Vector valued inequalities and Littlewood-Paley operators on Hardy spaces
aa r X i v : . [ m a t h . C A ] S e p VECTOR VALUED INEQUALITIES AND LITTLEWOOD-PALEYOPERATORS ON HARDY SPACES
SHUICHI SATO
Abstract.
We prove certain vector valued inequalities on R n related to Littlewood-Paley theory. They can be used in proving characterization of the Hardy spacesin terms of Littlewood-Paley operators by methods of real analysis. Introduction
We consider the Littlewood-Paley function on R n defined by(1.1) g ϕ ( f )( x ) = (cid:18)Z ∞ | f ∗ ϕ t ( x ) | dtt (cid:19) / , where ϕ t ( x ) = t − n ϕ ( t − x ). We assume that ϕ ∈ L ( R n ) and(1.2) Z R n ϕ ( x ) dx = 0 . If we further assume that | ϕ ( x ) | ≤ C (1 + | x | ) − n − ǫ for some ǫ >
0, then we have k g ϕ ( f ) k p ≤ C p k f k p , < p < ∞ , where k f k p = k f k L p (see [10] and also [1] for an earlier result). The reverse in-equality also holds if a certain non-degeneracy condition on ϕ is assumed in ad-dition (see [7, Theorem 3.8] and also [11]). This is the case for g Q with Q ( x ) =[( ∂/∂t ) P ( x, t )] t =1 , where P ( x, t ) is the Poisson kernel associated with the upperhalf space R n × (0 , ∞ ) defined by P ( x, t ) = c n t ( | x | + t ) ( n +1) / with c n = π − ( n +1) / Γ(( n + 1) /
2) (see [12, Chap. I]). Here we recall that ˆ Q ( ξ ) = − π | ξ | e − π | ξ | , where the Fourier transform is defined asˆ f ( ξ ) = F ( f )( ξ ) = Z R n f ( x ) e − πi h x,ξ i dx, h x, ξ i = x ξ + · · · + x n ξ n . Furthermore, it is known that(1.3) c k f k H p ≤ k g Q ( f ) k p ≤ c k f k H p for f ∈ H p ( R n ) (the Hardy space), 0 < p < ∞ , where c , c are positive constants(see [4] and also [14]). Recall that a tempered distribution f belongs to H p ( R n )if k f k H p = k f ∗ k p < ∞ , where f ∗ ( x ) = sup t> | Φ t ∗ f ( x ) | . Here Φ is in S ( R n )and satisfies R Φ( x ) dx = 1, where S ( R n ) denotes the Schwartz class of rapidly Mathematics Subject Classification.
Primary 42B25; Secondary 42B30.
Key Words and Phrases.
Vector valued inequalities, Littlewood-Paley functions, Hardy spaces.The author is partly supported by Grant-in-Aid for Scientific Research (C) No. 25400130,Japan Society for the Promotion of Science. decreasing smooth functions on R n ; it is known that any other choice of such Φgives an equivalent norm (see [4]).In this note we are concerned with the first inequality of (1.3) for 0 < p ≤ k f k H p ≤ c k g ϕ ( f ) k p , < p ≤ , for ϕ ∈ S ( R n ) satisfying (1.2) and a suitable non-degeneracy condition. Also,a relation between Hardy spaces on homogeneous groups and Littlewood-Paleyfunctions associated with the heat kernel can be found in [5, Chap. 7].On the other hand, it is known and would be seen by applying an easier version ofour arguments in the following that the Peetre maximal function F ∗∗ N,R can be usedalong with familiar methods to prove (1.4) when ϕ ∈ S ( R n ) with a non-degeneracycondition and with the condition supp( ˆ ϕ ) ⊂ { a ≤ | ξ | ≤ a } , a , a >
0, wherefor a function F on R n and positive real numbers N, R , the maximal function isdefined as(1.5) F ∗∗ N,R ( x ) = sup y ∈ R n | F ( x − y ) | (1 + R | y | ) N (see [8]).The purpose of this note is to prove (1.4) for a class of functions ϕ including Q and a general ϕ ∈ S ( R n ), without the restriction on supp( ˆ ϕ ) above, with (1.2)and an admissible non-degeneracy condition (Corollary 3.2) as an application of avector valued inequality which will be shown by using the maximal function F ∗∗ N,R (see Proposition 2.3, Theorem 2.10 below). The proof of Proposition 2.3 consistspartly in further developing methods of [13, Chap. V] and it admits some weightedinequalities. Theorem 2.10 follows from Proposition 2.3. Our proofs of Proposition2.3 and Corollary 3.2 are fairly straightforward and they will be expected to extendto some other situations.In Section 2, Proposition 2.3 will be formulated in a general form, while Theorem2.10 will be stated in a more convenient form for the application to the proof ofCorollary 3.2. In Section 3, we shall apply Theorem 2.10 and an atomic decom-position for Hardy spaces to prove Corollary 3.2. Finally, in Section 4, we shallgive proofs of Lemmas 2.1 and 2.5 in Section 2 from [13] and [8], respectively, forcompleteness; the lemmas will be needed in proving Proposition 2.3.2.
Vector valued inequalities
Let ϕ ( j ) , j = 1 , , . . . , M , be functions in L ( R n ) satisfying the non-degeneracycondition(2.1) inf ξ ∈ R n \{ } sup t> M X j =1 | F ( ϕ ( j ) )( tξ ) | > c for some positive constant c . We write ϕ = ( ϕ (1) , . . . , ϕ ( M ) ), ˆ ϕ = ( F ( ϕ (1) ) , . . . , F ( ϕ ( M ) )). Lemma 2.1.
