Applicable Analysis. Vol. 65, pp. 69-94 Reprints available directly from the Publisher Photocopying permitted by license only

Q 1997 OPA (Overseas Publishers Association) Amsterdam B.V. Published in The Netherlands

under license by Gordon and Breach Science Publishers Printed in India

The Brothers Riesz Theorem for Conjugate Differential forms in En Communicated by R.F? Gilbert and F! Ricci

A. CIALDEA Dzpartimento di Matematica, Universith dells Basilicata Vza N. Sauro 85, Potenza, 85100, Italy

Abstract. It is proved that if u and v are two conjugate real differential forms in a bounded domain SZ in Rn and if u , v and their adjoint forms have traces on the boundary of SZ in the sense of the k-measures, then all these traces have to be absolutely continuous k-measures. This theorem provides a generalization in Rn of the 'famous theorem of F. and M. Riesz.

AMS: 31B25,28A99

KEY WORDS: Brothers Riesz theorem, differential forms, potential theory.

(Received for Publication November 1996)

To the Memory of my Teacher Prof. Gaetano Fichera.

1. Introduction. The classical theorem of F. and M. Riesz [15] states that if p is a complex measure

defined on the circle T = { z E @ I lzl = 1) which satisfies

then p is absolutely continuous, i.e. there esists a function f E L1(T) such that

for any Bore1 set B c T. A similar theorem for simply connected domains in C can be obtained by means of conformal mappings. For applications of this theorem to Fourier series see [21], and for a generalization in the theory of multiple Fourier series, see 121. Recently a mechanical interpretation of the Brothers Riesz theorem was given in [3].

There are several generalizations of this theorem in the theory of holomorphic functions of several complex variables (see [I , 161 and the references therein). Another generalization has been obt,ained in the theory of uniform algebras (see [19]).

The Brothers Riesz theorem can be stated also in the following interesting way: if u(x, y) and v(x, y) are two conjugate real harmonic functions in a domain 52 and both of them have traces on 852 in the sense of measures (for the precise meaning of that, see 5 3), then these measures have to be absolutely continuous. In order to obtain

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70 ALBERT0 CIALDEA

generalizations of this statement in Rr', at first one has to give a concept of conjugate harmonic functions and this can be done in several ways (see e.g. 1131 and [14]). In the paper [17] Stein and Weiss proved (among other things) a generalization of the Brothers Riesz theorem for a system of conjugate harmonic functions (ul , . . . , u,) in the half-space, i.e. for a system of functions such that

In the present paper we use a different concept of conjugate functions which is based on some ideas due to Volterra [20]. It can be easily described by means of differential forms. We say that a k-form u (i.e. a differential form of degree k) and a (k + 2)-form v are conjugate in R c lRn if du = bv, 6u = 0, dv = 0, where d is the differential operator and S is the co-differential (actually this concept is slightly different from the one given by Volterra: in fact u and v are conjugate in the sense of Volterra if du = bv, see [20], pp.87-90). If n = 2, f (2) = u(x, y) + iv(x, y) is a holomorphic function and we identify v with a 2-form, then du = 6v is just the Cauchy-Riemann equation, while bu = 0 and dv = 0 are automatically satisfied.

This concept of conjugate forms is more general than the one considered by Stein and Weiss; in fact, if we identify ( u l , . . . ,.I',) with the 1-form u = uhdxh, the system (1.1) is nothing but du = 0, 6u = 0. In other words Stein and Weiss have considered only the forms which are of degree 1 and conjugate to v = 0.

We shall prove a general Brothers Riesz theorem for conjugate forms of any order in a bounded domain R which is a generalization of the classical one. We shall use the concept of k-measure introduced in [7,8]. Roughly speaking a k-measure is a differential form of degree k whose coefficients are measures. If the k-measure is defined on a Cm manifold then a k-measure can be considered as a current of order 0 (see [6]). But if we do not need to consider general currents, the concept of k-measure can be introduced in a direct and more natural way on a differentiable manifold (see [7, 8, 111).

The generalization of the Brothers Riesz Theorem we shall prove is the following: if the k-form u and the ( k + 2)-form v are conjugate in a bounded domain R and if u, v and their adjoint forms have traces on C in the sense of the h-measures, then all these h-measures are absolutely continuous. The proof of this result hinges on one side on some integral representations of differential forms (see 5 3) and on the other side on a harmonic majorization of certain subharmonic functions, which is a basic tool introduced in [17].

In the last section we give also a version of this theorem for non homogeneous differentialforms.

One of the interesting aspects of the classical Brothers Riesz Theorem is that the absolute continuity of p follows from some "orthogonality" conditions. Theorem X provides a result of this kind in Rn which generalizes the original statement in C.

