A NEW PROOF OF THE ATOMIC DECOMPOSITION OF HARDY SPACES

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1 A NEW PROOF OF THE ATOMIC DECOMPOSITION OF HARDY SPACES S. DEKEL, G. KERKYACHARIAN, G. KYRIAZIS, AND P. PETRUSHEV Abstract. A new proof is given of the atomic decomposition of Hardy spaces H p, 0 < p 1, in the classical setting on. The new method can be used to establish atomic decomposition of maximal Hardy spaces in general and nonclassical settings. 1. Introduction The study of the real-variable Hardy spaces H p, 0 < p 1, on was pioneered by Stein and Weiss [6] and a major step forward in developing this theory was made by Fefferman and Stein in [3], see also [5]. Since then there has been a great deal of wo done on Hardy spaces. The atomic decomposition of H p was first established by Coifman [1] in dimension n = 1 and by Latter [4] in dimensions n > 1. The purpose of this article is to give a new proof of the atomic decomposition of the H p spaces in the classical setting on. Our method does not use the Calderón- Zygmund decomposition of functions and an approximation of the identity as the classical argument does, see [5]. The main advantage of the new proof over the classical one is that it is amenable to utilization in more general and nonclassical settings. For instance, it is used in [2] for establishing the equivalence of maximal and atomic Hardy spaces in the general setting of a metric measure space with the doubling property and in the presence of a non-negative self-adjoint operator whose heat kernel has Gaussian localization and the Maov property. Notation. For a set E we will denote E + B(0, δ) := x E B(x, δ), where B(x, δ) stands for the open ball centered at x of radius δ. We will also use the notation cb(x, δ) := B(x, cδ). Positive constants will be denoted by c, c 1,... and they may vary at every occurrence; a b will stand for c 1 a/b c Maximal operators and H p spaces. We begin by recalling some basic facts about Hardy spaces on. For a complete account of Hardy spaces we refer the reader to [5]. Given φ S with S being the Schwartz class on and f S one defines (1.1) M φ f(x) := sup φ t f(x) with φ t (x) := t n φ(t 1 x), and t>0 (1.2) M φ,af(x) := sup t>0 sup y, x y at We now recall the grand maximal operator. Write φ t f(y), a 1. P N (φ) := sup (1 + x ) N max x α N+1 α φ(x) 2010 Mathematics Subject Classification. Primary 42B30. Key words and phrases. Hardy spaces, Atomic decomposition. 1

2 2 S. DEKEL, G. KERKYACHARIAN, G. KYRIAZIS, AND P. PETRUSHEV and denote F N := {φ S : P N (φ) 1}. The grand maximal operator is defined by (1.3) M N f(x) := sup φ F N M φ,1f(x), f S. It is easy to see that for any φ S and a 1 one has (1.4) M φ,af(x) a N P N (φ)m N f(x), f S. Definition 1.1. The space H p, 0 < p 1, is defined as the set of all bounded distributions f S such that the Poisson maximal function sup t>0 P t f(x) belongs to L p ; the quasi-norm on H p is defined by (1.5) f H p := sup P t f( ) L p. t>0 As is well known the following assertion holds, see [3, 5]: Proposition 1.2. Let 0 < p 1, a 1, and assume φ S and φ 0. Then for any N n p + 1 (1.6) f H p M φ,af L p M N f L p, f H p Atomic H p spaces. A