MULTIPLIERS ON WEIGHTED HARDY SPACES OVER CERTAIN TOTALLY DISCONNECTED GROUPS

In this note, we consider the multipliers on weighted H1 spaces over totally disconnected locally compact abelian groups with a suitable sequence of open compact subgroups (Vilenkin groups). We first show an (H1,L1) multiplier result from which Onneweer's theorem follows. We also give an (H1,H1) multiplier result under a condition of Baernstein-Sawyer type.

extended interpolation theorem for weighted H and L p spaces.We do not know whether this multiplier is also a weighted (HI,H multiplier.But we are able to show that a Baernsteln-Sawyer type condition [2] which is stronger than Onneweer's, implies a weighted (HI,H result.This is also a generalization of Theorem 2 [2].
Throughout this note, G will denote a locally compact abelian group with a sequence {Gn}... such that (i) each G is an open compact subgroup of G, n C G and order (Gn/Gn+I) < (ii) Gn+l n (iii) Gn G and G n {0}.
To KITADA Moreover we shall assume that G is order-bounded, i. is an open co.act subgroup of I', n (if) rn rn+l and order (Fn+1 /rn order (Gn/Gn+1 )' (iii)' U r r and r [1}.
n n We choose Haar measures p on G and X on r so that p(G 0) x(r 0) I, then p(G n) (X(rn)) := (ran) for each n Z.For an arbitrary set A we denote its indicator function by A" The symbols and w will be used to denote the Fourier and inverse Fourier transform respectively.
It is easy to see that for each n Z we ' -lr have (G) ((rn)) We set Dn := ((Gn)) IG for each n Z.
n n n We now define the weighted L p spaces.For R, we define the function v on G by v(x) (ran)if x GnGn+l (n&Z); 0 if x O.We denote the L p spaces with respect to the measure d:= vd on G by LaP(G), simply L p. Also for p < , we set G Let S(G) be the set of all functions on G such that has compact support and is constant on the cosets of some subgroup %(n depends on ) of G. >-I, then S(G) is dense in L p for p < .
In order to define the weighted Hardy spaces on G, we first define weighted atoms on G. Let < q =.A function a(x) on G is a (l,q) atom if there exists an interval I I (x): The weighted Hardy space HI'q(G), simply H l'q is the space of all functions f on on G such that f(x) I a(x), 0 where the ak's are (1,q)a atoms and 7-IAkl <-We set I II inf I X k[, where the infi is taken over all such decompositions.en H l'q is a subspace of L and that It also follows easily from the definition HI, I, q2 l,ql C H C H HI, H whenever < ql < q2 < " We denote a by a" In the following section, we show that H l'q H if -I < a 0 and < q < -.
We say that m L (r) is an (X,Y) multiplier (or a multiplier on X, when X Y) if there exists a constant C > 0 so that for all XS(C) p where X and Y are equal to Hu or Lu.
According to [I], we say that EL (r) satisfies condition C(k,r) for some k Z and r [I,") if there exist C, g > 0 so that for all , n Z with and there exists C > 0 so that for all Z we have sup I( k) (x-y) (k)V(x)IdB; y C} C, if r-I, G where k F k for each k Z and r' denotes the conjugate exponent of r.Let < a < 0o, p < and 0 < q o.A function f on G belongs to the Herz space Ka'q(G), simply K a'q if P P p with the usuaI modification if q '