Finally we remark that the general theorems obtained in the theory of uniform algebras do not seem to imply the results proved in this paper.

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2. Some results in potential theory. Let s(x, y ) be the fundamental solution for Laplace equation:

( 2 ~ ) - ' log Is - yl n = 2 -[(n - 2)cn]-I 1x - y 1 2 - n n = 3,4, . . .

c, being the hypersurface measure of the unit sphere in Rn . Let R be a bounded domain of the real cartesian space Rn such that its boundary

OR is a Lyapunov hypersurface C, i.e. C has a uniformly Holder continuous normal field of some exponent 1 (0 < 1 5 1). v = (vl , . . . , v,) denotes the inward unit normaI to C. We want to describe the boundary behavior of the potential

where p E M(C), i.e. /I is a measure defined on the family of all the Borel sets of C. We recall that, because of the Lebesgue-Radon- Nykodim theorem, there exists a function f E L1(C) such that:

for any Borel set B c C, p* being a singular measure, i.e. a measure concentrated on a set of zero (n - 1)-dimensional Lebesgue measure of C. The measure p is said to be absolutely continuous if p* = 0.

The point xo is called a Lebesgue point of p if:

where I . I is the total variation of a measure, a is the (n - 1)-dimensional Lebesgue measure on C and C R ( x O ) = {x E C I Ix - xO1 5 R) . It is classically known (see e.g. [9]) that the set of points which are not Lebesgue points has Lebesgue zero measure on C.

Let us construct now a family of approximating domains 0,. Let X(x) be a unit vector defined and continuously differentiable on C such that X(x).v(x) 2 po > 0, Vx E C (see [12], p.273-275). By C, we denote the hypersurface xQ = x + eX(x), x E C, where lpl 5 po ( p o small enough). i2, is the bounded domain whose boundary is C,. The domains Re with 0 < Q 5 eo are contained in R, while R is contained in 0, with -Po 5 p < 0. We call C, a famaly of approxi~r~ating hypersurfaces.

Even if in the sequel we shall suppose that C is a Lyapunov hypersurface, the next results hold in the more general case of a C1 hypersurface.

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I. Let C E C1. If xo is a Lebesgue point for p , then we have:

(2.3) lim J ( s ~ ~ ( ~ ~ - Y) - s ~ ~ ( x - ~ - Y)) dpy = Y ~ ( X O ) I(XO) e-o+ C

where f E L1(C) is the function in (2.1).

The proof of this theorem in the case of an absolutely continuous measure and with v instead of X can be found in [lo]. It is well known that

uniformly for s o varying on C (see [lo]). This implies (see e.g. 1121, ~.269-271) that the function which is equal to J, s,, (x - y) do, in 52 and to a ( z ) + S, sZk (X - Y) doy on C is continuous on a. Then

1 iim s,, (x, - y) day = ~ * ( x o ) + ~ z k (XO - 8) d g ~ , p + O + d

In the same way we have

and therefore (2.3) holds when p = a . Now all we need to prove is that

(2.4) dly+ l ( s z k ( x e - 9) - ~=k(x-e - Y)) day = 0

where a = p - f (xo)a , s o being a Lebesgue point for p. Let ( T ~ , . . . , ~ ~ - 1 , v,,) be an orthonormal basis and consider the relevant coordinate

system (7;t) = ( v l , . . . , qn-1,t) with the origin in s o . We can find a constant d > 0 (independent of s o ) such that Ed G Cd(xo) admits the representation t = c(q) with

where Bd(0) = (77 E Rn-I 1 1171 5 d). It is obvious that

As far as the integral over Cd is concerned, we may write

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cz1[Ixk - ~ k l 1 [xQ - Y I - ~ - I x - ~ - Y I - " I + e ( Ize - YI-" + 11-Q - Y I - ~ )I . Since there exists a constant C such that Izj-- - yI2 2 C (lqI2 + e2 + C2(q)) = C (15 - yI2 + e2) 1: C (1qI2 + e2) (see e.g. [4], p.196), we have

and then

Because of this inequality, we have

where G is the measure defined on the Bore1 sets of Rn-': G(B) = (aI(C(B)), C(B) being the part of C d which corresponds to B n {(ql < d } through t = [(q).

On the other hand

where a' is the distribution function :

Because of (2.2) we may suppose d such that Z{(ql < 6) 5 E r n { ( ~ [ < 6) (m being the Lebesgue measure in Rn- ' ) for any 6 < d. Therefore, since

it follows

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and since the last integral is less then or equal to

This shows that

which, together with (2.6), gives (2.4), i.e, the result. We remark that we have stated this theorem by using the vector A, but since we

did not use the fact that the vector X is of class C1, the theorem is still true if we take v instead of A.