Let ϕ ( j ) , j = 1 , , . . . , M , be functions in L ( R n ) satisfying (2.1) .Then, there exist b ∈ (0 , and positive numbers r , r with r < r such that if b ∈ [ b , , we can find η = ( η (1) , . . . , η ( M ) ) which satisfies the following :(1) η ∈ C ∞ ( R n ) , where η ∈ C k ( U ) means η ( j ) ∈ C k ( U ) for all ≤ j ≤ M ; ECTOR VALUED INEQUALITIES 3 (2) supp F ( η ( j ) ) ⊂ { r < | ξ | < r } , ≤ j ≤ M ;(3) each F ( η ( j ) ) is continuous, ≤ j ≤ M ;(4) P ∞ j = −∞ h ˆ ϕ ( b j ξ ) , ˆ η ( b j ξ ) i = 1 for ξ ∈ R n \ { } , where h z, w i = P Mj =1 z j w j , z, w ∈ C M ( the Cartesian product of M copies of the set of complex numbers ) .Further, if ˆ ϕ ∈ C k ( R n \ { } ) , then ˆ η ∈ C k ( R n ) . See [13, Chap. V] and also [2].We assume that M = 1 for simplicity. Suppose that ψ ∈ L ( R n ) and there existΘ ∈ C ∞ ( R n ) and A ≥ ψ ( ξ ) = ˆ ϕ ( ξ )Θ( ξ ) on {| ξ | < r A − } .Suppose that b ∈ [ b ,
1) and let η be as in Lemma 2.1 with M = 1. For J > ζ J by(2.3) ˆ ζ J ( ξ ) = 1 − X j : b j ≤ J ˆ ϕ ( b j ξ )ˆ η ( b j ξ ) . We note that supp(ˆ ζ J ) ⊂ {| ξ | ≤ r J − } , ˆ ζ J = 1 in {| ξ | < r J − } . By (2.2) itfollows that ˆ ψ ( ξ ) = X j : b j ≤ A ˆ ψ ( ξ ) ˆ ϕ ( b j ξ )ˆ η ( b j ξ ) + ˆ ζ A ( ξ ) ˆ ψ ( ξ )= X j : b j ≤ A ˆ ϕ ( b j ξ ) F ( α ( b j ) )( b j ξ ) + ˆ ϕ ( ξ ) ˆ β ( ξ ) , where α ( b j ) ( x ) = ψ b − j ∗ η ( x ) and ˆ β ( ξ ) = ˆ ζ A ( ξ )Θ( ξ ).Let E ( ψ, f )( x, t ) = f ∗ ψ t ( x ), f ∈ S ( R n ). Then we have(2.4) | E ( ψ, f )( x, t ) | ≤ X j : b j ≤ A | E ( α ( b j ) ∗ ϕ, f )( x, b j t ) | + | E ( β ∗ ϕ, f )( x, t ) | . Also, let E ψ ( x, t ) = E ( ψ, f )( x, t ), when f is fixed.Define(2.5) C ( ψ, t, L, x ) = (1 + | x | ) L (cid:12)(cid:12)(cid:12)(cid:12)Z ˆ ψ ( t − ξ )ˆ η ( ξ ) e πi h x,ξ i dξ (cid:12)(cid:12)(cid:12)(cid:12) , t > , L ≥ . Consequently, | α ( b j ) s ( x ) | = C ( ψ, b j , L, x/s ) s − n (1 + | x | /s ) − L for j ∈ Z (the set of integers). Likewise, we have | β s ( x ) | = D (Θ , A, L, x/s ) s − n (1 + | x | /s ) − L , where(2.6) D (Θ , J, L, x ) = (1 + | x | ) L (cid:12)(cid:12)(cid:12)(cid:12)Z ˆ ζ J ( ξ )Θ( ξ ) e πi h x,ξ i dξ (cid:12)(cid:12)(cid:12)(cid:12) . Here ˆ ζ J is as in (2.3). We also write C ( ψ, j, L, x ) = C ( ψ, b j , L, x ), j ∈ Z . Let C ( ψ, j, L ) = Z R n C ( ψ, j, L, x ) dx, j ∈ Z , (2.7) D (Θ , J, L ) = Z R n D (Θ , J, L, x ) dx. (2.8) SHUICHI SATO
We also write C ( ψ, j, L ) = C ϕ ( ψ, j, L ), D (Θ , J, L ) = D ϕ (Θ , J, L ) to indicate thatthese quantities are based on ϕ . See Lemma 2.8 below for a sufficient conditionwhich implies C ( ψ, j, L ) < ∞ , D (Θ , J, L ) < ∞ .The maximal function in (1.5) is used in the following result. Lemma 2.2.
Let ϕ, ψ ∈ L ( R n ) . Suppose that ϕ satisfies (2.1) . Let b ∈ [ b , .We assume that ψ and ϕ are related by (2.2) with Θ ∈ C ∞ ( R n ) and A ≥ . Let N > . Then for f ∈ S ( R n ) , we have (2.9) | E ( ψ, f )( x, t ) | ≤ C X j : b j ≤ A C ( ψ, j, N ) E ( ϕ, f )( · , b j t ) ∗∗ N, ( b j t ) − ( x )+ CD (Θ , A, N ) E ( ϕ, f )( · , t ) ∗∗ N,t − ( x );(2.10) E ( ψ, f )( · , t ) ∗∗ N,t − ( x ) ≤ C X j : b j ≤ A C ( ψ, j, N ) b − jN E ( ϕ, f )( · , b j t ) ∗∗ N, ( b j t ) − ( x )+ CD (Θ , A, N ) E ( ϕ, f )( · , t ) ∗∗ N,t − ( x ) . Proof.