function a L ( ) is called an atom if there exists a ball B such that (i) supp a B, (ii) a L B 1/p, and (iii) x α a(x)dx = 0 for all α with α n(p 1 1). The atomic Hardy space H p A, 0 < p 1, is defined as the set of all distributions f S that can be represented in the form (1.7) f = λ j a j, where λ j p <, j=1 j=1 {a j } are atoms, and the convergence is in S. Set (1.8) f H p := inf A f= j λjaj ( j=1 λ j p) 1/p, f H p A. 2. Atomic decomposition of H p spaces We now come to the main point in this article, that is, to give a new proof of the following classical result [1, 4], see also [5]: Theorem 2.1. For any 0 < p 1 the continuous embedding H p H p A is valid, that is, if f H p, then f H p A and (2.1) f H p A c f H p, where c > 0 is a constant depending only on p, n. This along with the easy to prove embedding H p A Hp leads to H p = H p A and f H p f H p A for f Hp. Proof. We first derive a simple decomposition identity which will play a central rôle in this proof. For this construction we need the following

3 A NEW PROOF OF THE ATOMIC DECOMPOSITION OF HARDY SPACES 3 Lemma 2.2. For any m 1 there exists a function φ C 0 ( ) such that supp φ B(0, 1), ˆφ(0) = 1, and α ˆφ(0) = 0 for 0 < α m. Here ˆφ(x) := φ(x)e ix ξ dx. Proof. We will construct a function φ with the claimed properties in dimension n = 1. Then a normalized dilation of φ(x 1 )φ(x 2 ) φ(x n ) will have the claimed properties on. For the univariate construction, pick a smooth bump ϕ with the following properties: ϕ C0 (R), supp ϕ [ 1/4, 1/4], ϕ(x) > 0 for x ( 1/4, 1/4), and ϕ is even. Let Θ(x) := ϕ(x + 1/2) ϕ(x 1/2) for x R. Clearly Θ is odd. We may assume that m 1 is even, otherwise we wo with m + 1 instead. Denote m h := (T h T h ) m, where T h f(x) := f(x + h). We define φ(x) := 1 x m 1 h Θ(x), where h = 8m. Clearly, φ C (R), φ is even, and supp φ [ 7 8, 1 8 ] [ 1 8, 7 8 ]. It is readily seen that for ν = 1, 2,..., m ˆφ (ν) (ξ) = ( i) ν x ν 1 m h Θ(x)e iξx dx and hence ˆφ (ν) (0) = ( i) ν On the other hand, ˆφ(0) = φ(x)dx = 2 R R R x ν 1 m h Θ(x)dx = ( i) ν+m 0 R Θ(x) m h x ν 1 dx = 0. 3/4 x 1 m h Θ(x)dx = 2( 1) m Θ(x) m h x 1 dx. However, for any sufficiently smooth function f we have m h f(x) = (2h)m f (m) (ξ), where ξ (x mh, x + mh). Hence, m h x 1 = (2h) m m!( 1) m ξ m 1 with ξ (x mh, x + mh) [1/8, 7/8]. Consequently, ˆφ(0) 0 and then ˆφ(0) 1 φ(x) has the claimed properties. With the aid of the above lemma, we pick φ C0 ( ) with the following properties: supp φ B(0, 1), ˆφ(0) = 1, and α ˆφ(0) = 0 for 0 < α K, where K is sufficiently large. More precisely, we choose K n/p. Set ψ(x) := 2 n φ(2x) φ(x). Then ˆψ(ξ) = ˆφ(ξ/2) ˆφ(ξ). Therefore, α ˆψ(0) = 0 for α K which implies x α ψ(x)dx = 0 for α K. We also introduce the function ψ(x) := 2 n φ(2x) + φ(x). We will use the notation h k (x) := 2 kn h(2 k x). Clearly, for any f S we have f = lim j φ j φ j f (convergence in S ), which leads to the following representation: For any j Z [ f = φ j φ j f + φk+1 φ k+1 f φ k φ k f ] Thus we arrive at (2.2) f = φ j φ j f + = φ j φ j f + k=j 1/4 [ ] [ ] φk+1 φ k φk+1 + φ k f. k=j ψ k ψ k f, f S j Z (convergence in S ). k=j Observe that supp ψ k B(0, 2 k ) and supp ψ k B(0, 2 k ).