Let us now consider the Hardy-Littlewood mazimal function. Given g E L1(C), its maximal function is defined as

It is known that

for any g E LP(C). Even if nowadays this result is standard, for sake of completeness we give a sketch

of the proof. At first, let us observe that there exists a constant A such that

for any x E C and for any R > 0. In fact, if 3R < d ( d being such that (2.5) holds), then

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On the other hand, if 3R 1 d then

o(S) So (2.9) holds with A = max{\/i 3"-', -1. Now we can repeat word by word the proofs of Lemma 5.2.3 and Theorem 5.2.4 of

[16], p.68-69 (replacing the unit sphere S with our C) in order to show that

for any g E L1(C), i.e. that M is of weak type (1, l ) . Because of this inequality and since IIMgllco < llgllOO, keeping in mind that M is subadditive, we ca.n obtain (2.8) by means of the classical Marcinkiewicz interpolation theorem (see e.g. [21], Vol.11, p.111-115).

From now on we shall suppose C E C1+'. Also the next lemma has to be considered as known.

11. Let g E LP(C) and w be the double layer potential

There exists a function G E LP(C) such that I w ( x ~ ) ~ 5 G(z), Vx E C, 0 < e < PO.

Note that there exists a positive constants C1 such that

Since

On the other hand, if we denote by O the set C n { 2 j P 5 Ix - y 1 < 2j+l we have

do, = Q j=1

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These integrals can be estimated by means of the Hardy-Littlewood maximal func- tion:

and

e

cl"

hy 5 C:Q 2 / do, j j= 1 @ . - @ . ( ( x - y l 2 + e2)-i J I - 1

where all the constants can be chosen independent of Q. Therefore (2.11) gives

I w ( ~ e ) l I G ( x )

Due to the weak singularity, the integral belongs to LP(C) and then G ( x ) E LP(C), because of (2.8).

The following lemma of functional analysis will be useful.

111. Let {T,) be sequence of linear and compact operators from a reflexive Banach space B into itself. Let us suppose that T,+T, i.e. (IT, - TI( 4 0 and N(I + T) = ( 0 ) . If U , + Tnun = v, and v , - v (i.e. v, weakly tends to v ) , then u, - u , where u + T u = v .

At first let us show that the sequence j)u, 1 1 is bounded. Assume the contrary. There exists a subsequence of {u, ) , which we still denote by {u, ) , such that llunll + CO. Then - we have G, + TnGn = G,, where G , = &, v, = a. Since { G , ) is bounded, there exists a subsequence {G,,) weakly tending to an element u E B. We may write

Since T is compact, Tii,, 4 T u ; moreover JI(T,,, - T)G,,, 11 5 IITnk - TI( + 0 and Gn, -+ 0. Therefore G,, -+ u and u + T u = 0. This implies that Cnk -, 0 and this is a contradiction because )121,, 11 = 1.

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The sequence I(unll being bounded, there exists a subsequence unk - u. Since u,, +T,,u,, = v,,, Tnkunk = (T,, - T ) u n k + T u n k + TU and v,, - v , it follows that u + T u = v. The same arguments show that, for any subsequence {u,,), there exists a subsequence {u,,,) weakly tending to u , which is the only solution of the equation u + Tu = v. This implies that u , - u .

Let us consider now the double layer potential on C,:

If 1C, is a function defined on C , and we set $ ( y ) = $(ye ) , we can consider $ as a function defined on C. Then the double layer potential on C, can be considered as a linear and continuous operator T, from LP(C) into itself. If we denote by T the double

the following theorem is true:

IV. T,*T

We can cover C with a finite number of { & ( x i ) } in such a way (2.5) holds. Let { E i ) be a partition of unity subordinate to this cover. We have that

for y E Ci = C d ( x i ) , where do, = Jigdo. This can be written as

In addition to (2.10), the following inequalities hold for any x , y E C and for any O<e<eo:

I2 - Y I I C Ixe - Y ~ I

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with C and H independent of e. The kernel Kel (x , y) defines a singular integral operator S, whose kernel on Ci is

given by:

if we denote by S h the singular integral operator whose kernel is (xh - yh) ( x - yl-" , we may write IISe9llp = IIS~(;O[Y~ - uh~Jtg])lIp 5 l l S l l l l ~ h -~heJieIlcoll9llp Since llvh - vhpJi,ll, -) 0, it follows l/Sell -) 0.

The vector X(x) can be taken as a COo vector defined all over Rn (see 1121, p.273- 275) and so we may write

X ( 5 ) - X(y) - X(x) - X(y) - grad X(x) . ( 2 - y) - x - Y + grad X(x) . ----- . 1.2: - yln 12 - yl" lx - yIn '

the first kernel has a weak singularity on T, while the second defines a singular integral operator. Therefore the sequence of integral operators whose kernels are K,z(x, y ) uniformly tends to 0 as p -+ 0.