GAGn+
We now state the main theorems: THEOREM I.
Let L (r)and suppose that satisfies condition C(k,r) for some k Z and r [I,).
Then k is an (H a' La) multiplier for -I/r' < a 0.
As a Corollary we obtain Theorem of [I]: COROLLARY.Let L (). (1) Suppose that condition C(k,r) holds for all k Z, for some r (I,), and with constants C and independent of k Z.If is a L 2 multiplier on for some 0 with-I/r' < 0 < I/r', then is a multiplier on L p for all p, , such tat < p < and (ii) If C(k,l) holds for all k & Z, and with C independent of k Z, then is a multiplier on L p for < p <-.A slight modification of the argument in [2] establishes the result, so we omit the proof.
Hl,q H THEOREM B. Let -I < = O.Then =, for < q < .PROOF.We have already seen that H is continuously included in H l'q for each < q < (R).In order to establish the opposite inclusion, it suffices to show that a (1,q) atom a has the representation a(x) S aj (x)   (3.I) 0 where each aj is a <l,=)a atom and E IXjl C, C independent of a. Like the non- weighted case, thls can be done by using the Calderon-Zygmnd decoposftion [4], [5].
For t > 0 (we shall be expllct later), we denote the open set (x G: Mq (b) > t}: [x G; Ma(Iblq)(x) > t q} by U t.We note that U t C I for t > I. (This is easily seen from the fact that for any two intervals in G, they are disjoint or one contains the other).Lemma (b) implies that and a(Gk) as k [l, Lemma (a)].Thus we have the decomposlton Ut: lj;j.where the lj's are maximal disjoint sub-intervals of U t.The Calderon- Zygmund decomposition is now that b(x)--gO(x) + hi, where gO(X) b(x) tf x Ut; m(b,lj) if x E lj and hi(x) (b(x) g0(x)) lj(X), and where m(b,lj) denotes the average of b over I] with respect to .Then the maximality of the lj's and Lemma (a), (b) imply that Ig0(x) Cot, Da-a.e. and ,,thjlqd=)l/q 2CC0t: Clt Lemma (c).If we set (Clt)-lha bj, then bj is supported in lj, bj d 0 and by q (lj) for each J.
The idea will be now to do for each bj the same kind of decomposition that we performed for b (with the same t) and to build an induction process which will eventually lead to the decomposition (3.1).We shall use multl-indices for the successive decomposition, in the following way: b(x) gO(x) + I hj (x) gO(x) + C It r. bj (x) Jo 0 JO 0 go(X) + c it z (gj (x) + r.
for every J0' 'jn By using (i), (ii) and (iv), we see that the L -norm of the last term in the right hand side of (3.3) is bounded by (Ct that Ct l-q < I, we have that 1-q)n+l a(1).llence for large t > 0 SO Let -I < a 0 and < Pl " Suppose that T is a sublinear H by which we mean that there exists B 0 such that operator of weak-type (I,I) on , for every f H and t > O; H L PI with constant B Then for < p < PI' T is of type and T is of weak-type on a (p,p) on L P a with constant depending only on B O, BI' Pl and p.
Let f L p and choose a q so that < q < p < Pl < " As in the proof of Theorem B, we consider the open set Et:= {Mq f > t} {Ms(Ill q) > tq}, for t > O.
Then we have the same kind of decomposition; E t From this we obtaln a Calderon-Zygmund decomposition f gt + ht' where gt f if x Et; m(f,lj) if x , lj for each j, and h: h t Z hi,where hj :-(f gt I We then have Igt (x) C0t and J J f lhjl q dBa)I/q C1 t (Ba(lj) I J for each J E N. Hence aj:= (Clta(lj))-lh j is a (l,q) a atom and h Clt E Ba(l )a E H l'q And Theorem B implies that h H wlth norm bounded j j a a by Cta(Et).The rest of proof proceed as in [4], [5] with a few modifications, so we omit the details. 4. PROOFS OF THE MAIN RESULTS.
On the other hand, B f If K(x-y)a(y) dl (y)l d(x) f f K(x-y)-K(x-x0))a/y)dB(Y)I dlJ C(mn )-S (m) g / C la(x0+Y) va(x0+Y) d(y) C(mn)-e (mn_l)e)) ll..a..l,a C n This completes the proof L 2 PROOF OF COROLLARY (i) Since L (r) is amultiplier on it follows from a classical interpolation theorem for weighted spaces [6] and [I, Proposition I] L 2 that is a multiplier on for all-laOl a < laOl.As in the proof of [I Theorem I], the case where < p < 2 and -laOl a 0, has to be proved L 2 Let < p < 2 and laOl a 0. Since each k, k Z is a multiplier on and also a (H I,LI k a a) multiplier by Theorem I, it follows from Theorem C that is a multiplier on L p.The assumption that the constants C and are independent of k, implies that is a multiplier on L p.
(ii) This is already seen in the proof of Theorem I.
PROOF OF THEOREM 2. According to Theorem A, it suffices to show that ll(@,a )v *lli,a C for all (1,)a atom a.Let avbe a (1,') atom, supported on an interval l:=x 0 + Gn(X 0 G, n Z).
We set *a f. C. ( n K E + I/r' , n nce, we have that ilf ill, C by (4.1) and (4.4). is completes the proof.
RESULTS.To prove Theorem 2, we need the "maximal function" characterization of H locally in LL(G) we define the maximal function M f of f by f Mf(x) sup (I) f If(Y) Idea (y)}' I I where the l's are intervals containing x.When O, we denote M by M, simply.type (p,p) on L p for < p < , H a is of weak-type (1.1) on L a a If a 0, then for all interval I Fo r (d) la(1) C l(1) fnf{va(y); y I, y 0}.L If 0, then M is of weak-type (I,I) on a PROOF. (a) and (c) are Lemmas l(b) and (c) in [I].(b) follows from (a).By (c), we have that Mr(x) C M f(x) for each x E G. Then (d) follows from (b). a L belongs to H ff and only if f*: Mf LI THEOREM A. Let a > -I.An f a Moreover llfllHl fs equivalent to " 'in for each n N, where, bj0,...,jn_ := (Clt)-lhJo..... Jn-I and (i) licl({Mq a (bjo,...,jn_ > t} c ua(jo.... ,in_l)/t q Jn-l' n N, then these are (I,) atoms by (fit) and (v).Thus we obtain that -I a(x) a(1) b(x) Cota(1)-l(a(1)aO(x) + Cl t r. a lj aj (x) +...), O' "''Jn-I O'''''Jn-I JO Jn-I which is the desired representation (3.1).For, the sum of the absolute value of the coefficients of the right hand side is bounded by Cot (ctl-q)k: C, C independent of a.This cgmpletes the proof.0 THEOREM C.
y) d (y)f lK(x-y)-K(x-x0) dr(x) I G I f a(xo+Y) d.(y) f jg(x-y)-g(x)ld<x) , C f la(Xo+y)l d(y) follows, when r I. is together with eorem C implies the conclusion of rollary Next, let r > I. To estite A, we use rollary (ll) and Lem (c).en C a(1) inf{va(x) x 6 I, x O} Ca(1)-I pa(1) -C.On the other hand, using Lemma (c) again, y la(xO+Y))d.(y)y#K(x-y)-K() v(x+xO) d.(x) lf*llr Cllfl[r Cllall "r Thus as in the proof of Theorem I, we have that A y (f*)rdla)I/r y Var, dl)i/r' c l.ll .<>z='r' f la(Y)idB(y) inf{v (x); e I,