4 4 S. DEKEL, G. KERKYACHARIAN, G. KYRIAZIS, AND P. PETRUSHEV In what follows we will utilize the grand maximal operator M N, defined in (1.3) with N := n p + 1. The following claim follows readily from (1.4): If ϕ S, then for any f S, k Z, and x (2.3) ϕ k f(y) cm N f(x) for all y with y x 2 k+1, where the constant c > 0 depends only on P N (ϕ) and N. Let f H p, 0 < p 1, f 0. We define (2.4) Ω r := {x : M N f(x) > 2 r }, r Z. Clearly, Ω r is open, Ω r+1 Ω r, and = r Z Ω r. It is easy to see that (2.5) 2 pr Ω r c M N f(x) p dµ(x) c f p H p. r Z From (2.5) we get Ω r c2 pr f p Hp for r Z. Therefore, for any r Z there exists J > 0 such that φ j φ j f c2 r for j < J. Consequently, φ j φ j f 0 as j, which implies (2.6) f = lim K K k= ψ k ψ k f (convergence in S ). Assuming that Ω r we write := { x Ω r : dist(x, Ω c r) > 2 k+1} \ { x Ω r+1 : dist(x, Ω c r+1) > 2 k+1}. By (2.5) it follows that Ω r < and hence there exists s r Z such that E rsr and = for k < s r. Evidently s r s r+1. We define (2.7) F r (x) := ψ k (x y) k s E ψ k f(y)dy, x, r Z, r and more generally (2.8) F r,κ0,κ 1 (x) := κ 1 k=κ 0 ψ k (x y) ψ k f(y)dy, s r κ 0 κ 1. It will be shown in Lemma 2.3 below that the functions F r and F r,κ0,κ 1 are well defined and F r, F r,κ0,κ 1 L. Note that supp ψ k B(0, 2 k ) and hence ( (2.9) supp ψ k (x y) ψ ) k f(y)dy + B(0, 2 k ). On the other hand, clearly 2B(y, 2 k ) ( ) Ω r \ Ω r+1 for each y E, and P N ( ψ) c. Therefore, see (2.3), ψ k f(y) c2 r for y, which implies (2.10) ψ k ( y) ψ k f(y)dy c2 r, E. Similarly, (2.11) E E φ k ( y) φ k f(y)dy c2 r, E. We collect all we need about the functions F r and F r,κ0,κ 1 in the following

5 A NEW PROOF OF THE ATOMIC DECOMPOSITION OF HARDY SPACES 5 Lemma 2.3. (a) We have (2.12) E r k = if r r and = r Z, k Z. (b) There exists a constant c > 0 such that for any r Z and s r κ 0 κ 1 (2.13) F r c2 r, F r,κ0,κ 1 c2 r. (c) The series in (2.7) and (2.8) (if κ 1 = ) converge point-wise and in distributional sense. (d) Moreover, (2.14) F r (x) = 0, x \ Ω r, r Z. Proof. Identities (2.12) are obvious and (2.14) follows readily from (2.9). We next prove the left-hand side inequality in (2.13); the proof of the right-hand side inequality is similar and will be omitted. Consider the case when Ω r+1 (the case when Ω r+1 = is easier). Write U k = { x Ω r : dist(x, Ω c r) > 2 k+1}, V k = { x Ω r+1 : dist(x, Ω c r+1) > 2 k+1}. Observe that = U k \ V k. From (2.9) it follows that F r (x) = 0 for x \ k sr ( + B(0, 2 k )). We next estimate F r (x) for x k sr ( +B(0, 2 k )). Two cases present themselves here. Case 1: x [ k sr ( + B(0, 2 k )) ] Ω r+1. Then there exist ν, l Z such that (2.15) x (U l+1 \ U l ) (V ν+1 \ V ν ). Due to Ω r+1 Ω r we have V k U k, implying (U l+1 \ U l ) (V ν+1 \ V ν ) = if ν < l. We consider two subcases depending on whether ν l + 3 or l ν l + 2. (a) Let ν l + 3. We claim that (2.15) yields (2.16) B(x, 2 k ) = for k ν + 2 or k l 1. Indeed, if k ν + 2, then Ω r \ V ν+2, which implies (2.16), while if k l 1, then U l 1, again implying (2.16). We also claim that (2.17) B(x, 2 k ) for l + 2 k ν 1. Indeed, clearly (U l+1 \ U l ) (V ν+1 \ V ν ) (U k 1 \ U l ) (V ν+1 \ V ν+1 ) U k 1 \ V k+1, which implies (2.17). From (2.9) and (2.16)- (2.17) it follows that ν+1 F r (x) = ψ k (x y) ψ l+1 k f(y)dy = ψ k (x y) ψ k f(y)dy + ν 2 +2 ψ k (x y) ψ k f(y)dy + ν+1 k=ν 1 ψ k (x y) ψ k f(y)dy.

