. Soc. 72, 427465 (1966). G. H. H~DY, J. E. L I T ~ w o o D and G. PoLYA, Inequalities. Cambridge 1934. H. KOBF~, A note on Hilbert transforms. Quart. J. Math. Oxford Ser. 14, 4954 (1943). U. KVRAN, Extensions to halfspace of a theorem on conjugate functions. Proc. London Math. Soc. (3) 16, 590598 (1966). [5] G. O. Oxn~ioL~, On the infinitesimal generator of the Poisson Operator. Proc. Cambridge Phil. Soc. 62, 713718 (1966).
Vol. XXII, 1971
Conjugate harmonic functions
513
[6] M. RIESZ, Sur les functions conjugu6es. Math. Z. 27, 218244 (1927). [7] F~. C. TITCH~SH, Introduction to the theory of Fourier integrals. Oxford 1937. [8] 1~. G. TI~ICoMI, On the finite Hilbert transformation. Quart. J. Math. Oxford Ser. (2) 2, 199211 (1951). Eingegangen am21.8.1969 Anschrift des Autors: G. O. Okikiolu University of East Anglia Norwich England
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