Mem. Dierential Equations Math. Phys. 12(1997), 42{49
A. Cialdea
THE BROTHERS RIESZ THEOREM IN IRn AND LAPLACE SERIES
Abstract. In the present paper the concept of series conjugate to a
Laplace series on the (n ? 1)-dimensional sphere is introduced and a
Brothers Riesz Theorem for such series is proved.
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1. Introduction
It is well known that the classical Brothers Riesz theorem can be stated
in terms of Fourier series. Namely, if a trigonometric series and its conjugate
series
a0
1
1
X
X
(ak sin k# ? bk cos k#)
2 + k=1 (ak cos k# + bk sin k#) ;
k=1
are both Fourier-Stieltjes series, then they are ordinary Fourier series (see,
e.g., [8], p.285). This means that, if and are two real measures such
that
2
2
2
2
Z
Z
Z
Z
0
0
0
0
cos k# d = sin k# d;
sin k# d = ? cos k# d (k = 1; 2; : : : ); (1.1)
then and have to be absolutely continuous, i.e., there exist f; g 2
L1(0; 2) such that
(B ) =
Z
B
f (#) d#;
(B ) =
Z
g(#) d#
B
for any Borel set B [0; 2]. As far as generalizations of this theorem in
higher real dimensions are concerned, we recall that Bochner [1] proved a
result of this kind in the theory of multiple Fourier series. In [7], Muckenhoupt and Stein have deeply studied some series arising from ultraspherical
1991 Mathematics Subject Classication. 33C55, 31B05.
Key words and phrases. Laplace series, F. and M. Riesz Theorem.
43
expansions of functions having appropriate rotational invariance. This has
led to consider some series generalizing the trigonometric ones. In this context, they gave also a concept of conjugacy for such series and they proved a
result which is an analogue of the theorem of F. and M. Riesz. Because of the
rotational invariance, the series they have considered are one-dimensional.
The purpose of this paper is to prove theorem IV, which provides a
Brothers Riesz theorem for Laplace series. We remark that no rotational
invariance is supposed. The series conjugate to a Laplace series, which is
introduced in Section 3, is not a series of scalar functions, but of dierential
forms of degree n ? 2.
2. An Overview of Previous Results
Let be a domain in IRn . By Ck1 (
) we denote the space of dierential
forms of degree k (briey k-forms) dened in such that their coecients
are of class C 1 . The dierential and the co-dierential of u are denoted
by du and u, respectively, while u denotes the adjoint of u with respect
to the usual metric of IRn . Namely, if u = k1! ui1 :::ik dxi1 : : : dxik ; where
ui1 :::ik 2 C 1 (
) are the components of a skew-symmetric covariant tensor,
then
1 @ui1 :::ik dxj dxi1 : : : dxik ; u = (?1)n(k+1)+1 d u ;
du =
k! @xj
in?k :
i1
u = k!(n 1? k)! s11:::::::::::n
::sk i1 ::in?k us1 :::sk dx : : : dx
We say that non{homogeneous dierential form U 2 C01 (
) P: : : Cn1 (
)
is self-conjugate if dU = U in . This means that, if U = nk=0 uk (uk
being a k-form), then u1 = 0, duk = uk+2 (k = 0; : : : ; n ? 2), dun?1 = 0.
It is possible to show that holomorphic functions of one complex variable,
solutions of the Moisil-Theodorescu system, quaternionic hyperholomorphic
functions, harmonic vectors (i.e. vectors w such that div w = 0; curl w =
0) can be identied with particular self-conjugate forms (for the details,
see [4]).
In [3], [4] it is showed that several results of the theory of holomorphic
functions of one complex variable hold true for self-conjugate forms in IRn .
Hereafter denotes a bounded domain of IRn such that its boundary is
a Lyapunov boundary and Mk () is the space of k-measures on (see [5]).
I. If a self-conjugate form U 2 C01 (
) : : : Cn1 (
) is such that U and
U admit traces on in M0() : : : Mn?1 (), then these traces are
absolutely continuous.
This theorem provides a generalization of the classical Brothers Riesz
Theorem in IRn and in some domains it can be stated also in the following way, where !hi1 :::ik denotes the k-form !h (x)dxi1 : : : dxik and f!hg is a
complete system of homogeneous harmonic polynomials.