Finally let us observe that

and

I and then we have A similar inequality holds for 1 & -

This implies that

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where s is such that 0 < s < 1. This shows that also the sequence of integral operators whose kernels are K g ~ ( x , y) uniformly tends to 0 as Q --t 0.

The uniform convergences we have proved imply the result.

3. k-forms and k-measures. In this section we start recalling some definitions about k-forms and k-measures.

Let Vm C Rn be an oriented m-dimensiona.1 differentiable simple manifold of class p (i.e. V" is CP-homeomorphic to the fundamental polyhedron of Rm and an orientation has been fixed) . A differential f o rm of degree k (briefly a k-form) on Vm is represented in an admissible coordinate system ( t l , . . . , t m ) as

where us,. ..,, are the components of a skew-symmetric covariant tensor. By Ci(Vm) we denote the space of the k- forms whose components are continuously differentiable up to the order q in a coordinate system of class Cqtl (and then in every coordinate system of class Cq+'). L i (Vm) denotes the space of the k- forms whose components are LP functions in a coordinate system of c1a.s~ C1 (and then in every coordinate system of class C1).

Let us suppose now that Vm C Rn and u E Ct(A) is a continuous k-form defined in a domain A > Vm. If the representation of u in the natural coordinate system is

the trace of u on Vm is given by:

where x = f ( t ) is a pmametric equation of Vm (if u is a 0-form, then f:u is by definition 4f (t)l).

The d ~ f f e r e n t i a l of u E Ci(A) is the (k + 1)-form defined by

while the co-dzffcrentinl of u is the ( A : - 1)-form

*u being the adjoint of u:

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The laplacian of u E C,?(A) is the k-form A2u = -(d6 + 6d)u; a direct computation shows that

1 . . . AZu = A2~~sl , . .8kdxS1 dxSk

and then a k-form is harmonic (i.e. A2u = 0 ) if and only if all the coefficients u,,,..,, are harmonic functions.

A k-measure p is the object determined (in a fixed coordinate system (t', ..., tm) ) by the coefficients pa, ..,,, (B), where /la, . . .Sk is a measure defined on the family { B ) of all the Bore1 sets of Vm and pal ,,.,,(B) depends skew-symmetrically on the indices sl ... s k . Moreover if ...jk (B) are the components of the k- measure p in the coordinate system ( r l , . .., r m ) , we must have

for every Borel set B E {B). hfk(Vm) denotes the space of all the k-measures on Vm. A Lebesgue-Radon-Nykodim decomposition holds also for a k-measure: if p E Mk(Vm), there exists a k-form f E L i ( V m ) and a singular measure p* E Mk(Vm) such that

where x(B) is the in-dimensional Lebesgue measure on Vm. If p* = 0, p is said to be absolutely continuous.

If u is a k-form and v is an h-form, their exterior product is

If u E Ck(Vm) and p E Mr,(Vn') the exterior product 13 = u /\ p is the (k + h)-measure whose components are

for any Borel set B; p A u is by definition ( - l )kh(u /\ p). By using a partition of unity, the previous definitions can be extended to oriented

m-dimensional differentiable manifold. For further details we refer to 17, 8, 111. sk(x, y ) will denote the following double k-form introduced by Hodge:

. . . . . . . sk(x, y) = ~ ( x , y)d.xJ1 dxikdyJ1 dyik j~ < . . . < j k

It satisfies the following identities (see [ 5 ] , p.309):

The next lemma shows some properties of potentials having sk(x, y ) as kernel.

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V. Let p E C:(a). The following formulas hold:

uniformly for y varying on C.

At first let us prove (3.5); we may write (see (3.1))

If we denote by fil ,,,, (x) = * ~ ( p ( x ) A dxil . . . dxin-1-h ) from (2.3) it follows that

because vj(y)dyj = 0. In order to prove (3.2) we recall (see [5], p.308) that, if u E C;f(a),

- Lz[Sh(Y,i) A * ~ u ( x ) - ~ Y ( s ) A *SD(Y. x)] = z ~ ( Y ) Y E 0 Y @

If 1C, is a (n - 1)-form on C, say 1C, = +" do, is $o.