Using (2.4), we see that | E ψ ( z, t ) |≤ C X j : b j ≤ A Z | E ϕ ( y, b j t ) | (cid:18) | z − y | b j t (cid:19) − N C ( ψ, j, N, ( z − y ) / ( b j t ))( b j t ) − n dy + C Z | E ϕ ( y, t ) | (cid:18) | z − y | t (cid:19) − N D (Θ , A, N, ( z − y ) /t ) t − n dy. If we multiply both sides of the inequality by (1 + | x − z | /t ) − N and observe that (cid:18) | z − y | b j t (cid:19) − N (cid:18) | x − z | t (cid:19) − N ≤ C A,N b − Nj (cid:18) | x − y | b j t (cid:19) − N for all x, y, z ∈ R n and t > b j ≤ A , then we see that | E ψ ( z, t ) | (1 + | x − z | /t ) − N ≤ C X j : b j ≤ A b − Nj Z | E ϕ ( y, b j t ) | (cid:18) | x − y | b j t (cid:19) − N C ( ψ, j, N, ( z − y ) / ( b j t ))( b j t ) − n dy + C Z | E ϕ ( y, t ) | (cid:18) | x − y | t (cid:19) − N D (Θ , A, N, ( z − y ) /t ) t − n dy, and hence | E ψ ( z, t ) | (1 + | x − z | /t ) − N ≤ C X j : b j ≤ A b − Nj E ϕ ( · , b j t ) ∗∗ N, ( b j t ) − ( x ) Z C ( ψ, j, N, ( z − y ) / ( b j t ))( b j t ) − n dy + CE ϕ ( · , t ) ∗∗ N,t − ( x ) Z D (Θ , A, N, ( z − y ) /t ) t − n dy ≤ C X j : b j ≤ A C ( ψ, j, N ) b − Nj E ϕ ( · , b j t ) ∗∗ N, ( b j t ) − ( x ) + CD (Θ , A, N ) E ϕ ( · , t ) ∗∗ N,t − ( x ) . ECTOR VALUED INEQUALITIES 5
The estimate (2.10) follows by taking the supremum in z over R n . The proof of(2.9) is easier; putting z = x and arguing as above, we get (2.9). (cid:3) Let ϕ ∈ L ( R n ). Suppose that ϕ satisfies (2.1). Let L >
0. We consider thefollowing conditions. ϕ ∈ C ( R n ) , ∂ k ϕ ∈ L ( R n ) , ≤ k ≤ n ;(2.11) | ˆ ϕ ( ξ ) | ≤ C | ξ | ǫ for some ǫ > j ≥ C ϕ ( ∇ ϕ, j, L ) b − jL − ǫj < ∞ for some ǫ >
0, together with (2.11);(2.13) D ϕ ( L ) < ∞ , with (2.11),(2.14)where we write ∇ ϕ = ( ∂ ϕ, . . . , ∂ n ϕ ), ∂ k = ∂ x k = ∂/∂ x k and C ϕ ( ∇ ϕ, j, L ) = P nk =1 C ϕ ( ∂ k ϕ, j, L ); also we define D ϕ ( L ) = P nk =1 D ϕ (Ξ k , , L ) by taking Θ( ξ ) =Ξ k ( ξ ) = 2 πiξ k and J = 1 in (2.8). We note that (2.11) implies the following (with ǫ = 1):(2.15) | ˆ ϕ ( ξ ) | ≤ C | ξ | − ǫ for some ǫ > . Let ψ ∈ L ( R n ). We assume that ψ is related to ϕ as in (2.2) with Θ ∈ C ∞ ( R n )and A ≥
1. We also consider the conditions:sup j : b j ≤ A C ϕ ( ψ, j, L ) b − ǫj < ∞ for some ǫ > D ϕ (Θ , A, L ) < ∞ . (2.17)Let M be the Hardy-Littlewood maximal operator M ( f )( x ) = sup x ∈ B | B | − Z B | f ( y ) | dy, where the supremum is taken over all balls B in R n such that x ∈ B and | B | denotesthe Lebesgue measure of B . Let 1 < p < ∞ . We recall that a weight function w belongs to the weight class A p of Muckenhoupt on R n if[ w ] A p = sup B (cid:18) | B | − Z B w ( x ) dx (cid:19) (cid:18) | B | − Z B w ( x ) − / ( p − dx (cid:19) p − < ∞ , where the supremum is taken over all balls B in R n Also, we recall that a weightfunction w is in the class A if M ( w ) ≤ Cw almost everywhere. The infimum of allsuch C is denoted by [ w ] A .For a weight w , the weighted L p norm is defined as k f k p,w = (cid:18)Z R n | f ( x ) | p w ( x ) dx (cid:19) /p . We have the following vector value inequality.
Proposition 2.3.
Let ϕ ∈ L ( R n ) . We assume that ϕ satisfies (2.1) with M = 1 .Let N > , n/N < p, q < ∞ and w ∈ A pN/n . Suppose that ϕ satisfies (2.11) , (2.12) and (2.13) , (2.14) with L = N . Let ψ ∈ L ( R n ) . Suppose that ψ is related to ϕ asin (2.2) with Θ ∈ C ∞ ( R n ) , A ≥ and (2.16) , (2.17) hold with L = N . Then (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:18)Z ∞ | f ∗ ψ t | q dtt (cid:19) /q (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) p,w ≤ C (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:18)Z ∞ | f ∗ ϕ t | q dtt (cid:19) /q (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) p,w for f ∈ S ( R n ) with a positive constant C independent of f . SHUICHI SATO
We need the next result to show Proposition 2.3.
Lemma 2.4.
Suppose that < q < ∞ , N > and that ϕ ∈ L ( R n ) satisfies (2.1) , (2.11) , (2.12) and (2.13) , (2.14) with L = N . Then Z ∞ E ( ϕ, f )( · , t ) ∗∗ N,t − ( x ) q dtt ≤ C Z ∞ M ( | f ∗ ϕ t | r )( x ) q/r dtt , r = n/N. We need the following in proving Lemma 2.4.