44
II. Let be such that IRn ? is connected. If = (0 ; : : : ; n?1 ); e =
(en ; : : : ; e1 ) 2 M0 () : : : Mn?1 () are such that
Z
+
[k ^ d!hi1 :::ik ? !hi1 :::ik ^ ek + d!hi1 :::ik ^ ek+2 ? k?2 ^ !hi1 :::ik ] = 0
(where k = en?k = 0; k = ?2; ?1; n) for any 1 i1 < < ik n,
h = 1; 2; : : : , k = 0; 1; : : : ; n, then ; e are absolutely continuous.
For the details and the proofs of I and II we refer to [3].
3. The Series Conjugate to a Laplace Series
It is well known that if u is a harmonic function in the unit ball B = fx 2
IRn jxj < 1g, then it can be expanded by means of harmonic polynomials
u(x) =
1
X
jxjh
h=0
2) ((hn+?n2)!?3)!
h!
p
nh
X
k=1
ahk Yhk
x
jxj ;
where pnh = (2h + n ?
and fYhk g is a complete system of ultraspherical harmonics. We suppose fYhk g orthonormal, i.e.,
Z
(
Yhk Yrs d
= 1 if h = r and k = s,
= 0 otherwise.
The \trace" of u on = fx 2 IRn jxj = 1g is given by the expansion
1
p
nh
XX
h=0 k=1
ahk Yhk (x)
(jxj = 1):
If the coecients ahk are
ahk =
Z
f Yhk d
(ahk =
Z
(3.1)
Yhk d);
we say that (3.1) is the Laplace series of the function f (of the measure ).
Let us consider the 2-form
v=
1
p
nh
XX
ahk
dY
(
h
+
2)(
n + h ? 2) hk
h=0 k=1
x
jxj
^ d(jxjh+2 )
(3.2)
and its adjoint
v =
1
p
nh
XX
ahk
(
h
+
2)(
n + h ? 2)
h=0 k=1
dYhk jxxj ^ d(jxjh+2 ) : (3.3)
It is possible to show that dv = 0; v = du in B , i.e., the non{homogeneous
form u + v is self{conjugate.
45
If n = 2, then the series which is obtained by taking jxj = 1 in (3.3) is just
the series conjugate to (3.1); roughly speaking, it, represents the "trace" of
the harmonic conjugate function on . So in general, for any n, we say
that (3.3) (with jxj = 1) is the series conjugate to (3.1). It represents the
"restriction" of v on , while the "restriction" of v, provided it does exist,
is equal to 0, as it follows from (3.2).
In order to write the conjugate series (3.3) more explicitly, let us consider
the polar coordinates in IRn : xh = xh (%; '1 ; : : : ; 'n?1 ); h = 1; : : : ; n; and
the relevant metric tensor
gij =
@x @x
; g = g = @x @x ; i; j = 1; : : : ; n?1; gnn = @x
@x :
@'i @'j ni in @'i @%
@% @%
Let fgij g be the inverse matrix of fgij g (i.e., gij gjs = si ) and let us set
g = det(gij )i;j=1;::: ;n . We remark that we have gnn = gnn = 1; gij = 0 if
i 6= j and that g = det(gij )i;j=1;::: ;n?1 .
If v = vj d'j d%, we have
1:::::: :n pg g jj g nn vj d's1 : : : d'sn?2 =
v (n ?1 2)! jns
1 ::sn?2
= (n ?1 2)! s11::::::s::nn??11pg gsn?1 sn?1 vsn?1 d's1 : : : d'sn?2 ;
and then we may write
v =
nX
?1
j =1
(?1)n?1?j pg gjj vj d'1 : : : bj : : : d'n?1
(3.4)
(where bj indicates that d'j is omitted). Therefore the restriction vj is
given by (3.4), where g, gjj , vj are considered for % = 1. In particular,
taking (3.2) as v, we nd that the series conjugate to (3.1) can be written
as
nX
?1
ahk
hk
(?1)n?1?j pg gjj @Y
d'1 : : : bj : : : d'n?1 : (3.5)
(
n
+
h
?