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Let c(x) be the form kc; , ,,,,,, dzil . . . d:r'" where c,, ,.,,, are constants. From (3.6) we have:

and thus

= c(y)

On the other hand (3.5) shows that

Let now p E C;(n). In order to have (3.2), it is now sufficient to show that, for a fixed y E C:

where the coefficients of c(x) are p;,,,.i,, ( 9 ) It is obvious that

and that we can choose d (independent of y ) in such a way that ) p , , ... ;, ( x ) - p i , ,.,;, (y)l < E for any x E Ed and for any i l . . . i h . From (2.7) it follows that

and this gives (3.7). Let us consider now (3.3) and (3.4). If c ( r ) is a differential form with constant

coefficients, we have

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and this implies immediately that (3.3) and (3.4) are true for ~ ( x ) = c(x). These formulas for any p E C:(n) can now be established by repeating the proof of (3.7). It is not difficult to check that all the limits we have considered are uniform with respect to y E C.

If u is a k-form continuous in n, then the trace of u on C is defined as the k-form on C which in local coordinates is given by f:u. In order to precise in which sense a k-measure on C is the trace of a k-form defined in R, let us consider a family of approximating hypersurfaces {C,). We say that the k-form u E Ck(R) admits a trace a E Mk(C) in the sense of k-measures with respect to the family {C,) if

It is worthwhile to remark that this definition depends on the chosen family {C,}, as the following example shows: let n = 2 and C = {(x, y) E R2 I y = 0, 0 5 x 5 I ) , u(x,y) = yZ/(x2 + y6). If we take {C,} obtained by X(x) = (cos29,sin29), 29 being a constant 0 < zP < :, the function u has a null trace with respect to {C,). In fact, for any continuous p we have:

= M e --+ 0 (as p + 0') . (1 + e cos 29) cos 29

On the other hand, if we take X(x) = (0, I ) , we have

1 1 e2 dx = - arcta.n - -+ +ca (as e + o+) , e3 which shows that the function u does not admit a trace with respect to this new family.

The following theorem will play an essential role in the next section.

(2) If w is a h-form, then d* w = ( - l )h *6w. In fact d* w = ( - l ) ( n - h - l ) ( n + l ) * * d * ~ = (-l)(n-h-l)("+')(-l)"(h+l)+l * hw = (-l)h * Sw.

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VI. Let f E C:(R), g E C:+?(R), h E Ci-,(R) be such that

Let us suppose that f , * f , *g, h admit traces in the sense of h- measures with resgect to one approximating family {C,). Namely let a E Mk(C), G E Mn-k(C), E Mn-k-~(C)l -y E Mk-2(C) be the traces o f f , * f , * g , h respectively. Then the following formula holds:

Moreover f and * f admit the same traces a , i3 with respect to any approximating family {CQ)

F'rom (3.8) it follows that f is harmonic in 52: A2 f = -(d6+6d) f = -d2h-62g = 0. Then (3.6) gives:

On the other hand, from (3.8) we have *df = *6g = (- 1) kd * g (see (')) and then

Since d(sk ~ * g ) = dsk ~ * ~ + ( - l ) ~ s ~ ~ d * g and d(hA*sk) = d h ~ * s k f ( - l ) k - 2 h ~ d * ~ k , by means of an integration by parts the last integral is equal to:

Therefore we may write

Letting e + 0' and recalling the definition of trace, we obtain the sought integral formula.

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Let now {C,} be a different approximating family. - From the integral formula we have just proved, it follows that for any p E C r - l - k ( 0 )

We may write (see (3.1))

Moreover (3.10'1

The last integral tends to - p ( y ) unifor~nly for y E C because of lemma V (where h = n - 1 - k). Let us show now that the other integrals tend to 0, namely that

(3.12) lim LEO e-O+ P(x, ) A : d z [ ~ n - i - * ( ~ . ~ - * ) ] - J + c P ( X ) A : d z [ ~ n - ~ - k ( ~ e 7 ~ ) ] = 0 .

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In fact if we write p(xe) = pi,,.,i,(x,)dx~ . . . dx) and we consider it as a differential form on C we have

Since

we have that p(se) - p(x) = O(p). By means of easy calculations, it follows that [p(xe) - p(x)] A *,d,[sn.(y,x,)] = O(plx, - y/ l -n ) = O(e1/21x - YI'-n+1/2) and so

uniformly for y E C. On the other hand: * z d z [ s n - l - ~ ( y , ~ e ) - sn-l-k(y-,, x)] = u ( @ ' / ~ 1s- y 11-n+1/2) (this inequality follows from the fact that sy, (y, xe)-sy, ( Y - ~ , X ) = O(p Ix - yI (lx - yI2 + e2!)-n/2) (see [4], p.197)) and then

uniformly for y E C and so (3.11) holds wit,ll the sa.me kind of uniformity. Analogously (3.12) holds and then we have proved tha.t

uniformly for y E C. In the same way, from theorem V, it follows that

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= lim ( - I ) " ~ Q-O+

P(XQ) A ; S z [ ~ n - k + l ( ~ , x e ) - ~ n - k + l ( ~ , x - p ) ] = 0

uniformly for y E C. All the limits we have proved show that

i.e. the trace o f f is cu with respect to {Y,}. In the same way one can prove that the trace of * f is G.