Lemma 2.5 (see [8]) . If F ∈ C ( R n ) and R > , r > , then F ∗∗ N,R ( x ) ≤ Cδ − N M ( | F | r )( x ) /r + CδR − |∇ F | ∗∗ N,R ( x ) for all δ ∈ (0 , , where N = n/r and the constant C is independent of δ and R .Proof of Lemma . . By Lemma 2.5 we have(2.18) E ( ϕ, f )( · , t ) ∗∗ N,t − ( x ) ≤ Cδ − N M ( | f ∗ ϕ t | r )( x ) /r + Cδ | f ∗ ( ∇ ϕ ) t | ∗∗ N,t − ( x ) , where f ∗ ( ∇ ϕ ) t = ( f ∗ ( ∂ ϕ ) t , . . . , f ∗ ( ∂ n ϕ ) t ), r = n/N . We apply (2.10) of Lemma2.2 with ψ = ∂ k ϕ , Θ( ξ ) = 2 πiξ k , A = 1 in (2.2). Then | f ∗ ( ∇ ϕ ) t | ∗∗ N,t − ( x ) ≤ C X j ≥ C ϕ ( ∇ ϕ, j, N ) b − jN E ( ϕ, f )( · , b j t ) ∗∗ N, ( b j t ) − ( x )+ CD ϕ ( N ) E ( ϕ, f )( · , t ) ∗∗ N,t − ( x ) . Using this in (2.18) and applying H¨older’s inequality when q >
1, we see that(2.19) E ( ϕ, f )( · , t ) ∗∗ N,t − ( x ) q ≤ Cδ − Nq M ( | f ∗ ϕ t | r )( x ) q/r + C q δ q X j ≥ C ϕ ( ∇ ϕ, j, N ) q b − jNq b − τc q j E ( ϕ, f )( · , b j t ) ∗∗ N, ( b j t ) − ( x ) q + Cδ q D ϕ ( N ) q E ( ϕ, f )( · , t ) ∗∗ N,t − ( x ) q . where τ > c q = 1 if q > c q = 0 if 0 < q ≤ , ∞ ) with respect to themeasure dt/t and if we apply termwise integration on the right hand side, then wehave(2.20) Z ∞ E ( ϕ, f )( · , t ) ∗∗ N,t − ( x ) q dtt ≤ Cδ − Nq Z ∞ M ( | f ∗ ϕ t | r )( x ) q/r dtt + C q δ q X j ≥ C ϕ ( ∇ ϕ, j, N ) q b − jNq b − τc q j + D ϕ ( N ) q Z ∞ E ( ϕ, f )( · , t ) ∗∗ N,t − ( x ) q dtt . The condition (2.13) with L = N implies that the sum in j on the right hand sideof (2.20) is finite if τ is small enough. We can see that the last integral on the righthand side of (2.20) is finite for f ∈ S ( R n ) by (2.12) and (2.15). Further, we have(2.14) for L = N . Altogether, it follows that the second term on the right handside of (2.20) is finite. Thus, we can get the conclusion if we choose δ sufficientlysmall. (cid:3) ECTOR VALUED INEQUALITIES 7
Proof of Proposition . . By (2.9) we have | E ( ψ, f )( x, t ) | q ≤ C q X j : b j ≤ A C ( ψ, j, N ) q b − τc q j E ( ϕ, f )( · , b j t ) ∗∗ N, ( b j t ) − ( x ) q + CD (Θ , A, N ) q E ( ϕ, f )( · , t ) ∗∗ N,t − ( x ) q , where τ > c q is as in (2.19). Integrating with the measure dt/t over (0 , ∞ ),we have(2.21) Z ∞ | E ( ψ, f )( x, t ) | q dtt ≤ C q X j : b j ≤ A C ( ψ, j, N ) q b − τc q j + D (Θ , A, N ) q Z ∞ E ( ϕ, f )( · , t ) ∗∗ N,t − ( x ) q dtt . The sum in j on the right hand side of (2.21) is finite by (2.16) with L = N if τ is small enough; also we have assumed D (Θ , A, N ) < ∞ ((2.17) with L = N ). Let r = n/N < q, p and w ∈ A pN/n . By (2.21) and Lemma 2.4 we see that (cid:18)Z R n (cid:18)Z ∞ | E ( ψ, f )( x, t ) | q dtt (cid:19) p/q w ( x ) dx ! /p (2.22) ≤ C (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:18)Z ∞ M ( | f ∗ ϕ t | r )( x ) q/r dtt (cid:19) /q (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) p,w = C (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:18)Z ∞ M ( | f ∗ ϕ t | r )( x ) q/r dtt (cid:19) r/q (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) /rp/r,w ≤ C Z R n (cid:18)Z ∞ | E ( ϕ, f )( x, t ) | q dtt (cid:19) p/q w ( x ) dx ! /p , where the last inequality follows form the following lemma, which is a version of thevector valued inequality for the Hardy-Littlewood maximal functions of Fefferman-Stein [3] (see [9] for a proof of the ℓ µ -valued case, which may be available also inthe present situation). Lemma 2.6.
Suppose that < µ, ν < ∞ and w ∈ A ν . Then for appropriatefunctions E ( x, t ) on R n × (0 , ∞ ) we have (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:18)Z ∞ M ( E t )( x ) µ dtt (cid:19) /µ (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) ν,w ≤ C Z R n (cid:18)Z ∞ | E ( x, t ) | µ dtt (cid:19) ν/µ w ( x ) dx ! /ν , where E t ( x ) = E ( x, t ) . This completes the proof of Proposition 2.3. (cid:3)
We have an analogous result for general ϕ = ( ϕ (1) , . . . , ϕ ( M ) ), although Propo-sition 2.3 is stated only for the case M = 1.It is obvious that Q , ˆ Q ( ξ ) = − π | ξ | e − π | ξ | , satisfies all the requirements on ϕ inLemma 2.4 for all N >
0. To state results with more directly verifiable assumptionson ϕ and ψ , we introduce a class of functions. SHUICHI SATO
Definition 2.7.