6 6 S. DEKEL, G. KERKYACHARIAN, G. KYRIAZIS, AND P. PETRUSHEV However, ν 2 +2 ψ k (x y) ψ k f(y)dy = ν 2 +2 [ φk+1 φ k+1 f(x) φ k φ k f(x) ] = φ ν 1 φ ν 1 f(x) φ l+2 φ l+2 f(x) = φ ν 1 (x y)φ ν 1 f(y)dy φ l+2 (x y)φ l+2 f(y)dy. E r,ν 1 E r,l+2 Combining the above with (2.10) and (2.11) we obtain F r (x) c2 r. (b) Let l ν l + 2. Just as above we have ν+1 F r (x) = ψ k (x y) ψ l+3 k f(y)dy = ψ k (x y) ψ k f(y)dy We use (2.10) to estimate each of these four integrals and again obtain F r (x) c2 r. Case 2: x Ω r \ Ω r+1. Then there exists l s r such that x (U l+1 \ U l ) (Ω r \ Ω r+1 ). Just as in the proof of (2.16) we have B(x, 2 k ) = for k l 1, and as in the proof of (2.17) we have (U l+1 \ U l ) (Ω r \ Ω r+1 ) U k 1 \ V k+1, which implies B(x, 2 k ) for k l + 2. We use these and (2.9) to obtain F r (x) = ψ k (x y) ψ k f(y)dy l+1 = ψ k (x y) ψ k f(y)dy + ψ k (x y) ψ k f(y)dy. +2 For the last sum we have ψ k (x y) ψ ν k f(y)dy = lim ψ k ψ k f(x) +2 ν +2 ( = lim φν+1 φ ν+1 f(x) φ l+2 φ l+2 f(x)) ν ( ) = lim φ ν+1 (x y)φ ν+1 f(y)dy φ l+2 (x y)φ l+2 f(y)dy. ν E r,ν+1 E r,l+2 From the above and (2.10)-(2.11) we obtain F r (x) c2 r. The point-wise convergence of the series in (2.7) follows from above and we similarly establish the point-wise convergence in (2.8). The convergence in distributional sense in (2.7) relies on the following assertion: For every ϕ S (2.18) g, ϕ <, where g (x) := ψ k (x y) ψ k f(y)dy. k s r Here g, ϕ := g ϕdx. To prove the above we will employ this estimate: (2.19) ψ k f c2 kn/p f H p, k Z.

