THE ATOMIC DECOMPOSITION OF HARMONIC FUNCTIONS SATISFYING CERTAIN CONDITIONS OF INTEGRABILITY

Distributions on Euclidean spaces with derivatives of their Poisson integral satisfying certain natural conditions of integrability are represented as sums of weighted atoms. The atomic decomposition is obtained by means of the Calder6n reproducing formula.


INTRODUCTION.
In [1] Bloom and DeSouza introduced the weighted special atom spaces B(,) {f E D'(T) f ,__ cb,, ',,__ c, [< o}, where T is the unit circle in the plane and D'(T) are distributions on T. The weight function is a member of a class defined by Bloom and DeSouza in [1]. Each is a special atom, that is either bn(t) =-(2r) -1 or b,(t) ([ I I) -1 I 1-1 {XR(t) XL(t)}, T, where is an interval in T, with the left and right halves L and R. (Our notation is slightly different from that of [1].) As usual I denotes the length of and XE the indicator function of E. With the usual atomic norm, B() becomes a Banach space and yields an atomic characterisation of some well-known spaces on the unit disc. For example, given a . (-1, 1)  The purpose of this paper is to investigate analogous spaces on the real line and, more generally, on the Euclidean spaces of arbitrary dimension. We develop a technique based on the Calder6n reproducing formula to prove decomposition theorems in these cases. Also, we obtain a sufficient and necessary condition for the weight function w to admit the atomic decomposition of the space S defined below. The method of decomposition is different from that of Bloom and DeSouza [1], and we neither have to investigate the dual space nor the boundary values of elements of S. The decomposition of S in terms of the special atoms falls into three steps. The first consists in the decomposition of S into so called Poisson atoms, which are relevant to properties of S. Then 626 K. BOGDAN we inquire into a class of the Fourier multipliers on S'. These are finally used in the proof of the decomposition theorem for the special atoms. In the last part of the paper we sketch out some possible generalisations of the method.

PRELIMINARIES.
All the vector spaces considered in this paper are over the field of the real numbers. We denote by N the set of all the non-negative integers and by N+ the set of all the positive integers. DEFINITION 2.1. Let (X, ]1" ]1) be a Banach space and let K -q} be a bounded subset of X.
The set of all b E X, which are of the form b= Zc'b'' Z ]c, I< oo, b, K, n N+, (2.1) n=l n=l is the atomic space B B(K) spanned by K.
We endow the space with the atomic norm b lIB inf -],a c, I, where the infimum is taken over all the possible representations (2. tlln(t) if2<t<c, are admissible for every -1 < c,/3 < 1, and 7, d R. To check this we apply the above remark.
The notion of admissibility arises naturally from the investigation of atomic subspaces of S'. We define it separately in order to establish firstly some preliminary properties of S it induces.  Proof. From (3.1) we get I sup>0v(s)-sf(t/s + 1)-(t)dt < . Since f(t/s + 1)-v(t)dtfcv(t)dt > 0 as s c, it follows that limsup_..cv(s)-'s-' < , hence there are positive constants g and c such that v(s) > csfor s > $. We have > I > sup,>oW(S)-s-c f(t/s + 1)-t-dt. The integral f(t/s + 1)-t-dt tends to , so v(s)-'s-' As usual, we write Pt, to denote the Poisson kernel, that is P,(.) c.(llxll + t)-"+1/, x n-, > 0 c. r( ' + 2 Note that definition of P, agrees with the notation of the dilation (P) if we put P P.  where V depends only on n. This y}e]ds that T is bounded. The Posson integral T, P,_,(z), > s, x R", is equal to T in h as a function of z and (it can be easily defined for 0 < _< s).
Therefore sume that T itself satisfies Tt T Pt-,, > s, and T, L(R"). By the sumption and by (3.6) and (3.7), it follows that there is a constant C such that Tt(x) C uniformly in z R", as .
where {e," < < n} is the standard basis in Accordingly, let ' G ' Pt.
The set of the Poisson w-atoms is bounded in S if and only if w is admissible. Proof. It suffices to prove (4.1), the rest being immediate. Let G G,K; K. Note that the function t-Gt(x) OPt(x) is harmonic in R +. It follows easily, that for every s > 0, the function ,(x,t) s(t + s)-Gt+,(x), x q R", > 0, is equal to the Poisson integral of G,. Direct differentiation gives V,,(z,t) s(t + s)-lCt+,(x), x R", s,t > 0. Therefore , I1 f( / s)-2w(t)dt fR, Et+,(z)dz, so (4.1) holds with 0 < C -II To state our next result we need some preliminaries.
Assume that w is admissible. Let F S'. Fix i, < < n, and consider the function R + S given by (V,.s) ;,,sO, F(y, s). We claim that it is continuous. Indeed, it is enough to check that the map (V,.) ;, is continuous. Fix a > 0, 0 R". Let s and y . We have   The Riemann integral conimutes with continuous linear operators, hence the sane result is obtained when the map acts on the integrand. Since Vlr(x,t)= t-Gt, Gr(x), x,y R", s,t > 0, we h&ve + Suppose that F f, P for some bounded f. Note that Vvf(U,s) s-f, G,(U), > 0, U 6 ". Therefore, using (4.4), we get To Bochner integral. The integrals are then well-defined because the functions N + S given by (9, s) G,, are bounded and strongly measurable and the corresponding measures are finite (see Yosida [5] for details on the Bochner integral). The integrals nay be approximated in norm by integrals of simple functions all of whose non-zero values are Poisson w-atoms. Again by Lenma    [6] and Frazier et al. [7]. For example an atomic decomposition of the homogeneous Besov spaces B' is stated in [7] (see Frazier and Yawerth [8] for the proof). Taking advantage of the result and of Theorem 4.1, it may be verified that the Poisson integrals of the distributions in ", form the space S for (t) -, o (-1, 1). We note here that the atomic decomposition is understood more generally in Frazier and Yawerth [6] and Frazier et al. [7] than in tiffs paper (see also Feichtinger and Gr6chenig [9] for a group theoretical point of view). Our approach is rather in spirit of the decomposition of the space ' which is treated separately in [7], however there are changes due to the fact that special aoms defined below are not smooth nor radial and that we consider a wide class of weight functions. Also, recall that we did not assume that the boundary values of elements in S are distributions. Now this fact may be verified using the atomic decomposition of S into Poisson w-atoms (regard (3.2), (3.3) as estimates for the size of atoms). We omit the details since the result will not be exploited in this paper.  (5.) where m is a real or con,plex function homogeneous of degree 0.