Let ψ ∈ L ( R n ). Let l be a non-negative integer and τ a non-negative real number. We say ψ ∈ B lτ if ˆ ψ ∈ C l ( R n \ { } ) and | ∂ γξ ˆ ψ ( ξ ) | ≤ C γ | ξ | − τ −| γ | outside a neighborhood of the originfor every γ satisfying | γ | ≤ l with a constant C γ , where γ = ( γ , . . . , γ n ) is amulti-index, γ j ∈ Z , γ j ≥ | γ | = γ + · · · + γ n and ∂ γξ = ∂ γ ξ . . . ∂ γ n ξ n .Clearly, Q ∈ B lτ for any l, τ . This is also the case for ψ ∈ S ( R n ). Lemma 2.8.
Suppose that ϕ ∈ L ( R n ) and ϕ satisfies the condition (2.1) . Let τ ≥ , J > and let L be a non-negative integer. (1) Suppose that ψ ∈ B L +[ n/ τ and ˆ ϕ ∈ C L +[ n/ ( R n \ { } ) , where [ a ] denotes the largest integer not exceeding a . Then we have sup j : b j ≤ J C ϕ ( ψ, j, L ) b − jτ < ∞ , where C ϕ ( ψ, j, L ) = C ( ψ, j, L ) is as in (2.7) . (2) Suppose that Θ ∈ C ∞ ( R n ) and ˆ ϕ ∈ C L +[ n/ ( R n \ { } ) . Then D ϕ (Θ , J, L ) < ∞ , where D ϕ (Θ , J, L ) = D (Θ , J, L ) is as in (2.8) . (3) Let ψ ( k ) ∈ L ( R n ) and F ( ψ ( k ) )( ξ ) = 2 πiξ k ˆ ϕ ( ξ ) , ≤ k ≤ n . If ϕ ∈ B L +[ n/ L +1+ τ , then we have sup j : b j ≤ J C ϕ ( ψ ( k ) , j, L ) b − jL − jτ < ∞ , D ϕ (Ξ k , , L ) < ∞ for each k , where Ξ k ( ξ ) = 2 πiξ k as above.Proof. Part (3) follows from part (1) and part (2) since ψ ( k ) ∈ B L +[ n/ L + τ andˆ ϕ ∈ C L +[ n/ ( R n \ { } ) if ϕ ∈ B L +[ n/ L +1+ τ . To prove part (1), we note that(1 + | x | ) [ n/ C ( ψ, t, L, x ) ≤ C (cid:12)(cid:12)(cid:12)(cid:12)Z ˆ ψ ( t − ξ )ˆ η ( ξ ) e πi h x,ξ i dξ (cid:12)(cid:12)(cid:12)(cid:12) + C sup | γ | = L +[ n/ (cid:12)(cid:12)(cid:12)(cid:12)Z ∂ γξ h ˆ ψ ( t − ξ )ˆ η ( ξ ) i e πi h x,ξ i dξ (cid:12)(cid:12)(cid:12)(cid:12) , where C ( ψ, t, L, x ) is as in (2.5). We note that ˆ η ∈ C L +[ n/ ( R n ) by Lemma 2.1,since ˆ ϕ ∈ C L +[ n/ ( R n \ { } ). The assumption ψ ∈ B L +[ n/ τ implies (cid:12)(cid:12)(cid:12) ∂ γξ h ˆ ψ ( t − ξ )ˆ η ( ξ ) i(cid:12)(cid:12)(cid:12) ≤ C M t τ , < t ≤ M, for any M >
0, if | γ | = L + [ n/
2] + 1 or γ = 0. It follows that C ( ψ, t, L, x ) ≤ C (1 + | x | ) − [ n/ − G ( x )with some G ∈ L such that k G k ≤ Ct τ . Thus, since [ n/
2] + 1 > n/
2, by theSchwarz inequality we have(2.23) Z R n C ( ψ, t, L, x ) dx ≤ Ct τ . The conclusion of part (1) follows from (2.23) with t = b j .Likewise, we have Z R n D (Θ , J, L, x ) dx < ∞ ECTOR VALUED INEQUALITIES 9 under the assumptions of part (2), where D (Θ , J, L, x ) is as in (2.6), which provespart (2). (cid:3) By Lemma 2.8 and Proposition 2.3 we have the following.
Theorem 2.9.
Let ϕ ∈ L ( R n ) satisfy (2.1) with M = 1 . Suppose that ψ ∈ L ( R n ) and ˆ ψ ( ξ ) = ˆ ϕ ( ξ )Θ( ξ ) in a neighborhood of the origin with some Θ ∈ C ∞ ( R n ) . Let < p, q < ∞ and let N be a positive integer such that N > max( n/p, n/q ) . Let w ∈ A pN/n . Suppose that ϕ belongs to B N +[ n/ N +1+ ǫ for some ǫ > and satisfies (2.11) and (2.12) . Also, suppose that ψ ∈ B N +[ n/ ǫ for some ǫ > . Then wehave (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:18)Z ∞ | f ∗ ψ t | q dtt (cid:19) /q (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) p,w ≤ C (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:18)Z ∞ | f ∗ ϕ t | q dtt (cid:19) /q (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) p,w for f ∈ S ( R n ) , where C is a positive constant independent of f .Proof. If we have (2.11) and if ϕ ∈ B N +[ n/ N +1+ ǫ , then (2.13) and (2.14) hold with L = N by part (3) of Lemma 2.8 with J = 1, τ = ǫ , L = N . Since ψ ∈ B N +[ n/ ǫ and ϕ ∈ C N +[ n/ ( R n \ { } ), if ˆ ψ ( ξ ) = ˆ ϕ ( ξ )Θ( ξ ) on {| ξ | < r A − } , A ≥ L = N by part (1) of Lemma 2.8 with J = A , τ = ǫ , L = N and part (2) of Lemma 2.8 with J = A , L = N , respectively. ThusProposition 2.3 implies the conclusion. (cid:3) This immediately implies the following.
Theorem 2.10.