2)
@'
j
j =1
h=0 k=1
1
p
nh
XX
Let us consider now the space L2n?2 () endowed with the scalar product
(; ) =
Z
+
^ ;
where denotes the adjoint of on with respect to the usual metric
on . Namely, if = (n?1 2)! s1 :::sn?2 d's1 : : : d'sn?2 then
pg
= (n ? 2)! s11::::::s::nn??11 gs1 i1 : : : gsn?2 in?2
s
i1 :::in?2 d' n?1
:
(3.6)
46
Let us introduce the following system of (n ? 2)-forms
hk
1
(dYhk ^ d%h+2 ) =
(h + 2) h(n + h ? 2)
=
p
nX
?1
1
p @Y
%h+1 (?1)n?1?j g gjj hk d'1 : : : bj : : : d'n?1 :
@'j
h(n + h ? 2)
j =1
p
III. The system f
If
=
nX
?1
j =1
hk
g is orthonormal in L2n?2().
1 j : : : d'n?1 ;
j d' : : : b
e
=
nX
?1
j d'
e
j =1
1 : : : bj : : : d'n?1 ;
taking into account (3.6), we get
e =
=
nX
?1
j =1
nX
?1
j =1
(?1)n?1?j pg g11 ::bj::gn?1;n?1 ej d'j =
(?1)n?1?j pg1gjj
j d'
e
j
(3.7)
;
from which easily follows
Z
+
^ e =
Z
nX
?1
1
g gjj
j =1
j ej d
:
This implies
Z n?1
X
1
@Y @Y
( hk ; rs ) = p
gjj hk rs d :
hr(n + h ? 2)(n + r ? 2) j=1 @'j @'j
On the other hand,
Z
nX
?1
j =1
gjj
@Yhk @Yrs
d = ?
@'j @'j
Z
Z
nX
?1
hk
Yrs d =
p1g @'@ j pg gjj @Y
@'j
j =1
Z
= ? Yrs Yhk d = h(n + h ? 2) Yrs Yhk d
( being the Laplace-Beltrami operator on ) shows that f
thonormal in L2n?2 ().
hk
g is or-
47
We nally remark that the conjugate series (3.5) (or (3.3) with jxj = 1)
can be written also as
1
p
nh
XX
r
h=1 k=0
h
a
n + h ? 2 hk
(3.8)
hk :
4. The Brothers Riesz Theorem for Laplace Series
If (3.1) is a Laplace series of a function f 2 L2 (), then the conjugate
series (3.8) is a \Fourier series" of an (n ? 2)-form g 2 L2n?2 (). This follows
immediately from Fischer-Riesz theorem since
pnh
1 X
X
h
a2hk < +1
a2hk n
+
h
?
2
h=1 k=0
h=1 k=0
1
p
nh
XX
implies that there exists g 2 L2n?2 () such that
r
Z
n+h?2
a =
g^
hk
h
hk
+
and therefore the conjugate series is the \Fourier series" of g with respect to
1 pP
nh
P
(g; hk ) hk : Moreover, we may suppose g such
the system f hk g:
h=1 k=0
that (g; ) = 0 8 2 L2n?2 () : (; hk ) = 0 (h = 1; 2; : : : ; k = 1; : : : ; pnh ).
Because of a completeness theorem proved in [2] (p.195), these orthogonality conditions are equivalent to the following ones: (g; ) = 0 8 2
Cn1?2 (IRn ) : d = 0 on . A similar question for measures is more delicate
and the theorem II makes possible to prove the following result, which can
be considered as the Brothers Riesz theorem for Laplace series:
IV. Let (3.1) be a Laplace series of a measure 2 M (). If its conjugate
series (3.8) is a \Fourier series" of an (n ? 2)-measure, i.e., if there exists
2 Mn?2 () such that
r
ahk =
and if
Z
+
n+h?2
h
Z
+
^ ^ = 0
(h = 1; 2; : : : ; k = 1; : : : ; pnh); (4.1)
hk
8 2 Cn1?2 (IRn ) : d = 0 on ;
(4.2)
then and are absolutely continuous.
We remark that in the case n = 2, (4.1) are nothing but (1.1), while (4.2)
is not restrictive. Observe now that we may write
1 @ (%h Y ) d = 1 d(%h Y ) :
Y d =
hk
h @%
hk
%=1
h
hk
48
On the other hand, because of (3.7),
nX
?1 @Y
1
1
hk
d'j = p
dYhk ;
h(n + h ? 2) j=1 @'j
h(n + h ? 2)
hk = p
and the conditions (4.1) can be written as
Z
e ^ d! =
+
Z
+
d! ^ e =
Z
^ d! = (?1)n
+
Z
+
d! ^ (4.3)
for any harmonic polynomial !. Here e 2 M0 () is dened by
Z
+
e ^ w d =
Z
8 w 2 C 0 ():
w d
In order to apply the theorem II, we have also to show that
(?