7 A NEW PROOF OF THE ATOMIC DECOMPOSITION OF HARDY SPACES 7 Indeed, using (1.4) we get ψ k f(x) p inf sup ψ k f(z) p inf cm N (f)(y) p y: x y 2 k z: y z 2 k y: x y 2 k c B(x, 2 k ) 1 M N (f)(y) p dµ(y) c2 kn f p H p, B(x,2 k ) and (2.19) follows. We will also need the following estimate: For any σ > n there exists a constant c σ > 0 such that (2.20) ψ k (x y)ϕ(x)dx c σ 2 k(k+1) (1 + y ) σ, y, k 0. This is a standard estimate for inner products taking into account that ϕ S and ψ C, supp ψ B(0, 1), and x α ψ(x)dx = 0 for α K. We now estimate g, ϕ. From (2.19) and the fact that ψ C0 (R) and ϕ S it readily follows that ψ k (x y) ϕ(x) ψ k f(y) dydx <, k s r. Therefore, we can use Fubini s theorem, (2.19), and (2.20) to obtain for k 0 g, ϕ ψ k (x y)ϕ(x)dx ψ k f(y) dy (2.21) c2 k(k+1 n/p) f H p (1 + y ) σ dy c2 k(k+1 n/p) f H p, which implies (2.18) because K n/p. Denote G l := l k=s r g. From the above proof of (b) and (2.13) we infer that G l (x) F r (x) as l for x and G l c2 r < for l s r. On the other hand, from (2.18) it follows that the series k s r g converges in distributional sense. By applying the dominated convergence theorem one easily concludes that F r = k s r g with the convergence in distributional sense. We set F r := 0 in the case when Ω r =, r Z. Note that by (2.12) it follows that (2.22) ψ k ψ k f(x) = ψ k (x y)ψ k f(y)dy = ψ k (x y) ψ k f(y)dy r Z and using (2.6) and the definition of F r in (2.7) we arrive at (2.23) f = r Z F r in S, i.e. f, ϕ = r Z F r, ϕ, ϕ S, where the last series converges absolutely. Above f, ϕ denotes the action of f on ϕ. We next provide the needed justification of identity (2.23). From (2.6), (2.7), (2.22), and the notation from (2.18) we obtain for ϕ S f, ϕ = ψ k ψk f, ϕ = g, ϕ = g, ϕ = F r, ϕ. k k r r k r Clearly, to justify the above identities it suffices to show that k r g, ϕ <. We split this sum into two: k r = k 0 r + k<0 r =: Σ 1 + Σ 2.

8 8 S. DEKEL, G. KERKYACHARIAN, G. KYRIAZIS, AND P. PETRUSHEV To estimate Σ 1 we use (2.21) and obtain Σ 1 c f H p 2 k(k+1 n/p) (1 + y ) σ dy k 0 r c f H p 2 R k(k+1 n/p) (1 + y ) σ dy c f H p. n k 0 Here we also used that K n/p and σ > n. We estimate Σ 2 in a similar manner, using the fact that ψ k (y) dy c < and (2.19). We get Σ 2 c f H p 2 kn/p ψ k (x y) dy ϕ(x) dx k<0 r c f H p 2 R kn/p (1 + x ) n 1 dx c f H p. n k<0 The above estimates of Σ 1 and Σ 2 imply k the justification of (2.23). r g, ϕ <, which completes Observe that due to x α ψ(x)dx = 0 for α K we have R n (2.24) x α F r (x)dx = 0 for α K, r Z. We next decompose each function F r into atoms. To this end we need a Whitney type cover for Ω r, given in the following Lemma 2.4. Suppose Ω is an open proper subset of and let ρ(x) := dist(x, Ω c ). Then there exists a constant K > 0, depending only on n, and a sequence of points {ξ j } j N in Ω with the following properties, where ρ j := dist(ξ j, Ω c ): (a) Ω = j N B(ξ j, ρ j /2). (b) {B(ξ j, ρ j /5)} are disjoint. (c) If B ( ξ j, 3ρ ) ( ) j 4 B ξν, 3ρν 4, then 7 1 ρ ν ρ j 7ρ ν. (d) For every j N there are at most K balls B ( ξ ν, 3ρν 4 ) ( ) intersecting B ξj, 3ρj 4. Variants of this simple lemma are well known and frequently used. To