Let ϕ ∈ L ( R n ) satisfy (2.1) with M = 1 , (2.11) and (2.12) . Weassume that < p, q < ∞ and N is a positive integer satisfying N > max( n/p, n/q ) .Let w ∈ A pN/n . Suppose that ϕ ∈ B N +[ n/ N +1+ ǫ for some ǫ > . Then, if ψ ∈ S ( R n ) and ˆ ψ vanishes in a neighborhood of the origin, the inequality (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:18)Z ∞ | f ∗ ψ t | q dtt (cid:19) /q (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) p,w ≤ C (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:18)Z ∞ | f ∗ ϕ t | q dtt (cid:19) /q (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) p,w , f ∈ S ( R n ) , holds with a positive constant C independent of f .Proof. We see that ˆ ψ ( ξ ) = ˆ ϕ ( ξ )Θ( ξ ) in a neighborhood of the origin with Θ beingidentically 0. Obviously, ψ ∈ B N +[ n/ . So all the requirements for ϕ and ψ inTheorem 2.9 are satisfied. Thus the conclusion follows from Theorem 2.9. Thiscompletes the proof. (cid:3) We note that Q fulfills all the requirements on ϕ in Theorem 2.10 for every N .The same is true of ϕ ∈ S ( R n ) satisfying (2.1) (with M = 1) and (1.2).3. Littlewood-Paley operators and Hardy spaces
Let H denote the Hilbert space of functions u ( t ) on (0 , ∞ ) such that k u k H = (cid:0)R ∞ | u ( t ) | dt/t (cid:1) / < ∞ . We first recall Hardy spaces of functions on R n withvalues in H , which will be used to prove (1.4) by Theorem 2.10 (see Corollary 3.2below).The Lebesgue space L q H ( R n ) consists of functions h ( y, t ) with the norm k h k q, H = (cid:18)Z R n k h y k q H dy (cid:19) /q , where h y ( t ) = h ( y, t ). For 0 < p ≤
1, we consider the Hardy space H p H ( R n ) offunctions on R n with values in H . We take ϕ ∈ S ( R n ) with R ϕ ( x ) dx = 1. Let h ∈ L H ( R n ). We recall that h ∈ H p H ( R n ) if k h k H p H = k h ∗ k L p < ∞ , where h ∗ ( x ) = sup s> (cid:18)Z ∞ | ϕ s ∗ h t ( x ) | dtt (cid:19) / , with h t ( x ) = h ( x, t ).If a is a ( p, ∞ ) atom in H p H ( R n ), we have(i) (cid:0)R ∞ | a ( x, t ) | dt/t (cid:1) / ≤ | Q | − /p , where Q is a cube in R n with sides parallelto the coordinate axes;(ii) sup( a ( · , t )) ⊂ Q uniformly in t >
0, where Q is the same as in ( i );(iii) R R n a ( x, t ) x γ dx = 0 for all t > γ such that | γ | ≤ [ n (1 /p − γ = ( γ , . . . , γ n ) is a multi-index and x γ = x γ . . . x γ n n .We apply the following atomic decomposition. Lemma 3.1.
Let h ∈ L H ( R n ) . If h ∈ H p H ( R n ) , then there exist a sequence { a k } of ( p, ∞ ) atoms in H p H ( R n ) and a sequence { λ k } of positive numbers such that P ∞ k =1 λ pk ≤ C k h k pH p H with a constant C independent of h and h = P ∞ k =1 λ k a k in H p H ( R n ) and in L H ( R n ) . A proof of the atomic decomposition for H p ( R n ) can be found in [6] and [13].Similar methods apply to the vector valued case.In this section, we prove the following result as an application of Theorem 2.10. Corollary 3.2.
Let < p ≤ , N > n/p . Suppose that ϕ ∈ L ( R n ) satisfies (2.1) with M = 1 , (2.11) , (2.12) and suppose that ϕ ∈ B N +[ n/ N +1+ ǫ for some ǫ > . Thenwe have k f k H p ≤ C p k g ϕ ( f ) k p for f ∈ H p ( R n ) ∩ S ( R n ) , where C p is a positive constant independent of f . This can be generalized to an arbitrary f ∈ H p ( R n ) if ϕ = Q or if ϕ is a functionin S ( R n ) satisfying (2.1) and (1.2) (see [14]).In proving Corollary 3.2, we need the following. Lemma 3.3.
Suppose that η ∈ S ( R n ) , supp(ˆ η ) ⊂ { / ≤ | ξ | ≤ } , ˆ η ( ξ ) = 1 on { ≤ | ξ | ≤ } and that Φ ∈ S ( R n ) satisfies R R n Φ( x ) dx = 1 . Let ψ ∈ S ( R n ) and supp ˆ ψ ⊂ { ≤ | ξ | ≤ } . Then, for p, q > and f ∈ S ( R n ) we have (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:18)Z ∞ sup s> | Φ s ∗ ψ t ∗ f | q dtt (cid:19) /q (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) p ≤ C (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:18)Z ∞ | η t ∗ f | q dtt (cid:19) /q (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) p . Proof.
We note that ˆΦ( sξ ) ˆ ψ ( tξ ) = ˆΦ( sξ ) ˆ ψ ( tξ )ˆ η ( tξ ). Thus we have | Φ s ∗ ψ t ∗ f ( x ) | ≤ ( f ∗ η t ) ∗∗ N,t − ( x ) Z R n | Φ s ∗ ψ t ( w ) | (1 + t − | w | ) N dw = ( f ∗ η t ) ∗∗ N,t − ( x ) Z R n | Φ s/t ∗ ψ ( w ) | (1 + | w | ) N dw ≤ C N ( f ∗ η t ) ∗∗ N,t − ( x )for any N >
0, with a positive constant C N independent of s, t . The last inequalityfollows from the observation that Φ s/t ∗ ψ , s, t >
0, belongs to a bounded subset
ECTOR VALUED INEQUALITIES 11 of the topological vector space S ( R n ), since F (Φ u ∗ ψ )( ξ ) = ˆΦ( uξ ) ˆ ψ ( ξ ), u >
0, andˆ ψ ( ξ ) is supported on { ≤ | ξ | ≤ } . Therefore, we have(3.1) (cid:18)Z ∞ sup s> | Φ s ∗ ψ t ∗ f ( x ) | q dtt (cid:19) /q ≤ C (cid:18)Z ∞ | ( f ∗ η t ) ∗∗ N,t − ( x ) | q dtt (cid:19) /q . Thus (3.1) and Lemma 2.4 with η in place of ϕ imply (cid:18)Z ∞ sup s> | Φ s ∗ ψ t ∗ f ( x ) | q dtt (cid:19) /q ≤ C (cid:18)Z ∞ M ( | f ∗ η t | r )( x )( x ) q/r dtt (cid:19) /q , with N = n/r . By this and Lemma 2.6, the conclusion follows as in (2.22). (cid:3) We also use the following to prove Corollary 3.2.