4. The Brothers Riesz Theorem in R". We say that u E Ci(S1) and v E C;+,(R) are conjugate if

As remarked in the Introduction, if n = 2, k = 0 this definition coincides with the usual definition of conjugate harmonic functions.

VII. Let u E CL(52) and v E C;+2(R) be two conjugate forms. Let us suppose that they and their adjoint forms admit traces in the sense of h-measures with respect to o_ne approximatingfamily {C,). Namely let cr E Adk(C), G E Eln-k(C), ,b E Mk+z(C), P E Mn-h-2(C) be the traces of u , *u, v, *v respectively. Then the following formulas hold:

(3) From (3.1) it follows that

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.-.. Moreover u, *u, V , *v admit the same traces a, G , ,!3, with respect to any approximating family { C e )

The result follows immediately from Theorem VI taking f = u , g = v , h = 0 for the first formula and f = v, g = 0, h = u for the other one.

If n=2 and f ( 2 ) = ~ ( 2 ) + 2 2 4 2 ) is a holomorphic function, then these formulas are nothing but the real and imaginary part of the Cauchy integral formula.

We are now ready for the main result: the extension of Brothers Riesz theorem to higher dimension.

VIII. Let u E C:(R) and v E C;+2(S2) be two conjugate forms. Let us suppose that u , *u , v , *v admit traces on C in the sense of h-measures with respect to one approximat- ing family (and therefore to any approximating family). Namely let cu e Mk(C), E Mn-k(C), /3 E Mk+2(C), E M n 4 . 4 ( C ) be the traces of u ,*u ,v , *v respectively. Then all these h-measures are absolutely continuous.

Let u = utl ,, dx'l . . . dx'h. At first let us prove that there exist the following limits when x , -, x on X(x)

for a.e. x E C and that $ ; , , . , i , belong to L1(C). In fact from theorem VII it follows that

uil ... ik ( x e ) = L z { a , A ; d g [ s ( y , r e ) d y i 1 . . . d y i k ] -

Let us consider the first term in this integral. In the same way we can study the other ones. We may write

where p h ' l . . . i k is the exterior product a A d y h d y i l . . . dyik . Theorem I implies that

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CONJUGATE DIFFERENTIAL FORMS 89

a.e. on C with f h i ,..., E L1(C) . This proves (4 .2) . Let now F ( x ) be the vector whose ( i ) + (k;2) components are all the components

of u and v, i.e. ui ,.,, i k and v, ,,,, ,,+,. The system (4 .1) can be considered as a partial differential system of the first order in the unknown F ( x ) :

where Ah are matrices with constant coefficients. We know already that if u and v are solutions of (4 .1) then they must be harmonic forms, i.e. all their components are harmonic functions. This shows that all the solutions of the system (4.3) are harmonic functions. This implies that there exists 0 < a < 1 such that the function IF(x)la is subharmonic in E (see [18], p.231-234). On the other ha.nd, from theorem VII it follows that

P

This means that IF(x,)la is a family of functions bounded in LP(C), where p = a > 1. Then there exists a sequence {pi} tending to 0 such that JF(xe i ) la is weakly convergent to a function f ( x ) in LP(C).

Let now ggi be the solution of the following integral equation on Cei

The double layer potential

is therefore an harmonic function defined in !2, which is equal to IF(xQi)Ja on Spi. Because of lemma 111, the sequence { g Q , } is weakly convergent to a function g E LP(C). Let us fix x E St,,; since (F(x ) la is subharmonic, we have that

and letting ~i -+ 0

Inequality (4.4) holds for any x E. Theorem I1 shows that

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90 ALBERT0 CIALDEA

being [G(x)]P E L1(C). Because of the Lebesgue dominated convergence theorem, we have

where .li, = $; ,,,. ;, dxil . . . dxik Therefore

and this means that o! = II, is absolutely continuous. We have also

- which shows that i5 is absolutely continuous. In the same way we obtain that /3 and P are absolutely continuous.

It is natural to ask if this result holds if we suppose only du = 6v instead of (4.1). The answer is negative. For sake of simplicity we shall give a counterexample in the half-space. Let us consider the function y (z , y, z) = z2 (x2 + y2 + z2)-4 in the upper half-space {z > 0) and let u = dv. We have du = dzV = 0 and then du = 6v where v = 0. By means of a direct conlputation it is possible to show that

u(x, y, z ) = (x2 + y2 + z2)-4 {-3z2(xdr + ydy) + z[2(x2 + y2) - z2]dz)

admits a null trace on the plane { z = 0) and that the trace of *u is given by 2 ~ 6 ( ~ , ~ ) dxdy, 6(0,0) being the Dirac's measure concentrated in (0,O). Therefore u, *u, v, *v admit k- measures as traces, but not all of them are absolutely continuous. If we want an ex- ample with a compact support, we can consider, instead of the previous f , the function f (x , y, z) = w(x, y, z)z2 (x2 + Y2 + z2)-4 where w is a CDO function which is equal to 1 in a neighborhood of (O,0,0) and has a compact support.