FOURIER MULTIPLIERS ON
There are (only) formal reasons for introducing the gradient V in (5.1) since we did not give a meaning to Ft()   Proof. Let us write Ca rather than C to distinguish between different numbers C depending on d.
We are now in a position to prove that S is equal to the atomic space generated by the functions b,. Note that these functions give a natural extension of the notion of the special atom on T. Note that fit,,/3(.r)dx 0 and that the .al) R" S y G E L(R'') s Ll)scltz continuous, that s c G I[,tR,,) C y 1, y G R", for a constant C. By Fubini-Tonelli theorem, this yields Note also tl,at B G IIc,R.)=ll B G/ ]lc,(R-), r > 0. Changing the roles of B and G above, we get * a I1,=11 * a/ I1(,, ",-, , > 0, witl, a constant c". Take c max(c', c") to obtain (6.3). Next, we prove (6.4) Clearly, it is enough to verify that % state the next result we need some preliminary steps. DEFINITION 6.3. Let w be admissible. Fixs>0andi: <i<n. For FSlet +* Each term in tim s,vm is understood as the Rienann integral of the function l"+t ..+ L (R")given by (, r) G, b,.Oj F(, r). The symbol of convoluLion in F0 * 6,, my be fully justified because limt_o+ Ft exists as a distribution. However, we do .or develope this poin here, and the notation Fo* b,, is purely formal. In pargicular le symbol Fo alone has no generMly prescribed meaning. To prove that l'b*b,, is well-defined, fix s > 0, and j _< i, j _< n. Clearly, the map R ''++ 9 t=l +1 is a well-defined Fourier multiplier with symbol being a negative, even function homogeneous of degree 0, which is 2n times continuously differentiable on R"{0}. Proof. First we check that (6.7) is well-defined. Fix i, n. An easy verification shows that -,y R the function n"+*-+ 9 (y, s) ,, S is continuous.  where W is certain trigonometric polymomial with smooth coeNcients depending only on (. In consequence, is continuously differentiable on N{0}. Similar arguments apply to all the derivatives up to the (2n-1)-th order. Finally, note that the integrand in (6.8) is non-positive for every > 0, and is negative if ( 0 and s is suNciently small. Therefore is negative for every ( 0. Since the integrand is an even function of , the same is true for . Proof. By Lemma 6.2 the integral in (6.9) is well-defined. Since, by definition M -, it is evident that right hand side of (6.9) is the identity of S and so (6.9) holds. We clearly have fR 'rF * b'(y)dyds fR ,Fo , b, ,(y)s-'dyds {f} .tis th bov conditions. The co,dition (6.12) may be somewhat relaxed since we can "convolve" (see Definition 6.a) F0 with both f., and (f), f e , several times (we write ](z) f(-x)). It follows that we can replace the symbol f(()in (6.12) by f()I o even by f(()I t with k e N+. Therefore -,1 '= I is the integral representation of the identity of T , and the arguments used before in this paper show that T GT. The atomic decomposition of T is also a consequence of the isomorphism between S and T. We shall sketch a proof of this isomorphism. Consider the Poisson integrals ' of the functions P', R", s > 0. They form a bounded set in S. We claim that the following linear operators are continuous: T U J(V)= [ #'U(y,s)w(s)dyds e S + s K,()= -a,, 0,(, )()@a e , ,.
For U , let V -4 = K,J(U). A method similar to that of the proof of Theorem 4.1, gives (() U(() for every ( e N and > 0, hence V U in , and -4,, K,J is the identity of . We check also that -4J = K, is the identity of S , and so the spaces S and are isomorphic. In fact, the spaces are isometric if we endow S with the norm .;( see Remark 4.2). The corresponding isometry is the operator 2 = K, which proves to be the map S F 0F .Weo mit the details.
ACKNOWLEDGEMENT. The author would like to express his gratitude to Dr. Krzystof Samotij Technical University of Wroctaw for his inspiration and guidance.