Lemma 3.4.
Let ˆ ψ ∈ S ( R n ) be a radial function supported on { ≤ | ξ | ≤ } suchthat Z ∞ | ˆ ψ ( tξ ) | dtt = 1 for all ξ = 0 .Let f ∈ H p ( R n ) ∩ S ( R n ) , < p ≤ , and put E ( y, t ) = f ∗ ψ t ( y ) . Then E is in H p H ( R n ) and we have k f k H p ≤ C k E k H p H . Let ψ be a function in L ( R n ) satisfying (1.2). Suppose that h ∈ L H . Let h ( ǫ ) ( y, t ) = h ( y, t ) χ ( ǫ,ǫ − ) ( t ), 0 < ǫ <
1, where χ S denotes the characteristic func-tion of a set S . Put F ǫψ ( h )( x ) = Z ∞ Z R n ψ t ( x − y ) h ( ǫ ) ( y, t ) dy dtt . To prove Lemma 3.4 we apply the following.
Lemma 3.5.
Let < p ≤ . Suppose that ψ ∈ S ( R n ) and supp ˆ ψ ⊂ { ≤ | ξ | ≤ } .Then sup ǫ ∈ (0 , k F ǫψ ( h ) k H p ≤ C k h k H p H . Proof.
Let a be a ( p, ∞ ) atom in H p H ( R n ) with support in the cube Q of thedefinition of the atom. We denote by y the center of Q . Let e Q be a concentricenlargement of Q such that 2 | y − y | < | x − y | if y ∈ Q and x ∈ R n \ e Q . LetΦ be a non-negative C ∞ function on R n supported on {| x | < } which satisfies R Φ( x ) dx = 1. Let Ψ s,t = Φ s ∗ ψ t , s, t >
0. Then Ψ s,t = (Φ s/t ∗ ψ ) t and Φ u ∗ ψ , u >
0, belongs to a bounded subset of the topological vector space S ( R n ), as in theproof of Lemma 3.3.Let P x ( y, y ) be the Taylor polynomial in y of order M = [ n (1 /p − y forΦ s/t ∗ ψ ( x − y ). Then, if | x − y | > | y − y | , we see that | Φ s/t ∗ ψ ( x − y ) − P x ( y, y ) | ≤ C | y − y | M +1 (1 + | x − y | ) − L , where L > n + M + 1 and the constant C is independent of s, t, x, y, y , and hence | Ψ s,t ( x − y ) − t − n P x/t ( y/t, y /t ) | ≤ Ct − n − M − | y − y | M +1 (1 + | x − y | /t ) − L . Therefore, by the properties of an atom and the Schwarz inequality, for x ∈ R n \ e Q we have (cid:12)(cid:12) Φ s ∗ F ǫψ ( a )( x ) (cid:12)(cid:12) = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)Z Z R n × (0 , ∞ ) (cid:0) Ψ s,t ( x − y ) − t − n P x/t ( y/t, y /t ) (cid:1) a ( ǫ ) ( y, t ) dy dtt (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ Z Q (cid:18)Z ∞ (cid:12)(cid:12) Ψ s,t ( x − y ) − t − n P x/t ( y/t, y /t ) (cid:12)(cid:12) dtt (cid:19) / (cid:18)Z ∞ | a ( y, t ) | dtt (cid:19) / dy ≤ C | Q | − /p Z Q (cid:18)Z ∞ (cid:12)(cid:12) Ψ s,t ( x − y ) − t − n P x/t ( y/t, y /t ) (cid:12)(cid:12) dtt (cid:19) / dy ≤ C | Q | − /p Z Q | y − y | M +1 | x − y | − n − M − dy ≤ C | Q | − /p +1+( M +1) /n | x − y | − n − M − . We note that p > n/ ( n + M + 1). Thus(3.2) Z R n \ e Q sup s> (cid:12)(cid:12) Φ s ∗ F ǫψ ( a )( x ) (cid:12)(cid:12) p dx ≤ C | Q | − p + p ( M +1) /n Z R n \ e Q | x − y | − p ( n + M +1) ≤ C. Since R ∞ | ˆ ψ ( tξ ) | dt/t ≤ C , by duality we havesup ǫ ∈ (0 , k F ǫψ ( h ) k ≤ C k h k L H , h ∈ L H ( R n ) . Thus, applying H¨older’s inequality, by the properties (i), (ii) of a we see that Z e Q sup s> (cid:12)(cid:12) Φ s ∗ F ǫψ ( a )( x ) (cid:12)(cid:12) p dx ≤ C | Q | − p/ (cid:18)Z e Q | M ( F ǫψ ( a ))( x ) | dx (cid:19) p/ (3.3) ≤ C | Q | − p/ (cid:18)Z Q Z ∞ | a ( y, t ) | dtt dy (cid:19) p/ ≤ C. The estimates (3.2) and (3.3) imply(3.4) Z R n sup s> (cid:12)(cid:12) Φ s ∗ F ǫψ ( a )( x ) (cid:12)(cid:12) p dx ≤ C. Using Lemma 3.1 and (3.4), we have Z R n sup s> (cid:12)(cid:12) Φ s ∗ F ǫψ ( h )( x ) (cid:12)(cid:12) p dx ≤ C k h k pH p H . This completes the proof. (cid:3)