The conditions (4.1) seem to be natural for the following reason. Let us consider n = 2; it is easy to see that if f ( z ) = u(z, y) + zv(x, y) is a holomorphic function, then there exists, at least locally, a (real) harmonic 1-form U(x, y) such that 6U = u and dU = - * v. In other words all the conjugate harmonic functions can be obtained by considering 6U and d U , U being an harmonic 1-form. Now, if we take an harmonic (k+l)-form U(x) in Rn and we consider u = 6U and v = -dU, then u and v are such that du = d6U = -6dU = 6v and moreover Su = 6'U = 0, dv = -@U = 0, i.e. (4.1) is satisfied.

Another question is if it is necessary to consider the traces of u, *u, v,*v and it is not sufficient to consider only the traces of u, v. The next example shows that

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CONJUGATE DIFFERENTIAL FORMS 9 1

we need to consider all the traces of u, *u, 21, *v. Let U = (x2 + Y2 + z2)-fdz and u = 6U, v = -dU. u and v are conjugate in the half-space {z > 0), because U is a harmonic form. Since

z 1 u = v = , (-ydydz + xdzdx)

(r2 + y2 + 9); (x2 + y2 + 2 2 ) ~

we have that the trace of u on the plane {z = 0) is S(o,o) and the trace of v is 0. Then the trace of u and v are measures, but not all of them are absolutely continuous. Note that, if n = 2 and f(z) = u(x, y) + iv(x, y) is a holomorphic function, then *U is a 2-form and then its trace on C is necessarily 0. Analogously, if v is identified with a 2-form, its trace is 0 and then we need only to consider the trace of *v. This is why in the classical Brothers Riesz theorem only two traces appear.

Now we want to give a generalization of theorem VIII for non homogeneous differ- ential forms. Let U € C,'(R) $ . . . $ CA(S1) be a non homogeneous differential form, i.e. U = C;=, uh, where uh is a differential form of degree h. We say that U is self-conjugate if dU = 6U, i.e if

(note that 6uo = 0 and dun = 0 are automatically satisfied). If U is self-conjugate then all the differential forms uh are harmonic. In fact, if we set A2U = x;=, &uh, then A2U = -(6d + db)U = -b2U - d2U = 0.

We may also consider Mo(C)$. . . $ MnPI(C) whose elements are non homogeneous k-measure O = fib, where dh E Mh (C).

It is obvious that, if we say that U admits trace in the sense of k- measures on C, this means that uh ( h = 0, . . . n - 1) admits a trace a h E Mh(C) in the sense introduced in 5 3 (the trace of u, is necessarily 0).

IX. If the self-conjugate form U E C;(R) @ . . . $ Ci(R) is such that U and *U admit traces on C in the sense of k-mea.sures with respect to one approximating family (and therefore to any approximating family), then these traces are absolutely continuous.

If we set uh = O for h = -2, -I , n + 1, n + 2, we may write duk = Suk+2, buk = duk-2 for any k = 0,. . . , n . Let ak , Gk be the trace of uk, *uk respectively. Theorem VI gives

Now the proof of the present theorem can be established step by step by repeating the proof of theorem VIII.

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92 ALBERT0 CIALDEA

Let us consider the syst,em of homogeneous harmonic polynomials:

where Y,k(w), . . . Yki,,(w) is a complete system of (surface) spherical harmonics of degree k (see e.g. . [18], . Ch.IV, § 2). Let us denote by {wh) this system ordered in one sequence and by w:""' the k-form wh(x)dxll . . . d z t h .

X. Let R be such that Rn -32 is connected. If a = ( a 0 , . . . , an- ' ) , G = ( P , . . . , G1) E Mo(C) $ . . . $ Mn-I(',) are such that

(where ak = 2"'k - - 0 , k = - 2 , - l , n ) f o r a n y 1 < i l < . . . < i k ~ n , h = 1 , 2 , . . . , k = 0,1 , . . . , n then a , G are absolutely continuous.