1)n
Z
+
(!dxp dxs )
^+
Z
+
e ^ (!dxp dxs ) = 0
(4.4)
for any harmonic polynomial ! and for any 1 p < s n. Since
(!dxp dxs ) = !;s dxp ? !;p dxs ; (!dxp dxs ) = xp !;s ? xs !;p (on ) (4.5)
@! ) and taking into account that if ! is a harmonic homoge(where !;p = @x
p
neous polynomial of degree h, then xp !;s ? xs !;p is a harmonic homogeneous
polynomial of the same degree, it follows from (4.3)
Z
= h1
+
Z
+
e ^ (!dxp dxs )
Z
= (xp !;s ? xs !;p ) d =
e ^ d(xp !;s ? xs !;p ) =
1 Z ^ d(x ! ? x ! ):
p ;s
s ;p
h
+
Moreover (s and p are xed),
d(xp !;s ? xs !;p ) = !;s dxp ? !;p dxs + [xp !;js ? xs !;jp ]dxj =
= !;s dxp ? !;pdxs + [xp !;ps ? xs !;pp ]dxp +
+[xp !;ss ? xs !;sp ]dxs +
Since
xp !;ps ? xs !;pp = xj !;js +
n
X
j =1
j 6=p;s
n
X
k=1
k6=p;s
[xp !;js ? xs !;jp ]dxj :
xs !;kk ?
n
X
k=1
k6=p;s
xk !;ks =
(4.6)
49
= (h ? 1)!;s +
and analogously
n
X
k=1
k6=p;s
(xs !;kk ? xk !;sk );
xp !;ss ? xs !;sp = ?(h ? 1)!;p ?
we may write d(xp !;s ?
=
n
X
j =1
j 6=p;s
n
X
?
(xp !;kk xk !;pk )
k=1
k6=p;s
p
xs !;p) = h(!;s dx !;p dxs ) + , where
;
?
[(xp !;sj ? xs !;pj )dxj +(xs !;jj ? xj !;sj )dxp +(xj !;pj ? xp !;jj )dxs ] :
Because of (4.5) and (4.6), we have
Z
+
e ^ (!dxp dxs ) =
Z
+
^ (!dxp dxs ) +
1 Z ^ :
h
+
(4.7)
Let us x j , s, p and consider the 1-form = (xp w;s ? xs w;p ) dxj +
(xs w;j ? xj w;s ) dxp + (xj w;p ? xp w;j ) dxs , where w is a scalar function. A
direct computation (we omit for lack of space) shows that, up to a multi1:::::: ::n dw ^ dxi1 ^ : : : ^ dxin?3 . Then d = 0
plicative constant, = jspi
1 ::in?3
R
and therefore d = 0. By virtue of (4.2), we must have + ^ = 0.
Now (4.4) follows from (4.7). By theorem II, e and (?1)n are absolutely
continuous and this completes the proof. Finally notice that, without (4.2),
the theorem is false (if n > 2).
References
1. B. Bochner, Boundary values of analytic functions in several variables and of
almost periodic functions. Ann. of Math. 45(1944), 708{722.
2. A. Cialdea, Sul problema della derivata obliqua per le funzioni armoniche e
questioni connesse. Rend. Acc. Naz. delle Scienze detta dei XL 12(1988), 181{200.
3. A. Cialdea, The Brothers Riesz theorem for conjugate dierential forms in IRn .
Appl. Anal. (to appear).
4. A. Cialdea, On the theory of self-conjugate dierential forms. (to appear).
5. G. Fichera, Spazi lineari di k-misure e di forme dierenziali. In: Proc. of Intern.
Symposium on Linear Spaces, Jerusalem 1960, Israel Ac. of Sciences and Humanities.
Pergamon Press, 1961, 175{226.
6. G. Fichera and L. De Vito, Funzioni analitiche di una variabile complessa.
Veschi, Roma, 1971.
7. B. Muckenhoupt and E. M. Stein, Classical expansions and their relation to
conjugate harmonic functions. Trans. Amer. Math. Soc. 118(1965), 17{92.
8. A. Zygmund, Trigonometric Series. Cambridge University Press, 1979.
(Received 25.06.1997)
Author's address:
Dipartimento di Matematica, Universita della Basilicata
Via N. Sauro 85, Potenza, 85100, Italy