Proof of Lemma . . The fact that E ∈ H p H ( R n ) can be proved similarly to theproof of Lemma 3.5 by applying the atomic decomposition for H p ( R n ) (see [14,Lemma 3.6]).We write F ǫ e ¯ ψ ( E )( x ) = Z ǫ − ǫ Z R n ψ t ∗ f ( y ) ¯ ψ t ( y − x ) dy dtt = Z R n Ψ ( ǫ ) ( x − z ) f ( z ) dz, ECTOR VALUED INEQUALITIES 13 where ¯ ψ denotes the complex conjugate, e g ( x ) = g ( − x ) andΨ ( ǫ ) ( x ) = Z ǫ − ǫ Z R n ψ t ( x + y ) ¯ ψ t ( y ) dy dtt . We note that d Ψ ( ǫ ) ( ξ ) = Z ǫ − ǫ ˆ ψ ( tξ ) b ¯ ψ ( − tξ ) dtt = Z ǫ − ǫ | ˆ ψ ( tξ ) | dtt . From this and Lemma 3.5 we have k f k H p ≤ C lim inf ǫ → k F ǫ e ¯ ψ ( E ) k H p ≤ C k E k H p H . (cid:3) Proof of Corollary . . We take a function η as in Lemma 3.3. Then by Lemma3.3 with q = 2 and Lemma 3.4, it follows that k f k H p ≤ C k g η ( f ) k p for f ∈ H p ( R n ) ∩ S ( R n ). If we use this and Theorem 2.10 with q = 2, w = 1 andwith η in place of ψ , we can reach the conclusion of Corollary 3.2. (cid:3) We can also prove discrete parameter versions of Proposition 2.3 and Corollary3.2 by analogous methods.
Proposition 3.6.
Let
N > , n/N < p, q < ∞ . Suppose that w ∈ A pN/n and that ϕ and ψ fulfill the hypotheses of Proposition . with N . Then, for f ∈ S ( R n ) wehave (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) ∞ X j = −∞ | f ∗ ψ b j | q /q (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) p,w ≤ C (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) ∞ X j = −∞ | f ∗ ϕ b j | q /q (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) p,w . Corollary 3.7.
Let < p ≤ and N > n/p . Suppose that ϕ fulfills the hypothesesof Corollary . with N . Then, for f ∈ H p ( R n ) ∩ S ( R n ) we have k f k H p ≤ C (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) ∞ X j = −∞ | f ∗ ϕ b j | / (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) p . Also, from Proposition 3.6 we have discrete parameter analogues of Theorems2.9 and 2.10. 4.
Proofs of Lemmas . and . . . Proof of Lemma . . There exist a finite family { I j } Lj =1 of compact intervals in(0 , ∞ ) and a positive constant c such thatinf ξ ∈ S n − max ≤ j ≤ L inf t ∈ I j M X i =1 | F ( ϕ ( i ) )( tξ ) | ≥ c, where S n − = { ξ : | ξ | = 1 } . This follows from a compactness argument, since each F ( ϕ ( j ) ) is continuous. Let b = max ≤ h ≤ L ( a h /b h ), where I h = [ a h , b h ]. Then b ∈ (0 ,
1) and if b ∈ [ b , t > ≤ h ≤ L , h ∈ Z , we have b j t ∈ I h for some j ∈ Z .Let [ m, H ] be an interval in (0 , ∞ ) such that ∪ Lj =1 I j ⊂ [ m, H ]. We take a non-negative function θ ∈ C ∞ ( R ) such that θ = 1 on [ m, H ], supp θ ⊂ [ m/ , H ].Then ∞ X j = −∞ θ ( b j | ξ | ) M X i =1 | F ( ϕ ( i ) )( b j ξ ) | =: Ψ( ξ ) ≥ c > ξ = 0 . We have Ψ( b k ξ ) = Ψ( ξ ) for k ∈ Z . Define F ( η ( j ) )( ξ ) = θ ( | ξ | ) F ( ϕ ( j ) )( ξ )Ψ( ξ ) − for ξ = 0and F ( η ( j ) )(0) = 0. Then, η has all the properties required in the lemma. Also,from the construction, we can see that ˆ η ∈ C k ( R n ) if ˆ ϕ ∈ C k ( R n \ { } ). Thiscompletes the proof. (cid:3) Proof of Lemma . . Let − R B ( x,t ) f ( y ) dy = | B ( x, t ) | − R B ( x,t ) f ( y ) dy , where B ( x, t )denotes a ball in R n with center x and radius t . Then, for u, r > x, z ∈ R n , | F ( x − z ) | = − Z B ( x − z,u ) | F ( y ) + ( F ( x − z ) − F ( y )) | r dy ! /r ≤ C r − Z B ( x − z,u ) | F ( y ) | r dy ! /r + C r − Z B ( x − z,u ) | F ( x − z ) − F ( y ) | r dy ! /r , where C r = 1 if r ≥ C r = 2 − /r if 0 < r <
1. Therefore(4.1) | F ( x − z ) | ≤ C r − Z B ( x − z,u ) | F ( y ) | r dy ! /r + C r sup y : | x − z − y |
From (4.1), (4.2) and (4.3), we see that | F ( x − z ) | ≤ C r δ − n/r (1 + R | z | ) n/r M ( | F | r )( x ) /r + 2 N C r u |∇ F | ∗∗ N,R ( x )(1 + R | z | ) N . If N = n/r , it follows that | F ( x − z ) | (1 + R | z | ) N ≤ C r δ − N M ( | F | r )( x ) /r + 2 N C r δR − |∇ F | ∗∗ N,R ( x ) . Thus we have the conclusion of the lemma by taking the supremum in z over R n . (cid:3) References [1] A. Benedek, A. P. Calder´on and R. Panzone,
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