Let D be an open ball centered in 0 and containing D. We have

where wlk . . .wpnkk are linearly independent harmonic polynomials of degree k. For any fixed x E Rn - D this series is uniformly convergent for y E C. Then (4.7) implies

- for any x E Rn - D and therefore for any n. E R" - R. This means that

Since (d6 + 6d)sh(y,z) = 0 (;c # y) and t,aking into account (3.1), we have

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CONJUGATE DIFFERENTIAL FORMS

Moreover

Let now uk(x) be the k-form given by the left hand side of (4.8). From (4.9) and (4.10) it follows that

for k = 0 , . . . , n - 2 and 6ul = 0, = 0, i.e. U = (uo , . . . u n ) is self-conjugate in fl (see (4.5)). (4.8) shows that for uk(x) (4.6) is true. As in Theorem IX, this implies that LY and i5 are absolutely continuous.

We remark that , if n = 2, this result is equivalent to the original Brothers Riesz theorem. Theorem X contains several results which seem to be interesting in themselves. If n = 3, for example, the next one is a particular case of X: XI. Let R be such that R3 - is connected. If a0 E Mo(C), a' E Ml(C), LY' E Mz(C) are such that

[a0 A *dwh + dwh A a'] = 0

for any h = 1,2,. . . and

for any 1 5 il < i2 5 3, h = 1,2 , . . . , then a", cvl, cv2 are absolutely continuous.

We only need to take cr = ( ao , 0, a'), G = (0, a ' , 0) in theorem X.

References. 1. Aizenberg, A., Dautov, Sh. A,: Differential Forms Orthogonal to Holomorphic

Functions or Forms, and their Properties, Trans. of Math. Monogr. 56, AMS, 1983.

2. Bochner, B.: Boundary Values of Analytic Functions in Several Variables and of Almost Periodic Functions, Ann. of Math., 45 (1944), 708-722.

3. Cassisa, C., Fichera, G.: A mechanical interpretation of the Brothers Riesz The- orem, in Elasticity: Mathematical Models and Applications, edited by Eason & Ogden, Ellis Horwood, 1990.

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94 ALBERT0 CIALDEA

4. Cialdea, A.: Sul problema della clerivata obliqua per le funzioni armoniche e que- stioni connesse, Rend. Accad. Naz. Sci. XL, 12 (1988), 181- 200.

5. Colautti, M. P.: Sui problemi varia.zionali di un corpo elastic0 incompressibile, Mem. Acc. Naz. Lincei, (8) 8 (1967), 291-343.

6. De Rham, G.: Varikte's diffhentiables (formes, courants, formes harmoniques), Act. Scient. Industr., Hermann & C., 1955.

7. Fichera, G.: k-misure su una varieti differenziabile, Rend. Sem. Mat. e Fis. Milano, 30 (1960), 45-58.

8. Fichera, G.: Spazi lineari di k-misure e di forme differenziali, in Proc. of Intern. Symposium on Linear Spaces, Jerusalem 1960, Israel Ac. of Sciences and Humani- ties. Pergamon Press, 1961, 175-226.

9. Fichera, G.: Lezioni sulle trasformazioni lineari, 1st. Matem. Univ. Trieste, Veschi, Roma 1962.

10. Fichera, G.: Unification of Global and Local Existence Theorems for Holomorphic Functions of Several Complex Variables, Mem. Acc. Naz. Lincei, (8) 18 (1986), 61-83.

11. Fichera, G.: Conditions for holomorphy of L:,, functions, in Partial differential equations and related subjects, edited by M. Mira.nda, Pitman Res. Notes in Math. Series 269, Longman, 1992.

12. Fichera, G., De Vito, L.: Funzioni analitiche di una variabile complessa, I11 Ediz., Veschi, Roma, 1971.

13. Kuran, U.: n-dimensional ext,ensions of theorems on conjugate functions, Proc. London Math. Soc., (3) 15 (1965), 713-730.

14. Muckenhoupt, B., Stein, M.: Classical expansions and their relation to conjugate harmonic functions, Trans. Am.er. Math. Soc., 118 (1965), 17-92.

15. Riesz, F. and M., ~ b e r Randwerte einer analytischen Funktion, Quatridme Congrds des mathe'maticiens scandinaves, Stockolm (1916), 27-44.

16. Rudin, W.: Function Theory in the Unit Ball of en, Springer Verlag, New York Heidelberg Berlin, 1980.

17. Stein, E. M., Weiss, G.: On the Theory of Harmonic Functions of Several Variables, Acta Math., 103 (1960), 25-62.

18. Stein, E. M., Weiss, G.: Introduction to Fourier Analysis on Euclidean Spaces, Princeton University Press, Princeton, 1971.

19. Stout, E. L.: The Theory of Uniform Algebras, Bogden and Quigley, Tarrytown on Hudson, New York, 1971.

20. Volterra, V.: Theory of Functionals and of Integral and Integro-Differential Equa- tions, Dover Publ. Inc., New York, 1930.

21. Zygmund, A.: Trigonometric Series, Cambridge University Press,1979.

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