prove it one simply selects {B(ξ j, ρ(ξ j )/5)} j N to be a maximal disjoint subcollection of {B(x, ρ(x)/5)} x Ω and then properties (a)-(d) follow readily, see [5], pp We apply Lemma 2.4 to each set Ω r, r Z. Fix r Z and assume Ω r. Denote by B j := B(ξ j, ρ j /2), j = 1, 2,..., the balls given by Lemma 2.4, applied to Ω r, with the additional assumption that these balls are ordered so that ρ 1 ρ 2. We will adhere to the notation from Lemma 2.4. We will also use the more compact notation B r := {B j } j N for the set of balls covering Ω r. For each ball B B r and k s r we define (2.25) E B := ( B + 2B(0, 2 k ) ) if B and set E B := if B =. We also define, for l = 1, 2,..., (2.26) R B l := EB l \ ν>le B ν and

9 A NEW PROOF OF THE ATOMIC DECOMPOSITION OF HARDY SPACES 9 (2.27) F Bl (x) := k s r R B l ψ k (x y) ψ k f(y)dy. Lemma 2.5. For every l 1 the function F Bl is well defined, more precisely, the series in (2.27) converges point-wise and in distributional sense. Furthermore, (2.28) supp F Bl 7B l, (2.29) and x α F Bl (x)dx = 0 for all α with α n(p 1 1), (2.30) F Bl c 2 r, where the constant c is independent of r, l. In addition, for any k s r (2.31) = l 1 R B l and R B l RB m =, l m. Hence (2.32) F r = B B r F B (convergence in S ). Proof. Fix l 1. Observe that using Lemma 2.4 we have B l Ω c r + B(0, 2ρ l ) and hence E B l := if 2 k+1 2ρ l. Define k 0 := min{k : 2 k < ρ l }. Hence ρ l /2 2 k 0 < ρ l. Consequently, (2.33) F Bl (x) := ψ k (x y) ψ k f(y)dy. k k 0 It follows that supp F Bl B ( ξ l, (7/2)ρ l ) = 7Bl, which confirms (2.28). To prove (2.30) we will use the following R B l Lemma 2.6. For an arbitrary set S let S k := {x : dist(x, S) < 2 k+1 } and set (2.34) F S (x) := ψ k (x y) ψ k f(y)dy S k k κ 0 for some κ 0 s r. Then F S c2 r, where c > 0 is a constant independent of S and κ 0. Moreover, the above series converges in S. Proof. From (2.9) it follows that F S (x) = 0 if dist(x, S) 3 2 κ 0 Let x S. Evidently, B(x, 2 k ) S k for every k and hence F S (x) = ψ k (x y) ψ k f(y)dy k κ 0 = k κ 0 B(x,2 k ) ψ k (x y) ψ k f(y)dy = F r,κ0 (x). On account of Lemma 2.3 (b) we obtain F S (x) = F r,κ0 (x) c2 r.

10 10 S. DEKEL, G. KERKYACHARIAN, G. KYRIAZIS, AND P. PETRUSHEV Consider the case when x S l \ S l+1 for some l κ 0. Then B(x, 2 k ) S k if κ 0 k l 1 and B(x, 2 k ) S k = if k l + 2. Therefore, l 1 F S (x) = ψ k (x y) ψ l+1 k f(y)dy + ψ k (x y) ψ k f(y)dy S k k=κ 0 l+1 = F r,κ0,l 1(x) + ψ k (x y) ψ k f(y)dy, S k where we used the notation from (2.8). By Lemma 2.3 (b) and (2.10) it follows that F S (x) c2 r. We finally consider the case when 2 κ 0+1 dist(x, S) < 3 2 κ 0. Then we have F S (x) = E rκ0 S κ0 ψ κ0 (x y) ψ κ0 f(y)dy and the estimate F S (x) c2 r is immediate from (2.10). The convergence in S in (2.34) is established as in the proof of Lemma 2.3. Fix l 1 and let {B j : j J } be the set of all balls B j = B(ξ j, ρ j /2) such that j > l and ( B ξ j, 3ρ ) ( j B ξ l, 3ρ ) l. 4 4 By Lemma 2.4 it follows that #J K and 7 1 ρ l ρ j 7ρ l for j J. Define { (2.35) k 1 := min k : 2 k+1 < 4 1 min { ρ j : j J {l} }}. From this definition and 2 k 0 < ρ l we infer (2.36) 2 k min { ρ j : j J {l} } > 8 2 ρ l > k 0 = k 1 k Clearly, from (2.35) (2.37) B j + 2B(0, 2 k ) B ( ξ j, 3ρ j /4 ), k k 1, j J {l}. Denote S := j J B j and S := j J B j B l = S B l. As in Lemma 2.6 we set S k := S + 2B(0, 2 k ) and Sk := S + 2B(0, 2 k ). It readily follows from the definition of k 1 in (2.35) that (2.38) R B l := EB l \ ν>le B ν = ( S ) ( ) k \ E S k for k k 1. Denote F S (x) := ψ k (x y) ψ k f(y)dy, and k k S k 1 F S(x) := ψ k (x y) ψ k f(y)dy. S k k k 1 From (2.38) and the fact that S S it follows that F Bl (x) = F S(x) F S (x) + ψ k (x y) ψ k f(y)dy. k 0 k<k 1 By Lemma 2.6 we get F S c2 r and F S c2 r. On the other hand from (2.36) we have k 1 k 0 7. We estimate each of the (at most 7) integrals above using (2.10) to conclude that F Bl c2 r. We deal with the convergence in (2.27) and (2.32) as in the proof of Lemma 2.3. R B l

11 A NEW PROOF OF THE ATOMIC DECOMPOSITION OF HARDY SPACES 11 Clearly, (2.29) follows from the fact that x α ψ(x)dx = 0 for all α with α K. Finally, from Lemma 2.4 we have Ω r j N B l and then (2.31) is immediate from (2.25) and (2.26). We are now prepared to complete the proof of Theorem 2.1. For every ball B B r, r Z, provided Ω r, we define B := 7B, a B (x) := c 1 B 1/p 2 r F B (x) and λ B := c B 1/p 2 r, where c > 0 is the constant from (2.30). By (2.28) supp a B B and by (2.30) a B c 1 B 1/p 2 r F B B 1/p. Furthermore, from (2.29) it follows that x α a B (x)dx = 0 if α n(p 1 1). Therefore, each a B is an atom for H p. We set B r := if Ω r =. Now, using the above, (2.23), and Lemma 2.5 we get f = F r = F B = λ B a B, r Z r Z B B r r Z B B r where the convergence is in S, and λ B p c 2 pr B = c 2 pr Ω r c f p H p, B B r B B r r Z r Z which is the claimed atomic decomposition of f H p. Above we used that B = 7B = 7 n B. Rema 2.7. The proof of Theorem 2.1 can be considerably simplified and shortened if one seeks to establish atomic decomposition of the H p spaces in terms of q-atoms with p < q < rather than -atoms as in Theorem 2.1, i.e. atoms satisfying a L q B 1/q 1/p with q < rather than a L B 1/p. We will not elaborate on this here. References [1] R. Coifman, A real variable characterization of H p, Studia Math. 51 (1974), [2] S. Dekel, G, Kyriazis, G. Keyacharian, P. Petrushev, Hardy spaces associated with non-negative self-adjoint operators, preprint. [3] C. Fefferman, E. Stein, H p spaces of several variables, Acta Math. 129 (1972), [4] R. Latter, A characterization of H p ( ) in terms of atoms, Studia Math. 62 (1978), [5] E. Stein, Harmonic analysis: real-variable methods, orthogonality, and oscillatory integrals, Princeton University Press, Princeton, NJ, [6] E. Stein, G. Weiss, On the theory of harmonic functions of several variables. I. The theory of H p -spaces. Acta Math. 103 (1960), Hamanofim St. 9, Herzelia, Israel address: Shai.Dekel@ge.com Laboratoire de Probabilités et Modèles Aléatoires, CNRS-UMR 7599, Université Paris VI et Université Paris VII, rue de Clisson, F Paris address: ke@math.univ-paris-diderot.fr Department of Mathematics and Statistics, University of Cyprus, 1678 Nicosia, Cyprus address: kyriazis@ucy.ac.cy Department of Mathematics, University of South Carolina, Columbia, SC address: pencho@math.sc.edu r Z